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http://arxiv.org/abs/2307.04201v1 | 20230709151149 | Bayesian estimation of the Kullback-Leibler divergence for categorical sytems using mixtures of Dirichlet priors | [
"Francesco Camaglia",
"Ilya Nemenman",
"Thierry Mora",
"Aleksandra M. Walczak"
] | stat.ME | [
"stat.ME",
"cs.IT",
"math.IT",
"physics.data-an"
] |
Laboratoire de physique de l'École normale supérieure,
CNRS, PSL University, Sorbonne Université and Université de
Paris, 75005 Paris, France
Department of Physics, Department of Biology, and
Initiative for Theory and Modeling of Living Systems, Emory University, Atlanta, Georgia, USA
Laboratoire de physique de l'École normale supérieure,
CNRS, PSL University, Sorbonne Université and Université de
Paris, 75005 Paris, France
Laboratoire de physique de l'École normale supérieure,
CNRS, PSL University, Sorbonne Université and Université de
Paris, 75005 Paris, France
In many applications in biology, engineering and economics, identifying similarities and differences between distributions of data from complex processes requires comparing finite categorical samples of discrete counts. Statistical divergences quantify the difference between two distributions. However their estimation is very difficult and empirical methods often fail, especially when the samples are small. We develop a Bayesian estimator of the Kullback-Leibler divergence between two probability distributions that makes use of a mixture of Dirichlet priors on the distributions being compared. We study the properties of the estimator on two examples: probabilities drawn from Dirichlet distributions, and random strings of letters drawn from Markov chains. We extend the approach to the squared Hellinger divergence. Both estimators outperform other estimation techniques, with better results for data with a large number of categories and for higher values of divergences.
Aleksandra M. Walczak
August 12, 2023
=========================
§ INTRODUCTION
Understanding of the structure and function of a large number of biological systems requires comparison between two probability distributions of their states or activities, generated under different conditions. For example, one may be interested in how the distribution of neural firing patterns underlying typical vocalizations in a song bird is different from patterns used to drive atypical, exploratory vocal behaviors <cit.>. One can similarly ask how different are the distributions of stimuli encoded by two different firing patterns; the difference then can be viewed as a measure of semantic similarity between these patterns <cit.>. In the context of immunology, one is often interested in information theoretic quantities in order to quantify diversity or to assess differences between distributions of immune receptors <cit.>. In these and similar examples, the Kullback-Leibler (KL) divergence , also known as relative entropy, is often used as a measure of dissimilarity. It is a non-symmetric measure of the difference between two probability distributions with a wide range of applications in information theory <cit.>. While not a distance in the mathematical sense, it is often the choice measure of dissimilarity since it can be applied to categorical (non-ordinal) data, when the usual statistical moments such as the mean and variance are not well defined. Indeed, like other “information theoretic quantities”, the KL divergence is not associated to the category itself, but rather to the underlying probability distribution <cit.>.
Estimation of information theoretic quantities is a hard problem, with a lot of attempts in the recent literature. Most of these have focused on the entropy and mutual information, but estimation of the KL divergence has also been investigated <cit.>. When faced with data without any knowledge of the true underlying distribution, empirical approaches (typically referred to as “naive” <cit.> or “plugin” <cit.>) are often used. These methods approximate the true probabilities of events with their empirical frequencies, with an optional pseudocount. These types of estimators have been investigated thoroughly. The consensus is that, for all entropic quantities, these estimates are typically strongly biased <cit.>. To overcome this limitation, other approaches have been proposed to estimate the Shannon entropy (or the mutual information) of categorical data. These techniques include Bayesian methods <cit.>, coverage adjusted methods <cit.> and bias corrected methods <cit.>. In the case of the KL divergence, the cross-entropy term, which diverges due to contributions where one distribution has samples and the other does not, makes it difficult to extend these methods in the absence of information about the joint distribution. The bias-corrected “Z-estimator” <cit.>, proposed for KL divergence estimation, tackles these issues. However, it has a strong dependence on the sample size.
Here we propose a Bayesian estimator of the DKL for systems with finite number of categories using a mixture of symmetric Dirichlet priors (Dirichlet Prior Mixture, or DPM). This approach is the generalization of the main idea from <cit.> that, to produce unbiased estimators, one needs to start with Bayesian priors that are (nearly) uniform not on the space of probability distributions, but directly on the quantity being estimated. Here we extend this idea beyond the estimation of entropy, for which it was first developed. We check that, for data distributed according to a Dirichlet prior, our new approach for estimation of the KL divergence consistently converges faster to the true value than other methods. We provide an algebraically equivalent expression for the Z-estimator (following <cit.>), which makes it applicable to large sample sizes. We also test the DPM technique on sequences generated by Markov chains, which are not typical within the DPM prior, obtaining better performance for datasets with many categories. We then focus our analysis on another measure of similarity between categorical distributions, the Hellinger divergence <cit.>, which, unlike the DKL, is a well defined bounded distance between distributions. To show the generality of our approach, we also develop a DPM estimator for the squared Hellinger divergence. In computational tests, we show the DPM approach to be accurate for this quantity as well. Since no estimation method can be guaranteed to estimate entropic quantities without a bias for an arbitrary underlying probability distribution, we finish by discussing the method's reliability when applied to real experimental data, where the true values of the divergences are not known a priori.
§ RESULTS
§.§ Bayesian framework for the estimation of the divergence
Our goal is to derive an estimate of the Kullback-Leibler divergence between the distributions of categorical variables t⃗ and q⃗, (q⃗‖t⃗).
We consider a discrete set of K categories labeled with i=1,…,K.
Examples of categorical variables include “words" defined as
sequences of neuron firing patterns (spike counts in time windows),
sets of coexisting ecological or molecular species or a sequence of
amino acids or nucleotides. Each category i has a certain (unknown)
probability q_i in the first condition, and t_i in the second
condition. We observe this category n_i times in an experiment done
in the first condition, and collect the data in the histogram
n⃗={ n_i}_i=1^K. An experiment in the second
condition returns the counts m⃗={ m_i}_i=1^K. We
want to estimate the Kullback-Leibler divergence between t⃗ and
q⃗ <cit.>, defined as:
(q⃗‖t⃗) = H(q⃗‖t⃗)-S(q⃗) =∑_i=1^K q_i logq_i/t_i,
where we defined the cross
entropy between t⃗, and q⃗,
H(q⃗‖t⃗) = - ∑_i=1^K q_i log t_i, and the Shannon
entropy,
S(q⃗) = - ∑_i=1^K q_i log q_i <cit.>.
Taking inspiration from Nemenman et al. <cit.>,
we choose to estimate the in a Bayesian framework.
The approach is summarized in Fig. <ref>. We do
not have access to the true probability distributions t⃗ and
q⃗, only to the empirical histograms n⃗ and m⃗.
The simple method consisting in approximating
q_i≈ n_i/∑_jn_j and likewise for t⃗ into
Eq. <ref> is known to work very
poorly <cit.>. The issue
comes from the presence of categories never observed in one sample,
while they are present in the other, resulting in divergence of the
logarithmic term. To go beyond that, we construct a prior of the
true distributions P_ prior (q⃗, t⃗) and weight the
estimate of the divergence by posterior over q⃗ and t⃗:
⟨ (q⃗, t⃗) |n⃗,m⃗⟩
=
∫dq⃗dt⃗ P_ post (q⃗, t⃗) (q⃗‖t⃗)
,
where
P_ post (q⃗, t⃗)
=
1/ZP_ prior (q⃗, t⃗) P(n⃗, m⃗|q⃗, t⃗)
,
with Z=P(n⃗,m⃗)= ∫dq⃗dt⃗ P_ prior (q⃗,t⃗) P(n⃗, m⃗|q⃗, t⃗) a normalization.
The empirical observations n⃗ and t⃗ are assumed to be independent samples of q⃗ and t⃗ respectively, and are thus distributed according to a multinomial distribution:
P( n⃗, m⃗|q⃗,t⃗ )
=
nqmt,
with
nq=
N !/∏_i=1^K n_i !∏_i=1^K q_i ^n_i
=
1/B(n⃗+1⃗)∏_i=1^K q_i ^n_i,
where N=∑_i=1^K n_i, and B(x⃗) is the multivariate Beta function: B(x⃗)=∏_i=1^K Γ (x_i)/Γ(∑_i=1^K x_i), where Γ(x) is the gamma function.
A natural choice for the prior on q⃗ and t⃗ is the Dirichlet distribution, which is the conjugate prior of the multinomial distribution, and is defined as
qα = δ(∑_i=1^Kq_i-1)/B(α⃗)∏_i=1^K q_i ^α-1,
where α∈ (0,∞) is the “concentration parameter”,
α⃗={α}_i=1^K and δ(x) is
the Dirac's delta function imposing normalization. Rank plots
associated to qα are shown in
Fig. <ref>A.
For α→∞, the prior tends to a uniform distribution q_i=1/K.
For small concentration parameters α, the distribution is peaked with weights given to just a few categories.
As noted in Ref. <cit.>, entropies of
distributions drawn from a Dirichlet with the same α all have
similar entropies, strongly biasing the Shannon entropy estimate,
especially in the large K limit. To reduce the bias, one then
uses a mixture of Dirichlet distributions at different α,
allowing substantially different values of the entropy a
priori. For a certain choice of the mixture distribution (the
prior over α, ρ(α)), one can achieve a
nearly-uniform a priori distribution of entropies and,
consequently, a much smaller estimation bias
<cit.>.
We expect also to have very similar values for all
distributions generated from the Dirichlet priors with fixed α
and β. We then expect that a good estimator may be produced by
using a mixture of Dirichlet
distribution that allows to span different values of the expected
:
P_ prior( q⃗,t⃗)=
∫_0^∞∫_0^∞dαdβ ρ(α,β) qαtβ
,
where ρ(α,β) is a “hyper-prior”, i. e., a prior over the hyper parameters α and β.
Plugging this prior into Eq. <ref> and <ref> gives:
⟨|n⃗,m⃗⟩
=
=
1/Z∫dαdβ
P( n⃗|α)P(m⃗|β)ρ(α,β)
⟨|n⃗,m⃗ ; α,β⟩
,
with
P( n⃗|α)
= ∫dq⃗ nq qα
=
B(n⃗+α⃗)/B(α⃗)B(n⃗+1⃗)
and likewise for P(m⃗|β).
The normalization now reads
Z = ∫dαdβ
P( n⃗, m⃗|α, β)
.
The expected value of the inside the integral may be computed analytically (see <ref>):
⟨|n⃗,m⃗ ; α,β⟩
=
=
∫dq⃗dt⃗ P(q⃗,t⃗|m⃗,n⃗,α,β)(q⃗‖t⃗)
=
∑_in_i+α/N+Kα{Δψ(M+Kβ , m_i+β)
-Δψ(N+Kα+1 , n_i+α+1) }
.
where Δψ(z_1,z_2)=ψ(z_1)-ψ(z_2) is the difference of digamma functions ψ (i. e., polygamma function of order 0, see Eq. <ref>).
Similarly we can calculate
⟨^2 |n⃗,m⃗⟩, which we can use to
compute the posterior standard deviation of our method
( <ref>). For a given choice of
ρ(α,β), the DPM estimate for in
Eq. <ref> can be computed numerically (same for
^2 in Eq. <ref>), as described in
detail in <ref>.
The code is available on github as specified in <ref>.
We expect that, in the limit of large data (N,M≫ K), the integral
of Eq. <ref> will be dominated by the values of
α and β that maximize the likelihoods
P(n⃗|α) and P(m⃗|β), regardless of
the
hyper-prior ρ(α,β).
The dominant role of the likelihood P(n⃗|α) for
increasing N was equivalently observed for the NSB entropy
estimator <cit.>. By contrast, we
expect the prior ρ(α,β) to play a role in the
low-sampling regime, as can be seen from Fig. <ref>.
A simplified approach for the estimation of the would then
be to provide a choice for the concentration parameters that
maximizes the likelihoods P(n⃗|α) and
P(m⃗|β) (see Eq. <ref>). We refer to the
application of Eq. <ref> with the
maximum-likelihood values of α and β
as the Dirichlet Prior (DP) estimator.
§.§ Choosing the hyper-prior
To finalize the estimation, we need to choose a functional form for the hyper-prior ρ(α,β) in such a way that the resulting ensemble has an evenly distributed .
In the limit of large numbers of categories (K≫ 1), both contributions of the , S(q⃗) and H(q⃗‖ t) are very peaked around their mean values, which can be computed analytically ( <ref>):
𝒜(α)≡⟨ S |α⟩=Δψ(Kα+1,α+1)≤log K,
and
ℬ(β) ≡⟨ H |α,β⟩=Δψ(Kβ,β) ≥log K
(which only depends on β), at fixed concentration parameters.
These mean values are shown in Fig. <ref>B, and
the corresponding =H-S in Fig. <ref>C as a
function of α and β.
We are interested in finding a hyper-prior such that the
resulting prior over is not peaked. This results in the
following inverse problem for finding the hyper-prior ρ_z(z),
where we denote by z:
ρ_z(z) ≈∫_0^∞dα∫_0^∞dβ ρ(α,β) δ( ℬ(β) - 𝒜(α)- z ),
with the choice ρ_z(z) to be made. Because we have a
one-dimensional target distribution ρ_z(z), but a 2-dimensional
hyper-prior ρ(α,β), there are infinitely many solutions
to this inverse problem. Without losing generality, we can make the
change of variable from α and β to 𝒜 and
ℬ:
ρ_z(z) ≈∫_0^log Kd𝒜∫_log K^+∞dℬ ρ_AB(𝒜,ℬ) δ( ℬ - 𝒜- z ),
with
ρ(α,β)=|∂_α𝒜||∂_βℬ|ρ_AB(𝒜(α),ℬ(β)).
Then a natural choice is to pick the Ansatz imposing that all values of 𝒜 and ℬ with the same are equiprobable:
ρ_AB(𝒜,ℬ)=ϕ(ℬ-𝒜).
Then ϕ(z) satisfies:
ρ (z) =
ϕ(z) ∫_0^log Kd𝒜 θ( z + 𝒜 -log K )
=
ϕ(z) { z θ(log K - z) + log K θ(z - log K) }
,
where θ(x)=1 if x≥ 1 and 0 otherwise (Heaviside function), or after inversion:
ϕ(z) =
ρ(z) z^-1 z < log K
ρ(z) 1log K otherwise.
Eqs. <ref>, <ref>, and <ref> give us
the final form of the hyper-prior ρ(α,β). We are
left with the choice of the distribution of the , ρ(z).
We pick a log-uniform (also known as “reciprocal”)
distribution,
ρ(z)∝ z^-1 <cit.>, allowing
to evenly span over different orders of magnitude of the .
The resulting hyper-prior is represented in
Fig. <ref>D.
§.§ Tests on synthetic Dirichlet samples
To assess the properties of the DPM estimator, we test it on data
generated from distributions drawn from Dirichlets
q⃗∼qα, t⃗∼qβ
(Eq. <ref>), for various values of α and
β. Having in mind applications to polypeptide sequences, we
perform our tests for three different numbers of categories K=20^2,
20^3, and 20^4, the numbers of all possible 2-mers, 3-mers
and 4-mers that can be produced with an alphabet of 20 letters
(e. g., amino acids). For each choice of q⃗ and t⃗,
samples n⃗ and m⃗ are generated from these distributions.
This application may be viewed as a the consistency check for te
estimator, since the estimator relied on the Dirichlet hypothesis,
which is satisfied by the data.
We know that standard Bayesian consistency applies, ensuring that
DPM (and DP) estimators converge to the true value in the limit of
large samples. To understand how DPM estimator converges to the
true value, we extract subsamples of increasing sizes N=M from a
larger sample of size 2· 10^7.
Fig. <ref> compares our
estimate to several state-of-the-art estimators: the additive
smoothing method with different values of the pseudo-count (see below
for details), the Z estimator, and the simplified version of our
method, the DP estimator, obtained by fixing α and β to
their maximum-likelihood values.
Additive smoothing estimators are defined as: (q⃗̂⃗‖t⃗̂⃗), with q̂_i=(n_i+a)/(N+Ka), and t̂_i=(m_i+b)/(M+Kb).
We use 4 choices for the pseudo-counts a and b, summarized in Table <ref>.
To avoid infinite values, in the case b=0 we set to zero the terms for which m_i=0.
It has been shown that naive estimators converge to the true
value in the limit of large samples, but have an infinite bias due
to low-probability categories <cit.>. The
“Z-estimator” <cit.> was introduced to
remove this bias asymptotically. Although its original definition was
given as a series, one can show following <cit.>
that its expression reduces to (Appendix <ref>):
D_ KL^(Z)
=
∑_i=1^K n_i/N[
Δψ (M+1, m_i+1)-Δψ (N, n_i)
],
where the first term in the sum corresponds to an estimator of
H(q⃗‖ t), and the second term is the classic
Schurmann-Grassberger estimator of the entropy
S(q⃗) <cit.>. In Appendix <ref> we
observe that
⟨|n⃗, m⃗, α, β⟩→D_ KL^(Z) in the limit α→ 0,
β→ 1, N ≫ K and M ≫ K.
Comparing the convergence of the different estimators to the true
value as a function of the subsample size N/K for
α=β=1 and K=20^2
(Fig <ref>A), we see that the DPM
performs better than other estimators. To assess how performance
depends on the concentration parameters, we repeated this convergence
analysis for different values of α and β. We measure
convergence through N^∗, defined as the sample size where the
estimator get within 5% of the true value
(Fig <ref>B). This measure
of accuracy has the advantage to be applicable to all considered
estimation methods.
Our estimator consistently performs well and compares favorably
to other methods when data was generated from distributions drawn
from symmetric Dirichlet. In most cases, the proposed DPM estimator
converges faster than all other considered methods
(Fig <ref>C).
The better performance is striking also for larger numbers of
categories, K=20^3 and 20^4
(Fig. <ref>).
§.§ Tests on synthetic Markov chain sequences
To test the performance of DPM on a different synthetic system that
does not satisfy the Dirichlet assumption, we generated L-long
sequences (or “L-grams”) from a Markov chain described by the
transition matrix Ŵ∈ℳ_20 with 20 states
μ=1,⋯,20. We choose each transition probability
P(μ→ν) from a uniform distribution in (0,1) and then impose
that the transition matrix is a right stochastic matrix,
P(μ→ν)=W_νμ by normalizing to 1 each column of the
transition matrix. An illustration where the states are the 20
amino acids is shown in Fig. <ref>A.
With this choice for the Markov transition matrix, all states
communicate and are non absorbing.
We verify there exists a stationary probability vector π⃗ = {π_μ}_μ=1^L that satisfies
π⃗ = Ŵπ⃗. The number of categories is K=20^L
and each category i corresponds to the L-gram
(x_1,⋯,x_L) with the stationary probability q_i equal to
q_i=π_x_1 W_x_2 x_1⋯ W_x_L x_L-1.
We analytically compute the entropy associated to the stationary distribution q⃗ of L-grams to get:
S^(L)(q⃗)
=
S(π⃗)
-
(L-1) ∑_μν W_νμπ_μlogW_νμ.
Typical values for the Shannon entropy of L-grams are shown in Fig. <ref>B along with the convergence curve of the NSB estimator.
We assume that the L-grams of a second system are generated by a similar Markov process but with a transition matrix V̂ and stationary probabilities of the σ⃗={σ_μ} states.
The cross-entropy between the t⃗ and q⃗ distributions reads:
H^(L)(q⃗‖t⃗)
=
H(π⃗‖σ⃗)
-
(L-1) ∑_μν W_νμπ_μlogV_νμ.
Similarly to the analysis in the previous section, we generate a large sample of L-grams from each distribution, with N=M=2·10^8.
We subsample this dataset at different sample sizes and estimate the and its standard deviation for L=2,3,4.
To study the average behavior, we divide the estimate by the expected result (Eq. <ref>) and we average over 30 simulations.
We observe that, in the case of small numbers of categories (K=20^2,
Fig. <ref>C top panel), DPM (and DP) perform
quite similar to the best alternative (Jeffreys), but with different
sign biases. However, the DPM estimator performance greatly
improves for larger K (Fig. <ref>C middle and
bottom panels). In all cases, the standard deviation associated to
the DPM estimator (red bars in Fig. <ref>C) captures
the spread across the different repetitions of the convergence curve
(red shade in Fig. <ref>C).
§.§ Estimator for the Hellinger divergence
Lastly, we extend the DPM method to estimate the Hellinger divergence between the discrete distributions q⃗ and
t⃗ <cit.>.
The Hellinger divergence is a symmetric statistical distance that satisfies the triangular inequality, making it a true distance in the mathematical sense <cit.>:
(q⃗,t⃗)^2
=
1/2∑_i=1^K(√(q_i)-√(t_i))^2=1-∑_i=1^K√(q_it_i).
Following the same approach as for the Kullback-Leibler divergence (details in <ref>), we obtain the DPM estimator for ^2:
⟨^2 |n⃗,m⃗⟩
=
=
1/Z∫dαdβ ρ_ H (α,β) P ( n⃗|α) P ( m⃗|β)
⟨^2 |n⃗, m⃗;α,β⟩,
with
⟨^2 |n⃗, m⃗;α,β⟩
=
=
1-∑_i=1^KB(1/2, N + Kα)/B(1/2,n_i + α)B(1/2, M + Kβ)/B(1/2,m_i + β),
where Z = ∫dαdβ ρ_ H (α,β) P ( n⃗|α) P ( m⃗|β) and B(x_1, x_2) = Γ(x_1)Γ(x_1)/ Γ(x_1+x_2) is the two-dimensional Beta function.
We test the Hellinger divergence estimator on the same synthetic datasets as in Fig. <ref>B (Fig. <ref>).
For datasets generated with Dirichlet-multinomial distributed samples, the DPM outperforms all considered plugin estimators
1 - ∑_i=1^K √(q̂_it̂_i), with q̂_i and t̂_i defined as before with pseudo-counts
a,b chosen according to Table <ref> (Fig. <ref>A).
As for the case of KL divergence, the performance improves for larger categories (Fig. <ref>).
Tests on the synthetic Markovian L-grams (see previous paragraph) show the DPM estimator performs better for larger numbers of categories K, with comparable performance to the best alternative (Jeffreys) for K=20^2 (Fig. <ref>B).
§ DISCUSSION
Correctly estimating statistical divergences between two distributions is an open problem in the analysis of categorical systems. Alongside the entropy, divergences such as the Kullback-Leibler and the Hellinger distance, are an important tool in the analysis of categorical data <cit.>.
We focused on categorical distributions with finite numbers of categories K (bounded domain), where K is a known quantity. We proposed a way (DPM) to extend the approach of Nemenman et al. <cit.> developed for Shannon entropy estimation, to Kullback-Leibler estimation. DPM introduces a
mixture of symmetric Dirichlet priors with a log-uniform a priori expected divergence distribution (Eq. <ref>).
We restricted our analysis to the case of the two finite samples of the same size N, although the method works for different sample sizes.
We also propose a simplified estimator (DP), which assumes a Dirichlet prior with concentration parameter fixed to the maximum value of the likelihood.
This estimator is faster to compute as it does not require to integrate over the concentration parameters.
We showed that the DPM method outperforms the tested empirical plugin techniques
in terms of estimation for synthetic data sampled from a Dirichlet-multinomial distribution with fixed concentration parameters. The estimation task gets harder for distributions with larger concentration parameters, i. e., closer to a uniform distribution, but easier for large numbers of categories K.
These convergence trends were confirmed by tests on sequences of L states generated by Markov chains.
In this case, DPM compares well to the best plugin estimator in the low sample size regime of K=20^2 and outperforms it for K≥ 20^3.
Similar results were obtained for the DPM estimate of the Hellinger divergence for both Dirichlet-multinomial and Markov chain datasets.
To our knowledge, DPM estimator of the Hellinger divergence is the first attempt to extend the ideas of Ref. <cit.> and to build a uniform prior estimator for a non-entropy-like quantity.
Our tests were restricted to categorical systems with rank distributions having exponentially decaying tails.
As previously discussed for the case of the NSB entropy estimator, the Dirichlet prior has major limitations in capturing the Shannon entropy if the system rank distribution is not decaying fast <cit.>.
Many real systems exhibit long-tailed rank distributions that decay as power-laws <cit.>, which are not well captured by a Dirichlet prior.
Preliminary (unpublished) tests of the DPM method for such systems show poor performance.
Similarly to the case of entropy estimates, we speculate that the limitations of this method are related to issues with the poor representation of long tails by Dirichlet priors.
Introducing a Pitman-Yor prior <cit.> could overcome this problem, as has been shown for entropy estimation by Archer et al. <cit.>, and offers a direction to generalize the applicability of the DPM method.
Extending the Pitman-Yor prior to the case of statistical divergences would require to compute expected values over the probabilities of both systems, but to the best of our knowledge this is not possible because of the lack of an analytical expression for the Pitman-Yor distribution.
Another difficulty may lie in the difficulty to encode correlations between the ranks of categories in the two distributions.
Our priors assume that the two unknown distributions q⃗ and t⃗ are drawn independently. However in real data they are generally correlated, which could have an impact on the quality of estimators when the distribution of frequencies becomes very broad.
In view of these complications, it is important to have practical criteria to ascertain if the output of the DPM estimator can be trusted for a specific dataset, or if it remains biased. Similar questions exist for estimation of many quantities, and specifically of entropic quantities, on categorial data since no estimator can be universally unbiased for them, and the decay of the bias with the sample size may be excruciatingly slow <cit.>. For entropy and mutual information, the standard approach is to observe if the empirical output drifts systematically as the sample size changes. One then declares the estimator trustworthy if the bias does not drift by more than the posterior standard deviation over about an order of magnitude change in the amount of data <cit.>. We expect this approach to transfer nearly verbatim to the and the Hellinger divergence context, easily detecting whether the DPM approach can be used for a specific dataset, or if other analysis methods should be sought.
§ ACKNOWLEDGEMENTS
We thank Antonio C. Costa for helpful discussions.
This work was partially supported by the
European Research Council Consolidator Grant n. 724208 and ANR-19-CE45-0018 “RESP-REP” from the Agence Nationale de la Recherche.
I.N. was supported in part by the Simons Foundation Investigator grant and by the U. S. NSF grant No. 2209996.
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*
§ SUPPLEMENTARY INFORMATION
§.§ Mathematical relations
We first introduce mathematical relations and notations that are used for the computation of the DP and DPM estimators for and ^2.
§.§.§ Wolpert-Wolf integrals
Given a vector x⃗={ x_i }_i=1^K, where x_i ∈ (0,∞) for all i=1,…, K, where K is a finite number of categories, the Wolpert-Wolf <cit.> integral is a multivariate Beta function B:ℝ_+^K→ℝ_+ in x⃗:
∫dq⃗ δ( ∑_i=1^K q_i-1) ∏_j=1^K q_j ^x_j-1
= ∏_j=1^K Γ (x_j)Γ(X) = B(x⃗),
where X=∑_i x_i and the function Γ is the Euler Gamma function Γ(x) = ∫_0^∞dt e^-tt^x-1.
All Bayesian calculations with multinomial likelihoods and multivariate Dirichlet priors involve the integral:
∫dq⃗ δ( ∑_i=1^K q_i-1) ∏_j=1^K f_j(q_j)
=
=
L^-1[ ∏_j=1^K L[ f_j(q) ](s)](q'=1),
where the f_i are regular functions, L is the Laplace transform in q (which is a function of s) and L^-1 is the inverse Laplace transform (which is a function of q').
§.§.§ Partial derivative operation
The “partial derivative operator” for the i-th dimension ∂_i = ∂/∂ x_i applied to the Beta function B returns
( ∂_i B ) (x⃗ )
=
∫dq⃗ δ(‖q⃗‖_1-1) ∏_j=1^K q_j ^x_j -1log q_i
= B(x⃗) [ ψ(x_i)-ψ(X) ],
where the function ψ is the polygamma function of order 0. The polygamma function of order ℓ is defined as
ψ_ℓ(x) = .d^ℓ/d y^ℓlogΓ(y)|_y=x.
In order to simplify the calculations, we define the following quantities related to the partial derivative operation (Eq. <ref>):
Λ_i(x⃗)
=
( ∂_i B ) (x⃗ )/B(x⃗)
=
Δψ(x_i,X) ,
where we make use of the contraction Δψ(z_1,z_2) = ψ(z_1)-ψ(z_2). Iterating this derivation on the function B, one can express the double derivatives as follows:
Λ_ij(x⃗)
=
(∂_ij B) ( x⃗)/B( x⃗)
=
(∂_i( B Λ_j))( x⃗)/B( x⃗)
=
Λ_i( x⃗)Λ_j( x⃗) + ( ∂_iΛ_j) ( x⃗)
,
where the derivative ∂_iΛ_j( x⃗) = δ_ijψ_1( x_i) -ψ_1( X) is a consequence of Eq. <ref>.
§.§.§ Shift Operation
The “shift operator” e^λ∂_i of parameter λ∈ℝ for the i-th dimension acts on the function B as follows:
( e^λ∂_iB )( x⃗)
=
B( x⃗+λi)
=
∫dq⃗ δ( ∑_i=1^K q_i-1) ∏_j=1^K q_j ^x_j-1q_i^λ
=
B( x⃗) B ( λ , X)B(λ, x_i)
,
where i = {δ_ij}_j=1^K indicates the i-th versor in the K-dimensional space of categories, with the condition x_i+λ>0. The function B( z_1,z_2) is the regular (two-dimensional) Beta function :
B( z_1,z_2)
=
Γ( z_1) Γ( z_2) Γ( z_1+z_2)
.
When λ =n ∈ℕ_+, the shift simplifies to
( e^n∂_i B ) ( x⃗)
=
B( x⃗) ∏ ^n-1_n'=0x_i+n'X+n'
as an immediate consequence of the recurrence relation Γ (x + 1) = x Γ(x).
Similarly to the case of partial derivatives, we introduce a class of functions to deal with the shift:
Ω _i( x⃗)
=
(e^∂_iB )( x⃗)/B ( x⃗)
=
x_iX
,
from which we compute two-dimensional shifts
Ω _ij( x⃗)
=
( e^∂_ie^∂_jB ) ( x⃗)/B( x⃗)
=
e^∂_i( B Ω_j) ( x⃗)/B( x⃗)
=
Ω_i( x⃗) Ω_j( x⃗ + i)
=
x_iX( x_j +δ _ij) /X+1.
§.§.§ Composed operations
For the sake of this work, another useful class of functions are the first order derivatives of the functions Ω defined as
∂_jΩ_i/Ω_i
=
∂_jlogΩ_i
=
δ_ij/x_j - 1/X
,
and, for the two-dimensional shift,
∂_kΩ_ij/Ω_ij
=
∂_klogΩ_ij
=
δ_ijk( 1/x_k + 1/x_k+1)
+ (δ_ik + δ_jk ) (1-δ_ij)1/x_k
+
- 1/X - 1/X+1
.
Similarly for the second order derivatives:
∂_jkΩ_i/Ω_i
=
∂_j∂_klogΩ_i + (∂_jlogΩ_i) (∂_klogΩ_i)
=
2/X^2 - δ_ij+δ_ik/x_i X
,
and
∂_khΩ_ij/Ω_ij
=
∂_k (Ω_ij∂_hlogΩ_ij)/Ω_ij
=
1/X^2 + 1/(X+1)^2
- δ_ijkh( 1/x_k^2 + 1/(x_k+1)^2)
+
-(δ_ikh + δ_jkh ) (1-δ_ij)1/x_h^2
+ (∂_klogΩ_ij) (∂_hlogΩ_ij).
Using all these definitions, we compute the following quantities:
e^∂_i∂ _j B/B
=
Ω_iΛ_j + ∂_jΩ_i,
e^∂_i e^∂_j∂ _k B/B
=
Ω_ijΛ_k + ∂_kΩ_ij,
e^∂_i∂ _jk B/B
=
Ω_i Λ_jk + (∂_j Ω_i) Λ_k
+
(∂_k Ω_i) Λ_j + ∂_jkΩ_i
and
e^∂_i e^∂_j∂ _kh B/B
=
Ω_ijΛ_kh + (∂_k Ω_ij) Λ_h
+
(∂_h Ω_ij) Λ_k + ∂_khΩ_ij.
which are used to reconstruct all estimators presented in this work.
§.§.§ A priori and a posteriori expected values
The operations presented in the previous sections are used compute the posterior expected values ⟨ F(q⃗,t⃗) |n⃗,m⃗;α, β⟩ for all the functions that can be expressed as :
F(q⃗,t⃗)
=
∑_i=1^K f_i(q⃗) g_i (t⃗).
Since the concentration parameters α, β are independent, for fixed concentration parameters the expected value of F factorizes:
⟨ F(q⃗,t⃗) |n⃗,m⃗;α, β⟩
=
∑_i=1^K ⟨ f_i |n⃗;α⟩⟨ g_i |m⃗;β⟩,
with
⟨ f_i |n⃗;α⟩B(n⃗+α⃗)/B(α⃗) B(n⃗+1⃗)
=
∫d q⃗ δ( ∑_i=1^K q_i-1)
qαnq
f_i(q⃗)
=
∫d q⃗ δ( ∑_i=1^K q_i-1)
∏_j^K q_j ^n_i+α-1/ B(α⃗) B(n⃗+1⃗)
f_i (q⃗).
For all functions f_i that can be expressed in terms of partial derivative (Eq. <ref>) and/or shift operators (Eq. <ref>),
a factor B(n⃗+α⃗) appears and the expected value is obtained explicitly simplifying the constant factors.
Specifically:
⟨ q_i |n⃗;α⟩
=
(e^∂_i B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨log q_i |n⃗;α⟩
=
(∂_i B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨ q_i log q_i |n⃗;α⟩
=
(e^∂_i∂_i B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨ q_i q_j |n⃗;α⟩
=
(e^∂_i e^∂_j B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨ q_i q_j log q_i |n⃗;α⟩
=
(e^∂_i e^∂_j∂_i B )(n⃗+α⃗)/B(n⃗+α⃗)
and
⟨ q_i q_j log q_i log q_j |n⃗;α⟩
=
(e^∂_i e^∂_j∂_i ∂_j B )(n⃗+ α⃗) /B(n⃗+α⃗)
.
The a priori expected values are computed in the same way, noticing that ⟨ f_j |α⟩ = ⟨ f_j |n⃗=0⃗;α⟩.
§.§.§ KL divergence estimation
We can use the previous results to compute the a posteriori expected value for the .
We start by computing the a posteriori expected value for the crossentropy H which is given by
⟨∑_i^K
H(q‖t)
|n,m, α, β⟩
=
∑_i^K⟨ q_i |n, α⟩⟨log t_i |m, β⟩
=
∑_i^K e^∂_i B(n+α) /B(n+α)∂_i B(m+β) /B(m+β)
=
∑_i^Kn_i+α/N+KαΔψ (M+Kβ, m_i+β)
,
where we took advantage of independence and used the relations Eq. <ref> and <ref> to obtain the explicit expressions in Eq. <ref> and <ref>.
Subtracting the a posteriori expected Shannon entropy ⟨ S |n,m, α, β⟩ = ∑_i n_i+αN+KαΔψ(N+Kα+1, n_i +α+1), we finally obtain the KL expected value in Eq. <ref>:
⟨|n⃗,m⃗ ; α,β⟩
=
∑_in_i+α/N+Kα{Δψ(M+Kβ , m_i+β)
- Δψ(N+Kα+1 , n_i+α+1) }.
§.§.§ Squared KL divergence estimation
In order to compute the posterior standard deviation of the Kullback-Leibler divergence estimator, we calculate the expected value of the squared KL divergence:
⟨^2 |n⃗,m⃗⟩
=
∫dαdβ
P(n⃗,α)P(m⃗|,β)ρ(α,β)
⟨^2 |n⃗,m⃗;α,β⟩
.
Similarly to the case of ,
we can compute explicitly
⟨^2 |n⃗,m⃗;α,β⟩
=
∑_ij⟨ q_i q_j logq_i/t_ilogq_j/t_j|n⃗,m⃗;α,β⟩
,
which requires to rewrite
q_iq_jlogq_it_ilogq_jt_j
=
q_iq_jlog q_ilog q_j
- 2 q_iq_jlog q_ilog t_j
+ q_iq_jlog t_ilog t_j
.
The explicit expression computed using Wolpert-Wolf properties (Eqs. <ref>, <ref>, <ref>, <ref>, <ref>) is:
⟨ q_i q_j logq_i/t_ilogq_j/t_j|n⃗,m⃗;α,β⟩
=
x_i(x_j+δ_ij)/X(X+1){δ_ijψ_1( x_i+2 ) -ψ_1( X+2 )
+ Δψ( x_i+1+ δ_ij,X+2) · (i↔ j)
- 2Δψ( x_i+1+δ_ij,X+2)
Δψ( y_j,Y)
+δ_ijψ _1(y_i) - ψ _1 (Y)
+
Δψ( y_i,Y)·(i↔ j)
}
,
where we have introduced the following notation to contract the expression: x⃗=n⃗+α⃗, X=N+Kα, y⃗=m⃗+β⃗ and Y=M+Kβ.
The factor (i↔ j) means taking the term that it multiplies with inverted indexes i and j.
§.§ Zhang-Grabchak divergence estimator
In Ref. <cit.> Zhang and Grabchak proposed an estimator for the Kullback-Leibler divergence, defined as:
D_ KL^(z)
=
∑_i=1^K n_i/N{ ∑_v=1^M-m_i1/v∏_s=1^v(1 - m_i/M-s+1)
-
∑_v=1^N-n_i1/v∏_s=1^v(1 - n_i-1/N-s)
}
,
where v and s are dummy variables.
§.§.§ Expression of the Z-estimator
Schurmann <cit.> has shown that, in the entropy term of Eq. <ref>, the summation in v of the i-th element can actually be expressed in a more concise way as
∑_v=1^N-n_i1/v∏_s=1^v(1 - n_i-1/N-s)
=
Δψ (N, n_i)
,
times a factor n_i/N.
The sum of these terms returns the Shurman-Grassberger entropy estimator S_ SG = ∑_i=1n_i/NΔψ (N, n_i) <cit.>.
If we simply plug N=M+1 and n_i = m_i+1 in Eq. <ref>, we can show that the analogous i-th crossentropy term in Eq. <ref> is equal to the following:
∑_v=1^M-m_i1/v∏_s=1^v(1 - m_i/M-s+1)
=
Δψ (M+1, m_i+1)
.
Finally, if we substitute Eq. <ref> and <ref> in Eq. <ref>, we obtain:
D_ KL^(z)
=
∑_i=1^K n_i/N[
Δψ (M+1, m_i+1)-Δψ (N, n_i)
]
.
which is the expression in Eq. <ref> of the main text.
§.§.§ Relation between the DP and the Z estimator
The Z-estimator can be expressed as an a posteriori expected value of the at α=0 and β=1, up to an additive constant.
We start by showing the following relation
lim_α→ 0⟨ S |n, α⟩
=
∑_i=1^K n_i/NΔψ (N+1, n_i+1)
=
1-K/N
+
∑_i=1^K n_i/NΔψ (N, n_i)
,
which makes use of the fact that ψ(x+1)=ψ(x)+1/x.
Considering now the crossentropy term with β=1, and performing the same limit as before, we observe that
lim_α→ 0⟨ H |n, m, α, β
=1 ⟩
=
∑_i=1^K n_i/NΔψ (M+K, m_i+1)
=
Δψ (M+K, M+1) +∑_i=1^K n_i/NΔψ (M+1, m_i+1)
,
where we used the fact that Δψ(x, x) = ψ(x)-ψ(x)=0 to add the term Δψ (M+1,M+1) in the sum.
Recognizing the two terms in Eq. <ref> we subtract Eq. <ref> and <ref> to obtain that
lim_α→ 0⟨|n, m, α, β=1 ⟩
=
Δψ (M+K, M+1)
+
K-1/N
+
D_ KL^(z)
.
§.§ The DPM squared Hellinger divergence estimator
We compute the DPM estimator for the squared Hellinger divergence ^2 (Eq. <ref>).
We do so by starting from the Bhattacharyya coefficient <cit.>
(q⃗,t⃗)
=
∑_i=1^K√(q_i)√(t_i)
=
1 - ^2(q⃗,t⃗)
.
Its a priori expected value under the assumption of the prior
P_ prior(q⃗,t⃗) =p( q⃗,t⃗|α,β)
=
qαtβ
is equal to :
⟨|α , β⟩
=
∑_i=1^K(e^1/2∂_iB)(α⃗)/B(α⃗)(e^1/2∂_iB)(β⃗)/B(β⃗)
=
K B(1/2, Kα)/B(1/2,α)B(1/2, Kβ)/B(1/2,β)
,
where we used the shift property (Eq. <ref>) with parameter λ=1/2. Following the derivation of the in the main text, we choose a metaprior ρ_ H (α,β) to control the a priori squared Hellinger divergence ⟨^2 |α , β⟩ = 1 - ⟨|α , β⟩:
ρ_ H (z) =
∫dαdβ ρ_ H (α,β) δ( ⟨^2 |α , β⟩ -z ).
We define g(x)= √(K) B(1/2, Kx) / B(1/2,x), which is a function g : ℝ→ [0,1).
Using a similar Ansatz of the one in the main text, we obtain ρ_ H (α,β) = |∂_α g (α) ||∂_β g (β) |ϕ( ⟨^2 |α , β⟩ ), where the condition in Eq. <ref> imposes
ϕ ( z ) = ρ_ H (z) (1-z)^2/z(2-z).
We choose ρ_ H (z) to be log-uniform.
Finally, knowing that the calculation for the posterior expected squared Hellinger divergence is analogous to the a priori expectation, we obtain the DPM squared Hellinger estimator in Eq. <ref> and <ref>.
§.§ Numerical implementation
§.§.§ Computations with multiplicities
In the low sampling regime (sparse data), there is a limited number of values the counts can take, which means that many categories will see the same pairs of values x⃗_⃗i⃗=(n_i,m_i).
To reduce the computational cost associated to summation over the K categories, we introduce a set of “multiplicities” <cit.> contained in the vector ν_x⃗, where each entry is the number of instances that appear n times in the first sample and m in the second.
Since by construction the dimension of ν_x⃗≤ K, we expressed all summation in terms of the multiplicities vector. Given a function of the two counts f, the sum over all categories is:
∑_i=1^K f(x⃗_⃗i⃗) = ∑_x⃗ν_x⃗ f(x⃗),
where the last sum runs over the ensemble of distinct pairs of observed counts. In the case of double sums (e.g. for ^2), one needs to re-express the function as:
f(x⃗_⃗i⃗, x⃗_⃗j⃗)
=
δ_ij f_∥(x⃗_⃗i⃗) + (1-δ_ij)f_(x⃗_⃗i⃗,x⃗_⃗j⃗),
where f_∥ and f_ is the function f for i=j and i≠ j.
The summation over the terms in δ_ij is calculated as before, and the double summation is
∑_i,j f_(x⃗_⃗i⃗, x⃗_⃗j⃗)
=
∑_x⃗, x⃗'⃗ν_x⃗ν_x⃗'⃗ f_(x⃗, x⃗'⃗).
These formulas allow us to exploit vectorial expressions in the numerical calculations.
§.§.§ Numerical integrations
Similarly to Ref. <cit.>, to compute numerically the quantities ⟨|n⃗,m⃗⟩ (Eq. <ref>) and ⟨^2 |n⃗,m⃗⟩ (Eq. <ref>), we first seek for the maximum (α_*,β_*) of the quantity ℒ(α,β) (see Fig. <ref> for further details).
For accuracy, we perform this computation in logarithmic space of logα and logβ.
Rescaling ℒ(α,β) by its maximum, integrands are 𝒪(1) for (α,β) ∼ (α_*,β_*).
To find the maximum of the log-evidence (minimum of the opposite), we use the “Limited-memory BFGS” optimization algorithm as coded in the function “minimize”, module optimize of the Python package scipy (version 1.7.3).
We evaluate the integrals using the trapezoidal rule. From the
Hessian at the maximum of the log-evidence, we compute the standard
deviation in the α and the β-direction as if the
posterior was Gaussian. We use this standard deviation to pick a
range of parameters spanning 3 standard deviations on both sides of
the maximum. We heuristically chose the number of bins within
the ranges for the integration, to be equal to
10(K/N)^2 for α
(10(K/M)^2 for β).
§.§ Code availability
The software for the DP, DPM and alternative estimators of the Kullback-Leibler and the Hellinger divergence presented in this article are collected in a Python package which can be found in the repository at <https://github.com/statbiophys/catede>.
In addition, the package provides a Python version for the NSB entropy estimator <cit.>, and a NSB estimator for the Simpson index <cit.>.
§ SUPPLEMENTARY FIGURES
|
http://arxiv.org/abs/2307.05132v1 | 20230711092210 | On the Use of Self-Supervised Speech Representations in Spontaneous Speech Synthesis | [
"Siyang Wang",
"Gustav Eje Henter",
"Joakim Gustafson",
"Éva Székely"
] | eess.AS | [
"eess.AS",
"cs.HC",
"cs.LG",
"cs.SD"
] |
plain
Coherent phonon and unconventional carriers in the magnetic kagome metal
E. Uykur
August 12, 2023
=========================================================================
Self-supervised learning (SSL) speech representations learned from large amounts of diverse, mixed-quality speech data without transcriptions are gaining ground in many speech-technology applications. Prior work has shown that SSL is an effective intermediate representation in two-stage text-to-speech (TTS) for both read and spontaneous speech. However, it is still not clear which SSL and which layer from each SSL model is most suited for spontaneous TTS. We address this shortcoming by extending the scope of comparison for SSL in spontaneous TTS to 6 different SSLs and 3 layers within each SSL. Furthermore, SSL has also shown potential in predicting the mean opinion scores (MOS) of synthesized speech, but this has only been done in read-speech MOS prediction. We extend an SSL-based MOS prediction framework previously developed for scoring read speech synthesis and evaluate its performance on synthesized spontaneous speech. All experiments are conducted twice on two different spontaneous corpora in order to find generalizable trends. Overall, we present comprehensive experimental results on the use of SSL in spontaneous TTS and MOS prediction to further quantify and understand how SSL can be used in spontaneous TTS. Audios samples: <https://www.speech.kth.se/tts-demos/sp_ssl_tts>
Index Terms: spontaneous speech synthesis, text-to-speech, self-supervised learning, mean-opinion-score prediction
§ INTRODUCTION
The availability of large amounts of data and computation has radically enhanced the capabilities of modern machine-learning systems.
One way that these developments can benefit ordinary applications with smaller amounts of data and computation is via “foundation models” <cit.>, publicly available pre-trained models created using self-supervised learning (SSL) on large amounts of unlabelled data.
Models that integrate representations of speech audio learned via SSL have demonstrated impressive results in areas such as speech recognition, speaker recognition, and voice conversion <cit.>.
Recently, these methods have also been considered for use as acoustic features in two-stage text-to-speech (TTS) <cit.>, showing promising results in replacing conventional mel-spectrogram features.
However, integrating SSL-based representations into TTS is still a novel concept, and it is not clear which representations are preferred for use in TTS, why they are preferred, nor what trade-offs are involved.
In addition to differences between different models, research into other applications has found that representations from different layers of the same SSL model may be preferred for different applications <cit.>.
Thus far, a handful of works have considered using SSL-based representations as TTS acoustic features [We use the term “acoustic features" in a broad sense to denote any intermediate features used between stages of a two-stage or multi-stage TTS system.]<cit.>, demonstrating advantages in creating TTS systems using SSL from mixed-quality audio.
SSL models have also been shown to be an effective mean opinion scores (MOS) predictor with minimal modification <cit.>. We aim to investigate both TTS and MOS prediction using SSL models, specifically the differences among SSLs and their layers, an under-explored aspect of prior studies.
Another important shortcoming of prior studies on using SSL in either TTS or MOS prediction is that they mostly only use speech read aloud as training data.
This contrasts against the majority of in-the-wild human speech, which tends to be spontaneous and unscripted.
Such speech involves unique verbal and nonverbal phenomena such as breathing <cit.>, disfluencies <cit.>, and discourse markers, which are seldom included or transcribed in conventional speech corpora, making them a blind spot of contemporary TTS research <cit.>.
In this paper, we compare representations derived from four different speech SSLs. We selected these models based on their high scores on the SUPERB benchmark <cit.>, similar dimensionalities and frame rates, and the availability of pre-trained weights. For some models, our comparisons consider multiple model versions, for example before and after ASR finetuning.
We study the utility of these models for two tasks in text-to-speech from spontaneous speech audio:
* As intermediate feature representations (“acoustic features”) in two-stage TTS.
* As backbone models for automatic prediction of speech quality (MOS) of synthetic speech.
We perform comprehensive experiments on two corpora previously used for spontaneous TTS.
Audio examples are available online at: <https://www.speech.kth.se/tts-demos/sp_ssl_tts>
§ BACKGROUND
Self-supervised representations learned from large amounts of untranscribed speech audio have recently found a large number of applications all across speech technology <cit.>.
In this section, we review the use of SSL representations in TTS and in speech-quality (MOS) prediction.
A particular focus of our survey (and, indeed, our entire paper) is spontaneous and conversational speech.
Despite accounting for the lion's share of human speech, and being considered vital for creating more-human like and convincing TTS for, e.g., conversational systems <cit.>, spontaneous speech and its challenges represent an underexplored topic in contemporary TTS research.
On the one hand, spontaneous speech exhibits increased acoustic and prosodic diversity <cit.>, and many spontaneous-speech databases rely on found or in-the-wild recordings, which may entail reduced audio quality, competing speakers, etc. <cit.>.
On the other hand, spontaneous speech cannot easily be partitioned into clean sentences, and the audio contains numerous phenomena that are difficult to transcribe.
This includes breathing <cit.>, disfluencies such as repetitions and filled pauses (uh/um) <cit.>, discourse markers (“like”, “you know”) <cit.>.
These phenomena are often missing from the input text, but they need to be generated by the TTS system.
It has been argued that these challenges of processing and synthesizing spontaneous speech can be effectively addressed by SSL <cit.>.
§.§ TTS Using SSL Models
The first systems using representations from modern self-supervised learning in TTS were likely WavThruVec <cit.> and VQTTS <cit.>.
Both proposed to replace the acoustic representations in between the acoustic model and the vocoder with a speech feature representation from an SSL model.
Compared to end-to-end TTS or traditional two-stage TTS based on mel-spectrogram features, this setup allows synthesizing high-quality audio even if the acoustic model is trained on mixed-quality audio material <cit.>.
While VQTTS trained a custom, discrete acoustic representation (turning acoustic modelling into a classification problem),
WavThruVec used a pre-trained wav2vec 2.0 model <cit.> as its intermediate acoustic representation.
There has also been work exploring the use of speech SSL models as added linguistic features instead of acoustic features <cit.>, and as basis for discrete coarse semantic tokens in two-stage discrete-token-based TTS approaches <cit.>.
Most relevant to the current work is the comparative study of Wang et al. <cit.>. They built a number of two-stage TTS systems using several publicly available speech SSL as intermediate representation, and contrasted these for synthesizing read as well as spontaneous speech.
They found that simply replacing traditionally used mel-spec with SSL representations improved both read and spontaneous TTS, but with the improvement being even more pronounced in the case of spontaneous TTS. They also found that intermediate SSL layers are better for TTS than the final layers, however they reached this comclusion by only comparing two layers in one SSL so it is limited in this respect. We focus exclusively on spontaneous speech in this work, and systematically expand the number of SSLs to 6 and the number of layers to 3 for each SSL, bringing the total number of TTS systems built to 36 (2 corpora × 6 SSLs × 3 layers) compared to only 4 in <cit.>
Another relevant work is MQTTS <cit.>. The model trains multiple discrete representations on the GigaSpeech corpus <cit.>, which is a very large corpus of in-the-wild speech that contains a lot of spontaneous speech. Through the self-supervised learned representations, a subsequently trained TTS model is able to generate high-quality spontaneous speech, demonstrating the advantage of SSL speech representations in synthesizing spontaneous speech.
§.§ Quality Prediction Using SSL Models
Apart from synthesizing speech, SSL models have also been considered for predicting quality scores (specifically MOS values) of natural and synthetic speech.
This was perhaps first done by <cit.> for predicting MOS values of different voice conversion systems. By building predictors from pre-trained SSL models and fine-tuning these end-to-end, they obtained a prediction accuracy surpassing previous state-of-the-art systems.
SSLs have subsequently become a hot topic in quality prediction.
The recent VoiceMOS Challenge considered predicting MOS scores of both voice conversion systems and of read-speech TTS <cit.>, and saw a very large portion of entries that made use of SSL models.
The main results saw pre-trained SSL models with fine-tuning outperform approaches that used such models without fine-tuning, in turn ahead of approaches that did not use SSL representations at all <cit.>.
Another recent challenge <cit.> focused on predicting speech quality in speech conferencing applications, and also saw several submissions, e.g. <cit.>, making use of SSL representations.
This task does involve spontaneous speech audio, but focuses only on assessing quality of speech transmission in online conferencing and not on asessing synthesized spontaneous speech from a TTS model.
Thus, none of the above works considered the use of SSL representations to predict the perceived quality of spontaneous TTS.
§ METHOD
The goal of this paper is to analyze the effect of using different SSL models in synthesizing and evaluating spontaneous speech; cf. Fig. <ref>.
In this section we describe the SSL models studied, the data, how we build TTS systems on these data, and how we use SSL representations for subsequent MOS score prediction.
Experimental results and discussion follow in Sec. <ref>.
§.§ Speech SSL Representations
Four speech SSL models were selected for our investigation.
These are summarized in Table <ref>.
All of these representations were investigated for spontaneous TTS, whereas only a subset were considered for the MOS-prediction task.
Our main reason for choosing these specific models was that they rank high on the SUPERB speech processing benchmark for speech SSLs <cit.>, and have a publically available implementation and weights.
Importantly, all chosen SSL models[Most models have several sizes, e.g. wav2vec2.0 base and large. We chose the base one in those cases.] have same dimensionality (765), number of layers (12 transformer layers), and frame rate (50), making them highly comparable.
The main differences between the models are the data and loss function/task used for training.
For each model, we considered the representations from three different layers (6, 9, and 12 out of 12), since prior work has shown that a middle layer of SSL models contains more prosodic information <cit.> that could benefit synthesis.
For some SSL models, we also found official ASR fine-tuned versions, which we include in the experiments in addition to the self-supervised pre-training-only models.
In total, we considered 18 different representations, 3 from each of 6 different SSL models.
We did not include mel-spectrogram baseline in this comparison because prior study has shown that it is much worse than SSL in two-stage spontaneous TTS <cit.>.
§.§ Spontaneous Speech Corpora
We trained our TTS voices on two corpora previously used in several different studies on spontaneous TTS.
The first corpus was created from the audio recordings of part 1 of the Trinity Speech-Gesture Dataset (TSGD) <cit.>, comprising 25 monologues, each on average 10.6 minutes long, spoken by a male speaker of Hiberno English in an impromptu, colloquial style.
During the recordings, the actor addresses a person seated behind the cameras, who is providing visual, but no verbal feedback.
To prepare the dataset for TTS, we segmented the corpus into stretches of speech delineated by breath events following <cit.>, and combined these segments in an overlapping fashion to form an utterance structure, with utterances no longer than 11 seconds, following <cit.>.
The second spontaneous corpus used in this work is the ThinkComputers Corpus (TCC) <cit.>, which is a 9-hour long corpus created from the speech of one of the hosts of the ThinkComputers podcast, which is available in the public domain.[<https://archive.org/details/podcasts_miscellaneous> Creator: ThinkComputers] The podcast recordings are approximately 50 min each, and
consist of two male speakers of American English discussing technology-related news and reviews. The speaking style can be described as extemporaneous, conversing freely around a prepared outline, meaning that the speakers use a prepared outline, but converse freely around the planned topics.
Both corpora were transcribed using ASR and subsequently corrected manually. All discourse markers, laughter, and filled pauses (uh, um) were transcribed orthographically, breath events were marked with a semi-colon, while pauses were transcribed using a comma. Other spontaneous speech phenomena such as tongue clicks were not part of the transcription.
§.§ TTS System
We used a similar system and training setup as WavThruVec <cit.>, with the difference being that we omitted the multi-speaker embeddings in our system. We illustrate the system in Fig. <ref>.
The stage-1 model is adapted from a parallel TTS model, FastPitch <cit.>, in which the alignment is learned automatically <cit.>.
[<https://github.com/NVIDIA/DeepLearningExamples/tree/master/PyTorch/SpeechSynthesis/FastPitch>. This model is called “FastPitch 1.1” in this official implementation.]
We used identical hyperparameters as in <cit.> to train stage-1 models, only changing the batch size to 128. We first trained on a read-speech corpus, namely LJ Speech[<https://keithito.com/LJ-Speech-Dataset/>], for 200 epochs, and then on each of the two spontaneous corpus for 200 epochs. This transfer-learning method has shown to be effective in allowing neural TTS to be trained on smaller spontaneous corpora <cit.>.
For the stage-2 model or the vocoder, we used HiFi-GAN <cit.>,[<https://github.com/jik876/hifi-gan>] trained with similar hyperparameters as in <cit.>. We used a batch size of 160, each datum being a 0.5 second random audio excerpt. All 36 stage-2 vocoders (for the 18 SSL representations in 2 corpora) were trained for 80k steps. We used the original audio sampling rate of 22 kHz. Models of both stages were trained on 1–2 Nvidia A100 GPUs depending on batch size.
§.§ MOS-Prediction System
We followed <cit.> to build a simple wav2vec 2.0 based MOS predictor. The predictor consists of a wav2vec 2.0 base model with a mean-pooling head on top and a linear projection to a scalar MOS value. We adapted the implementation of <cit.>[<https://github.com/nii-yamagishilab/mos-finetune-ssl.git>] and followed their training procedure and hyperparameters.
However, we tested different weight initializations and training-data splitting configurations for this fixed architecture, to probe how these factors affect performance of predicting MOS on spontaneous speech synthesis.
§ RESULTS
This section reports and discusses our three main experimental results, namely 1) vocoder error in copy synthesis achieved by tested SSLs, 2) the design and results of subjective listening test, and 3) the performance of SSL-based MOS predictors on MOS scores collected in the subjective listening test.
§.§ Vocoder Error in Copy Synthesis
We conducted copy synthesis of the audios in the validation set. SSLs were extracted from ground-truth audios in the validation set and then vocoded through the corresponding vocoders/second-stage models. Mel-spectrogram L1 error achieved by the vocoders are graphed in Fig. <ref> (blue squares), showing how these errors depend on the SSL model and layer.
There are clear trends in that, the deeper the layer, the greater the error, presumably because deeper layers are further removed from the original speech waveform.
In the two pairs of pre-training only and ASR fine-tuning models of data2vec and wav2vec2.0, ASR fine-tuning consistently leads to increased vocoding error over corresponding pre-trained models. However, we note that the lowest vocoding error overall is attained by representations from Whisper, which is a dedicated ASR model. In informal listening, we were impressed with the copy-synthesis performance from the Whisper-derived representations, consistent with its lowest vocoding error numbers.
Except for WavLM, the vocoding errors are very similar across the two corpora for the same model and layer. The two corpora are different in many aspects, so why are the achieved vocoding errors so close in the two corpora? This phenomenon deserves future investigation.
§.§ Subjective Evaluation of TTS Systems
We performed two MOS listening tests according to ITU standard P.800 <cit.>, one for each corpus, to evaluate the full two-stage TTS pipelines built with different SSLs.
Each evaluation used a pool of 20 utterances synthesised by each of the 18 different systems, for a total of 320 stimuli per corpus.
For each corpus we recruited 45 self-reported native English-speaking listeners via the Prolific crowdsourcing platform.
Each listener rated 51 randomly chosen stimuli from the pool balanced for SSL and layer.
Participants were asked to wear headphones, and they were requested to not take the test if they had a hearing impairment.
Ratings were integer values given on a scale from 1 through 5 with text labels as specified in aforementioned ITU standard <cit.>.
Attention checks in the form of multi-choice speech recognition tests were included to filter out unqualified test-takers.
Test-takers who completed their tests too quickly to have listened to all the audio were also disqualified. This resulted 44 valid completed tests for each corpus.
Participants were rewarded with an hourly wage of approximately 12 GBP with 15 or 20 minutes paid time [We used 15 minutes paid time for TCC test and 20 minutes paid time for TSGD test. We slightly underestimated completion time when conducting TCC test first, thus increased expected completion which is also the paid time for TSGD test.], thus 3 or 4 GBP each.
Results of the two listening tests are visualized in Fig. <ref> (red triangles).
We observe several prominent trends. First, the 9th layer outperforms the 12th (last) and 6th (middle) layers in 4 out of 6 SSL models on both corpora. Layer 9 outperforming layer 12 is consistent with prior study on SSL layer-wise TTS performance <cit.>, however layer 9 also outperforms layer 6 is an interesting new finding. We also find that SSLs after ASR fine-tuning obtained better ratings than corresponding SSLs prior to fine-tuning, i.e. underwent only self-supervised pre-training. For both corpora, the best performing representation is data2vec-base-asr layer 9 (TSGD: 3.90±0.18, TCC: 3.77±0.17). It is worth noting that MOS in the range of 3.90 is at the same level as current SOTA spontaneous TTS systems <cit.>, however we do not claim that our best system is as good as a SOTA system as it is difficult to make such comparison on MOS score alone while the settings of MOS tests could be very different.
We also note that consistency of the trends in SSL models and layers between the two corpora suggests that the results are likely to generalize to other spontaneous corpora.
C>p3.0cm
We also see that the vocoding errors do not correlate at all with perceived TTS quality. A lower vocoding error suggests that there is more acoustic information present in the representation, however, this does not lead to better overall two-stage TTS performance as measured in subjective MOS tests.
In fact, ASR fine-tuned data2vec, the best performing SSL model in two-stage TTS, consistently exhibited one of the highest vocoding errors, whereas Whisper underperformed for TTS despite having lowest vocoding errors. This suggests that there is a trade-off between the amount of acoustic information in the representation and how well can the first-stage acoustic model predict that representation from text, a phenomenon also observed in a prior study on using SSL in two-stage TTS <cit.>. Notably in that study, the authors found that mel-spec which achieves lowest vocoding error is the worst representation in two-stage spontaneous TTS. Another prior study reported similar results that regular TTS models have trouble findding alignemnt between mel-spec and text in spontaneous speech corpus. Our results provide further evidence to this hypothesis that there could be a trade-off between the amount of acoustic information an intermediate representation (SSL or otherwise) contains versus its achievable prediction accuracy (from text input in a TTS setting).
§.§ Evaluation of Automated MOS Prediction
Using MOS data obtained in our subjective listening tests, we probed two sets of factors that may affect the generalization ability of spontaneous-speech MOS prediction with SSL:
1) the starting weights used for fine-tuning and 2) the type of unseen data (dataset split), by specifically holding out either random audio samples, or data from specific utterances (input texts), or entire TTS models, or the full corpus (i.e., training on one corpus and predicting the scores on the other).
Except for at the corpus level, we performed 5-fold cross-validation for each of these experiments.
In addition to fine-tuning, we also tested the zero-shot performance of the predictor from <cit.>.
Results from the experiments on automated MOS prediction are reported in Table <ref>.
We make a number of observations from these results.
First, the zero-shot model from <cit.> pre-trained on read-speech MOS does not make meaningful predictions on this data as shown by its high MSE in all categories, however it achieves good linear correlation in some categories.
Fine-tuning improved performance, with fine-tuning on top of <cit.> or wav2vec2.0-base performing similarly and fine-tuning on top of wav2vec2.0-base-asr performing slightly worse.
Finally, although prediction MSE is low, correlations are not as strong as the numbers achieved by MOS predictors on read-speech data <cit.>.
Several factors may contribute to this, for example that the range of MOS values in our data is quite narrow, that we have less data available than for read speech MOS, and that predicting the scores of spontaneous TTS in general may be a more challenging task than for read speech.
§ CONCLUSION AND FUTURE WORK
We have compared various self-supervised speech representations in spontaneous text-to-speech and in MOS prediction on spontaneous speech synthesis, on two different corpora.
We used a total of 6 different SSLs and 3 layers from each SSL, totaling 18 representations, as intermediate features in two-stage TTS.
We found that representations from layer 9 of the SSL models provided better subjective TTS quality than layer 6 or layer 12 (the final layer), with the best spontaneous TTS quality achieved by layer 9 of data2vec with ASR fine-tuning.
We also found that TTS subjective MOS does not correlate with the vocoding loss obtained by the SSL representation, where the high-performing TTS representations obtained some of the worst vocoding loss, and vice versa. Our results could be used as reference for SSL selection in speech synthesis tasks that utilize SSL at any capacity, and for more in-depth analysis of inter-model and layer-wise differences of SSL models in TTS or other synthesis tasks.
We also studied the use of SSL models in predicting MOS of spontaneous speech synthesis using data obtained in our subjective listening tests.
We found that zero-shot prediction from a read-speech pre-trained SSL MOS predictor performs poorly, and that fine-tuning on spontaneous MOS data is crucial for a SSL MOS predictor to have any predictive value on synthesized spontaneous speech.
Compelling future work includes studying more SSLs on larger spontaneous corpora, as well as improving SSL and TTS architectures for spontaneous speech.
§ ACKNOWLEDGEMENTS
This work was partially supported by Digital Futures project “Advanced Adaptive Intelligent Systems”, the Swedish Research Council projects “Connected” (VR-2019-05003) and “Perception of speaker stance” (VR-2020-02396), and by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. The computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) at Chalmers e-Commons partially funded by the Swedish Research Council through grant agreement no. 2022-06725, and by the Berzelius resource provided by the Knut and Alice Wallenberg Foundation at the National Supercomputer Centre.
IEEEtran
|
http://arxiv.org/abs/2307.05669v2 | 20230711180001 | The Dependence of Iron-rich Metal-poor Star Occurrence on Galactic Environment Supports an Origin in Thermonuclear Supernova Nucleosynthesis | [
"Zachary Reeves",
"Kevin C. Schlaufman",
"Henrique Reggiani"
] | astro-ph.GA | [
"astro-ph.GA",
"astro-ph.SR"
] |
Zachary Reeves
[email protected]
0000-0002-0821-878X]Zachary Reeves
William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA
0000-0001-5761-6779]Kevin C. Schlaufman
William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA
0000-0001-6533-6179]Henrique Reggiani
Carnegie Fellow
The Observatories of the Carnegie Institution for Science,
813 Santa Barbara St, Pasadena, CA 91101, USA
It has been suggested that a class of chemically peculiar metal-poor stars
called iron-rich metal-poor (IRMP) stars formed from molecular cores with
metal contents dominated by thermonuclear supernova nucleosynthesis.
If this interpretation is accurate, then IRMP stars should be more
common in environments where thermonuclear supernovae were important
contributors to chemical evolution. Conversely, IRMP stars should
be less common in environments where thermonuclear supernovae were not
important contributors to chemical evolution. At constant [Fe/H]
≲ -1, the Milky Way's satellite classical dwarf spheroidal (dSph)
galaxies and the Magellanic Clouds have lower [α/Fe] than
the Milky Way field and globular cluster populations. This difference
is thought to demonstrate the importance of thermonuclear supernova
nucleosynthesis for the chemical evolution of the Milky Way's satellite
classical dSph galaxies and the Magellanic Clouds. We use data from
the Sloan Digital Sky Survey (SDSS) Apache Point Observatory Galactic
Evolution Experiment (APOGEE) and Gaia to infer the occurrence of
IRMP stars in the Milky Way's satellite classical dSph galaxies
η_dSph and the Magellanic Clouds η_Mag
as well as in the Milky Way field η_MWF and globular
cluster populations η_MWGC. In order of decreasing
occurrence, we find η_dSph=0.07_-0.02^+0.02,
η_Mag=0.037_-0.006^+0.007,
η_MWF=0.0013_-0.0005^+0.0006, and a 1-σ
upper limit η_MWGC<0.00057. These occurrences support
the inference that IRMP stars formed in environments dominated by
thermonuclear supernova nucleosynthesis and that the time lag between
the formation of the first and second stellar generations in globular
clusters was longer than the thermonuclear supernova delay time.
§ INTRODUCTION
Thermonuclear supernovae[In this article we use the phrase
“thermonuclear supernovae” to refer to the theoretical concept of
electron-degenerate carbon–oxygen white dwarfs experiencing runway
carbon fusion that releases enough energy to gravitationally unbind the
white dwarfs and thereby cause explosions.] are prolific producers of
iron-peak elements with a theoretically predicted average stable yield of
more than 0.6 M_⊙ of iron-peak elements but much less α
and light odd-Z elements. This is in sharp contrast to core-collapse
supernovae that produce α and light odd-Z elements in roughly
the solar ratios but with relatively little iron-peak production
<cit.>.
Based on a compilation of theoretical stable nucleosynthetic yields
predicted by a variety of thermonuclear supernova progenitor channels
(i.e., single degenerate and double degenerate) and explosion mechanisms
(e.g., pure detonations, pure deflagrations, delayed detonations,
double detonations, etc.), <cit.> proposed the existence of a
new class of chemically peculiar metal-poor stars with [Fe/H]
≲ -1 and [O,F,Ne,Na,Mg,Al,Cl,K,Co,Cu,Zn/Fe] < 0 formed
from molecular cores with metal contents dominated by thermonuclear
supernova nucleosynthesis.[The yields in that compilation came
from <cit.>, <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>, <cit.>, <cit.>,
<cit.>, and <cit.>.] They called stars with
these properties iron-rich metal-poor (IRMP) stars, as this part of
elemental abundance space is consistent with thermonuclear supernova
nucleosynthesis but rarely observed in metal-poor stars. They argued
that if their interpretation is correct, then IRMP stars should be more
common in environments where thermonuclear supernovae were relatively more
important contributors to chemical evolution relative to core-collapse
supernovae (e.g., environments with long star formation durations).
On the other hand, they argued that in environments where thermonuclear
supernovae were not important contributors to chemical evolution relative
to core-collapse supernovae (e.g., environments with short star formation
durations) IRMP stars should be less common.
One way to test this prediction would be to compare the relative
occurrence of IRMP stars in the Milky Way's field population, its
globular clusters, its satellite classical dwarf spheroidal (dSph)
galaxies, and the Magellanic Clouds. At constant spectroscopically
inferred metallicities [Fe/H] ≲ -1[In this
article metallicity [Fe/H] has its usual meaning [Fe/H] =
log_10(N_Fe/N_H)_∗-log_10(N_Fe/N_H)_⊙
where N_X are the logarithmic number densities
of atoms of an element X in a stellar photosphere and
N_H≡ 12.], both the Milky Way's classical dSph
satellites <cit.>
and the Magellanic Clouds <cit.> have
lower spectroscopically inferred ratios of the α
elements oxygen, magnesium, silicon, and calcium to iron
[α/Fe][In this article the α element-to-iron
ratio [α/Fe] has its usual meaning [α/Fe] =
log_10(N_α/N_Fe)_∗-log_10(N_α/N_Fe)_⊙
where α is the sum of some subset of the elements
oxygen, magnesium, silicon, and calcium.] than the Milky Way
<cit.>
and its globular clusters <cit.>.
These offsets in [α/Fe] between the Milky Way field
& globular cluster populations and its satellite classical dSph
galaxies & the Magellanic Clouds are usually understood to indicate
the longer durations of low-metallicity star formation and therefore the
relatively more important contributions of thermonuclear supernovae to
the chemical evolution of the latter two environments at low metallicities
<cit.>.
If the <cit.> interpretation of IRMP stars is valid, then
IRMP stars should be less common in the Milky Way field & globular
cluster populations than in its satellite classical dSph galaxies &
the Magellanic Clouds.
The nitrogen, sodium, and aluminum abundances of individual stars
in globular clusters have been found to be anticorrelated with
the carbon, oxygen, and magnesium abundances in the same stars.
These abundance anticorrelations have been interpreted as evidence
for multiple generations of star formation in globular clusters
<cit.>. The <cit.>
interpretation of IRMP stars also suggests the possibility that the
occurrence of IRMP stars in globular clusters can be used to constrain
the time lag between the formation of a globular cluster's first
and second stellar generations. For Type Ia supernovae[In
this article we use the phrase “Type Ia supernovae” to refer to the
electromagnetic transients empirically classified as Type Ia supernovae
based on their observed properties.] delay times τ_Ia≲ 100 Myr, the Type Ia supernova rate Φ_Ia has
the value Φ_Ia∼ 10^-12 yr^-1 M_⊙^-1
<cit.>. Assuming that rate and a
typical first generation initial globular cluster mass M_MWGC∼ 10^6 M_⊙ <cit.>, order 10 Type Ia
supernovae should occur in a newly formed globular cluster every 10 Myr
after the Type Ia delay time has elapsed. While the occurrence of the
short-period binaries necessary to produce most of the theoretically
predicted thermonuclear supernova progenitor systems is a factor of
about three lower in globular clusters' first generations than in the
field <cit.>, after the Type Ia supernovae delay time has
elapsed a few thermonuclear explosions should occur every 10 Myr during
the formation of a globular cluster. The <cit.> interpretation of
IRMP stars therefore implies that if the time lag between the formation
of first and second generation stars in globular clusters was shorter
than the typical thermonuclear supernova delay time, then the occurrence
of IRMP stars in globular clusters should be significantly lower than
in the Milky Way field. If the the time lag between the formation of
first and second generation stars in globular clusters is longer than the
typical thermonuclear supernova delay time, then the occurrence of IRMP
stars in globular clusters and the Milky Way field should be comparable.
We argue that if the <cit.> interpretation of the origin of
IRMP stars is accurate, then the occurrence of IRMP stars in the
Milky Way field population η_MWF, the Magellanic Clouds
η_Mag, and the Milky Way's satellite classical dSph
galaxies η_dSph should be ordered η_MWF
< η_Mag∼η_dSph. If the time lag
between the formation of the first and second generations in globular
clusters was shorter than the thermonuclear supernova delay time,
then the occurrence of IRMP stars in the Milky Way globular cluster
population η_MWGC should be η_MWGC≲η_MWF < η_Mag∼η_dSph. If the
time lag between the formation of the first and second generations in
globular clusters was longer than the thermonuclear supernova delay time,
then η_MWGC∼η_MWF < η_Mag∼η_dSph. In this article, we calculate the occurrence
of IRMP stars in the Milky Way's satellite classical dSph galaxies,
the Magellanic Clouds, the Milky Way field population, and in Milky Way
globular clusters. We describe in Section <ref> the assembly of our
analysis samples and quantify in Section <ref> the occurrence
of IRMP stars in each environment. We review the implications of those
occurrences in Section <ref> and conclude by summarizing our
findings in Section <ref>.
§ DATA
To calculate the occurrence of IRMP stars in the Milky Way's satellite
classical dSph galaxies, the Magellanic Clouds, the Milky Way field
population, and in Milky Way globular clusters we use data derived
from spectra that were gathered during the third and fourth phases
of the Sloan Digital Sky Survey <cit.> as
part of its Apache Point Observatory Galactic Evolution Experiment
<cit.>. These spectra were collected with the
APOGEE spectrographs <cit.> on the
New Mexico State University 1-m Telescope <cit.>, the Sloan
Foundation 2.5-m Telescope <cit.>, and the 2.5-m Irénée
du Pont Telescope <cit.>. As part of SDSS Data Release
(DR) 17 <cit.>, these spectra were reduced and analyzed
with the APOGEE Stellar Parameter and Chemical Abundance Pipeline
<cit.> using an H-band line list,
MARCS model atmospheres, and model-fitting tools optimized for the APOGEE
effort <cit.>.
We use the CasJobs
portal[<http://skyserver.sdss.org/casjobs/>] and the
query described in the Appendix to generate our initial sample of
photospheric stellar parameters and elemental abundances for giant stars
with logg < 3.8. As described in the Appendix, we use a carefully
curated set of data quality flags to ensure the accuracy and precision
of those photospheric stellar parameters and elemental abundances.
We set to any elemental abundance/elemental abundance
uncertainty pair that does not pass the data quality checks described
in the Appendix. Corrections for departures from local thermodynamic
equilibrium are usually small for H-band elemental abundance inferences
<cit.>, and we choose not to apply them in our analysis.
Following <cit.>, we define an iron-rich metal-poor star as
a star with [Fe/H] < -1 and [O,Na,Mg,Al,K,Co/Fe]
< 0. Those elemental abundance ratios form the intersection of the
IRMP criteria defined in <cit.> and the list of elemental
abundances reliably inferred for giant stars as part of APOGEE
DR17[<https://www.sdss.org/dr17/irspec/abundances/>].
For the purposes of the occurrence calculation described in the next
section, an IRMP star with [Fe/H] < -1 must have at least one
non- abundance ratio [O/Fe], [Na/Fe],
[Mg/Fe], [Al/Fe], [K/Fe], or [Co/Fe]
and all abundance ratios [O,Na,Mg,Al,K,Co/Fe] either sub-solar
or .
To accurately label stars with their correct galactic environments,
we first join the APOGEE data described above with data from Gaia
DR2 <cit.> and DR3
<cit.>.
We use the string to identify the
corresponding Gaia DR2 and DR3 long integers
by joining with the and
tables available in the
Gaia archive <cit.>. Occasionally multiple Gaia DR2 and DR3
long integers are matched to the same object in the
2MASS Point Source Catalog <cit.>. In those cases, we associate
a 2MASS object with the closest Gaia DR2 and DR3 object that has (1) an
absolute 2MASS K_s-band magnitude M_K < 2.31 assuming Gaia
DR2- or DR3-prior informed geometric distances <cit.> and
(2) Gaia–2MASS colors 0.5 < G-J < 2.3, 0.6 < G-H < 3.3, and 0.6 <
G-K_s < 3.4 predicted by the MESA Isochrones & Stellar Tracks
(MIST) grid for metal-poor giants in the range -2.5 < [Fe/H] <
-1.0 <cit.>.
We then use the Gaia DR2 to identify Milky Way
satellite classical dSph galaxies or Magellanic Clouds members using the
Gaia DR2-based membership lists published in <cit.>. We identify
globular cluster members using the Gaia DR3 and the
<cit.> lists of stars with globular cluster membership probability
greater than 0.5, and this procedure results in a sample with at least one
star from 41 globular clusters over the metallicity range -2.3 ≲
[Fe/H] ≲ -1.0.
Most stars observed as part SDSS-III/APOGEE and SDSS-IV/APOGEE-2 were
selected for observation by a procedure that sought to minimize age and
metallicity biases <cit.>, and we use the targeting
flag = 0 in the table to select
those stars for our Milky Way field sample. To ensure the cleanest Milky
Way field sample possible, we then remove from this sample any stars that
are identified as dSph, Magellanic Cloud, or globular cluster members
in <cit.> or <cit.>. Because the vast majority of stars
with [Fe/H] ≲ -1 observed as part of SDSS-III/APOGEE and
SDSS-IV/APOGEE-2 are on halo-like orbits <cit.>, our Milky
Way field sample can be thought of as a Milky Way halo sample. We list
in Table <ref> our entire analysis sample including IRMP status
and galactic environment. We report in Table <ref> the number of
stars classified as IRMP stars in each environment N_∗,IRMP
along with the number of stars in our analysis sample in each environment
N_∗,tot.
ccccc
Analysis Sample
0pt
APOGEE ID
Gaia DR3
Gaia DR2
IRMP
Environment
2M17165079-2422565 4111066558908989184 4111066558908989184 False Milky Way
2M17150296-2423503 4114045307692005632 4114045307692005632 False Milky Way
2M19154424-0604209 4211181280154376320 4211181280154376320 False Milky Way
2M19033822+1745138 4514220364269464832 4514220364269464832 False Milky Way
2M16574858-2156135 4126283868512500864 4126283868512500864 False Milky Way
2M18501947+2948368 2041393317228382848 2041393317228382848 False Milky Way
2M18132084+0112054 4275831399934633984 4275831399934633984 False Milky Way
2M17571005-3020262 4056215664653907456 4056215664653907456 False Milky Way
2M17415271-2715374 4060889448072712832 4060889448072712832 False Milky Way
2M19084424-0618527 4205916204326948736 4205916204326948736 False Milky Way
This table is published in its entirety in the
machine-readable format. A portion is shown here for guidance regarding
its form and content.
lccc
Occurrence of IRMP Stars as a Function of
Environment
0pt
Galactic Environment
N_∗,IRMP
N_∗,tot
Occurrence
Milky Way 5 4247 0.0013_-0.0005^+0.0006
Globular Cluster Sum 0 1998 <0.00057
LMC 4 203 0.023_-0.009^+0.012
SMC 24 572 0.043_-0.008^+0.009
Magellanic Clouds Sum 28 775 0.037_-0.006^+0.007
Sagittarius 0 27 <0.04
Ursa Minor 0 8 <0.12
Sextans 1 6 0.2_-0.1^+0.2
Sculptor 8 70 0.12_-0.03^+0.04
Draco 0 9 <0.11
Carina 1 31 0.05_-0.03^+0.05
dSph Galaxies Sum 10 151 0.07_-0.02^+0.02
The typical elemental abundance inference uncertainties in our analysis
sample are (0.04, 0.23, 0.03, 0.04, 0.08, 0.16) dex for ([O/Fe],
[Na/Fe], [Mg/Fe], [Al/Fe], [K/Fe],
[Co/Fe]). In any case, elemental abundance inference
uncertainties are irrelevant for the occurrence analyses presented in
Section <ref> if (1) the uncertainty distributions for each
elemental abundance inference for each individual star are symmetric and
(2) the uncertainty distributions have statistically indistinguishable
widths across all of the galactic environments we explored.
The individual elemental abundance inference uncertainties presented
in the SDSS DR17 version of the table are reported
as symmetric. Additionally, we confirmed that the individual elemental
abundance uncertainty distributions for oxygen, sodium, magnesium,
aluminum, potassium, and cobalt have statistically indistinguishable
widths across all of the galactic environments we explored. Both of
the conditions listed above are therefore met in our analysis sample.
Likewise, we argue that our procedure to handle data quality issues will
not bias the occurrence analyses we present in Section <ref>.
The reason is that values impact less than about 0.2% of
the magnesium abundance inferences in our sample. Because we require
all non- elemental inferences to meet our IRMP criteria,
we are able to exclude essentially all non-IRMP stars from our IRMP
sample using magnesium alone almost regardless of data quality issues.
The recent discovery of a very metal-poor star with elemental abundances
best explained by the nucleosynthesis expected in a pair-instability
supernova has focused attention on that explanation for stars with
significantly subsolar [Na/Fe] and [α/Fe]
abundance ratios <cit.>. While the stars in our analysis sample
have subsolar [Na/Fe] and the α-element abundance ratios
[O/Fe] and [Mg/Fe], none of the stars in our sample
have the strong odd–even abundance ratios predicted to be produced by
pair-instability supernovae <cit.>. We are therefore
confident that the IRMP stars we identify are related to thermonuclear
supernovae.
§ ANALYSIS
We model the number of IRMP stars N_∗,IRMP in a sample
of N_∗,tot candidates using a binomial distribution.
Following <cit.> we exploit the fact that a Beta(α,β)
distribution is a conjugate prior to the binomial distribution and will
result in a Beta distribution posterior for the occurrence of IRMP stars
in a sample. Bayes's Theorem guarantees
f(θ|𝐲) = f(𝐲|θ)f(θ)/∫ f(𝐲|θ)f(θ)dθ,
where f(θ|𝐲) is the posterior distribution of the model
parameter θ, f(𝐲|θ) is the likelihood of the data
𝐲 given θ, and f(θ) is the prior for θ.
In this case, the likelihood is the binomial likelihood that describes
the probability of a number of successes y in n Bernoulli trials
each with probability θ of success
f(y|θ) = ([ n; y ])
θ^y(1-θ)^n-y.
As shown by <cit.>, in this situation using a
Beta(α,β) prior on θ with hyperparameters α
and β results in a Beta posterior for θ of the form
Beta(α+N_∗,IRMP,β+N_∗,tot-N_∗,IRMP).
The hyperparameters α and β of the prior can be thought
of as encoding a certain amount of prior information in the form
of pseudo-observations. Specifically, α-1 is the number of
success and β-1 is the number of failures imagined to have already
been observed and therefore included as prior information on θ.
Taking any α = β = i where i ≥ 1 could be thought of as
an uninformative prior in the sense that the probability of success and
failure in the prior distribution are equally likely. However, if i
is large then there is imagined to be a lot of prior information and the
posterior distribution will mostly reflect the prior when n ≤ i.
On the other hand, if n ≫ i, then the posterior will be dominated
by the data. For that reason, we take α=β=1.
We provide the posterior median occurrence of IRMP stars in each
galactic environment in Table <ref>. We define the lower
uncertainty as the difference between the posterior median and its
16th percentile. Likewise, we define the upper uncertainty as the
difference between the posterior's 84th percentile and its median.
For an environment with no IRMP stars, we report 1-σ upper
limits as the 68th percentile of the posterior distribution.
We find that η_dSph=0.07_-0.02^+0.02,
η_Mag=0.037_-0.006^+0.007,
η_MWF=0.0013_-0.0005^+0.0006, and a 1-σ upper
limit η_MWGC<0.00057. In words, IRMP stars are much more
common in the Milky Way's classical dSph satellites and the Magellanic
Clouds than in the Milky Way field or globular cluster populations.
IRMP stars are less common in globular clusters than in any other
galactic environment. We find that the overlap probability between
the IRMP occurrences we observed in the Milky Way's classical dSph
satellites and the Magellanic Clouds is about one in 91, equivalent to
about 2.3 σ. We find that the overlap probabilities between the
IRMP occurrences we observe in the Milky Way's classical dSph satellites
& the Magellanic Clouds and the occurrence of IRMP stars we observe in
the Milky Way field population are about one in 2.8 × 10^15 and
one in 6.2 × 10^19, equivalent to about 8.1 and 9.2 σ.
The overlap probability between the IRMP occurrences we observe in the
Milky Way field and globular cluster populations is about one in 23,
equivalent to about 1.7 σ. We summarize these occurrence posterior
overlap probabilities in Table <ref>.
ccC
Occurrence Posterior Overlap Probabilities
0pt
Population
Population
P(Posterior Overlap)
dSph galaxies Magellanic Clouds 1.095 ×10^-2
dSph galaxies Milky Way field 3.596 ×10^-16
Magellanic Clouds Milky Way field 1.620 ×10^-20
Milky Way field Milky Way globular clusters 4.300 ×10^-2
The results presented in Table <ref> are averaged over metallicity.
To investigate IRMP occurrence as a function of metallicity, for each
class of galactic environment we divide into ten equal intervals the
metallicity range spanned by our analysis sample for that environment.
We then apply the same occurrence formalism in each individual metallicity
interval and plot occurrence as a function of metallicity for each
environment in Figure <ref>. We find no significant dependence
of the occurrence of IRMP stars on metallicity in our analysis sample.
§ DISCUSSION
We find that the occurrences of IRMP stars in the Milky Way's satellite
classical dSph galaxies, the Magellanic Clouds, the Milky Way field
population, and the Milky Way's globular cluster populations
have the values η_dSph=0.07_-0.02^+0.02,
η_Mag=0.037_-0.006^+0.007,
η_MWF=0.0013_-0.0005^+0.0006, and
η_MWGC<0.00057. The probability that the IRMP occurrences
in the Milky Way's satellite classical dSph galaxies and the Magellanic
Clouds overlap is about one in 91, equivalent to about 2.3 σ.
The probabilities that the IRMP occurrence posterior for the Milky
Way field overlaps with the IRMP occurrence posteriors for Milky Way's
satellite classical dSph galaxies and the Magellanic Clouds are about
one in 2.8 × 10^15 and one in 6.2 × 10^19, equivalent
to about 8.1 and 9.2 σ. The probability that the IRMP occurrences
in the Milky Way field and globular cluster populations overlap is about
one in 23, equivalent to about 1.7 σ. While the absolute values of
IRMP star occurrences may be difficult to interpret, as we argued in the
introduction their ordering η_MWGC∼η_MWF <
η_Mag∼η_dSph has two important implications.
The increased occurrence of IRMP stars in environments like the Milky
Way's satellite classical dSph galaxies and the Magellanic Clouds where
thermonuclear supernovae were important contributors to chemical evolution
supports the <cit.> scenario for IRMP star formation in molecular
cores with metal contents dominated by the thermonuclear supernova
nucleosynthesis. The confirmation of this <cit.> prediction
reinforces the idea that the elemental abundances of individual IRMP
stars can be used to investigate the progenitor systems and explosion
mechanisms responsible for the thermonuclear supernovae that produced
much of their metal contents.
The statistically indistinguishable occurrences of IRMP stars in the Milky
Way's field and globular cluster populations suggests that the time lags
between the formation of globular clusters' first and second stellar
generations were longer than the thermonuclear supernova delay time.
This is broadly consistent with the idea from the asymptotic giant
branch (AGB) scenario for globular cluster multiple populations that
the explosions of thermonuclear supernovae associated with a globular
cluster's first stellar generation quench the star formation event that
produced its second stellar generation <cit.>.
Given the suspected importance of thermonuclear supernovae for quenching
second generation star formation in globular clusters, it remains to
be explained why globular cluster second generation stars that bear the
imprint of thermonuclear supernova nucleosynthesis are so rare as to not
appear in a sample of nearly 2000 globular cluster members. Our result is
also consistent with scenarios for globular cluster multiple populations
invoking a thermonuclear supernova at the start of a cluster's evolution
<cit.>.
Contributions from both core-collapse and thermonuclear supernovae as well
as s- and r-process nucleosynthesis are required to explain the solar
abundance pattern. While the progenitors of core-collapse supernovae are
known to be massive stars and the s-process takes place in AGB stars,
the progenitors of thermonuclear supernovae and the astrophysical site of
the r-process are more uncertain. Observational constraints on either
the progenitors of thermonuclear supernovae or the astrophysical site of
the r-process are therefore valuable. The kilonova GW170817 confirmed
that neutron stars mergers are at least partially responsible for
r-process nucleosynthesis <cit.>. The relative
occurrences of IRMP and r-process enhanced stars in the same populations
can be used to compare the relative probabilities of the circumstances
that lead to the formation of IRMP and r-process enhanced stars.
We find that the occurrences of IRMP stars are an order of magnitude
lower than the occurrence of r-process enhanced stars in both the
Milky Way field and Magellanic Clouds. In the Milky Way field,
we find that η_IRMP,MWF=0.0013_-0.0005^+0.0006
while <cit.> found η_rII,MWF≈0.03
according to the definition of highly r-process enhanced
(i.e., r-II stars) from <cit.>.[More recent
estimates by the r-process Alliance have found similar occurrences
<cit.>.] In the Magellanic Clouds, we find
that η_IRMP,Mag=0.037_-0.006^+0.007 while <cit.>
found η_rII,Mag=0.38^+0.14_-0.13. We conclude that
the circumstances that lead to the formation of IRMP stars occur an
order of magnitude less frequently than the circumstances that lead to
the formation of r-process enhanced stars.
§ CONCLUSION
We conclude that iron-rich metal-poor stars with [Fe/H]
≲ -1 and [O,Na,Mg,Al,K,Co/Fe] < 0 are more common in
the Milky Way's satellite classical dSph galaxies and the Magellanic
Clouds than in the Milky Way field or globular cluster populations.
Because thermonuclear supernovae are thought to have been more important
contributors to the chemical evolution of the Milky Way's satellite
classical dSph galaxies and the Magellanic Clouds than to the Milky
Way field and globular cluster populations in the range [Fe/H]
≲ -1, our inferences confirm the interpretation of iron-rich
metal-poor stars put forward in <cit.> proposing that iron-rich
metal-poor stars formed from molecular cores with metal contents dominated
by thermonuclear supernova nucleosynthesis. Iron-rich metal-poor stars
can therefore be used to constrain the progenitor systems and explosion
mechanisms of the thermonuclear supernovae responsible for their elemental
abundances. We further find that the occurrences of iron-rich metal-poor
stars in the Milky Way's field and globular cluster populations are
statistically indistinguishable. This observation implies that the time
lag between the formation of a globular cluster's first and second stellar
generations was longer than the thermonuclear supernova delay time.
It is also consistent with explanations for globular cluster multiple
populations that require early enrichment by thermonuclear supernovae.
§ ACKNOWLEDGMENTS
We thank Yossef Zenati for sharing his expertise on Type Ia supernovae.
Support for this work was provided by Johns Hopkins University
through a Summer Provost's Undergraduate Research Award to Zachary
Reeves and by the Carnegie Institution for Science through a Carnegie
Postdoctoral Fellowship award to Henrique Reggiani. Funding for
SDSS-III has been provided by the Alfred P. Sloan Foundation,
the Participating Institutions, the National Science Foundation,
and the U.S. Department of Energy Office of Science. The SDSS-III
web site is <http://www.sdss3.org/>. SDSS-III is managed by the
Astrophysical Research Consortium for the Participating Institutions
of the SDSS-III Collaboration including the University of Arizona, the
Brazilian Participation Group, Brookhaven National Laboratory, Carnegie
Mellon University, University of Florida, the French Participation
Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre
Dame/JINA Participation Group, Johns Hopkins University, Lawrence
Berkeley National Laboratory, Max Planck Institute for Astrophysics,
Max Planck Institute for Extraterrestrial Physics, New Mexico State
University, New York University, Ohio State University, Pennsylvania
State University, University of Portsmouth, Princeton University, the
Spanish Participation Group, University of Tokyo, University of Utah,
Vanderbilt University, University of Virginia, University of Washington,
and Yale University. Funding for the Sloan Digital Sky Survey IV has
been provided by the Alfred P. Sloan Foundation, the U.S. Department
of Energy Office of Science, and the Participating Institutions.
SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
<www.sdss.org>. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution
for Science, Carnegie Mellon University, Center for Astrophysics |
Harvard & Smithsonian, the Chilean Participation Group, the French
Participation Group, Instituto de Astrofísica de Canarias, The Johns
Hopkins University, Kavli Institute for the Physics and Mathematics of
the Universe (IPMU) / University of Tokyo, the Korean Participation
Group, Lawrence Berkeley National Laboratory, Leibniz Institut für
Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching),
Max-Planck-Institut für Extraterrestrische Physik (MPE), National
Astronomical Observatories of China, New Mexico State University, New
York University, University of Notre Dame, Observatário Nacional /
MCTI, The Ohio State University, Pennsylvania State University, Shanghai
Astronomical Observatory, United Kingdom Participation Group, Universidad
Nacional Autónoma de México, University of Arizona, University
of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington,
University of Wisconsin, Vanderbilt University, and Yale University.
This work has made use of data from the European Space Agency (ESA)
mission Gaia (<https://www.cosmos.esa.int/gaia>), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
<https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for
the DPAC has been provided by national institutions, in particular the
institutions participating in the Gaia Multilateral Agreement.
This publication makes use of data products from the Two Micron All
Sky Survey, which is a joint project of the University of Massachusetts
and the Infrared Processing and Analysis Center/California Institute of
Technology, funded by the National Aeronautics and Space Administration
and the National Science Foundation. This research has made use of the
SIMBAD database, operated at CDS, Strasbourg, France <cit.>.
This research has made use of the VizieR catalog access tool, CDS,
Strasbourg, France. The original description of the VizieR service
was published in <cit.>. This research has made use of NASA's
Astrophysics Data System Bibliographic Services.
CDS, CTIO:2MASS, Du Pont (APOGEE), FLWO:2MASS, Gaia, Sloan
(APOGEE)
<cit.>,
<cit.>
<cit.>,
<cit.>,
<cit.>,
<cit.>
SDSS DR17 APOGEE data quality and targeting information are stored as
bitmasks.[<https://www.sdss.org/dr17/irspec/apogee-bitmasks>]
Since unreliable photospheric stellar parameters will lead to unreliable
elemental abundances, we exclude from our analysis stars with the bits
, , ,
, , ,
, , ,
, , ,
, , ,
, , or set
in the column in the table .
These flags correspond to binary digits 0, 1, 2, 3, 7, 8, 9, 10,
11, 16, 17, 18, 19, 23, 24, 25, 26, and 27. Note that though the
bit corresponding to should be set if any of the
bits , , ,
, , or are
set we choose to include all of these bits in our data quality checks.
Likewise, though the bit corresponding to should
be set if any of the bits , ,
, , ,
or are set we choose to include all of these bits in
our data quality checks. We remove duplicate observations from our
analysis sample by rejecting objects with binary digit 4 set in the
column in the table .
We exclude from our analysis any elemental abundances that
are indicated as suspect. We set to all
elemental abundances with the final value -9999 or with
any of the bits , ,
, , ,
, , ,
, , ,
or set in a column in the
table . These flags correspond to binary digits 0,
1, 2, 3, 4, 6, 8, 9, 10, 12, 14, and 16. We ultimately use the following
query in the CasJobs portal to generate our analysis sample.
aasjournal
|
http://arxiv.org/abs/2307.07565v1 | 20230714181236 | Soft resummation in processes with heavy quark: bridging the gap from 4-flavor to 5-flavor scheme | [
"Andrea Ghira"
] | hep-ph | [
"hep-ph"
] |
=6.0in =8.25in
=-0.3in =-0.20in
#1
Soft resummation in processes with heavy quark: bridging the gap from 4-flavor to 5-flavor scheme
#1
Andrea Ghira
#1
Dipartimento di Fisica, Università di Genova and INFN, Sezione di Genova,Via Dodecaneso 33, 16146, Italy
and
#1
Submitted to #1
Presented
PRESENTED AT
Andrea Ghira
Dipartimento di Fisica, Università di Genova and INFN, Sezione di Genova,Via Dodecaneso 33, 16146, Italy
In this work we present a new approach to threshold resummation in processes with heavy quarks. In particular we will focus on the differential decay rate of a color-singlet particle into a b b̅ pair and we will show how to resum in a consistent way both the logarithms of the mass and the logarithms of the heavy flavor energy fraction. Within this framework, we match the two different approaches existing in literature to threshold resummation, the main difference of which is the way in which the mass is treated (5-flavor scheme vs 4-flavor scheme).
DIS2023: XXX International Workshop on Deep-Inelastic Scattering and
Related Subjects,
Michigan State University, USA, 27-31 March 2023
< g r a p h i c s >
§ INTRODUCTION
We consider the production of a bb̅ pair from the decay of a color-singlet particle which can be a Higgs or Z boson,
plus undetected radiation:
h(q)→ b(p_1)+b̅(p_2)+X(k)
(four-momenta are indicated in brackets).
We are interested in the differential decay rate Γ/ x with respect to the dimensionless variable
x=2 p_1· q/q^2,
which coincides with the fraction of the total available energy carried away by the b quark in the centre of mass frame. In the threshold limit, which means either soft or collinear, x→ 1.
In order to perform the calculation of the decay rate Γ/ x two main approaches are mainly employed: we can refer to the first as the massive-scheme approach (or 4-flavor) and to the second as the massless one (or 5-flavor).
In the first framework, the spectrum Γ/ x is computed, up to some finite order in perturbation theory, taking into exact account the finite value of the heavy quark mass m. Collinear singularities are regularized by the heavy quark mass and therefore logm^2/q^2 appear in the perturbative coefficients: such logarithms may eventually spoil the convergence of the series. On the other hand, the kinematics of radiation is treated correctly at every perturbative order.
In the second approach, the heavy flavour mass is only used as a regulator of collinear divergences, while contributions proportional to powers of m^2/q^2 are systematically neglected. This means that we do not have control on the kinematics of the emission outside the collinear region.
This framework exploits a factorization theorem: the decay rate is written as a convolution of process independent fragmentation functions 𝒟_i and of process-dependent partonic cross-sections:
1/Γ_0Γ/ = ∑_i∫_x^1 z/z𝒞_i(x/z,, μ^2/q^2) 𝒟_i (z,μ^2,m^2) +m^2/q^2,
Here μ^2 is the factorization scale typically chosen oh the order of q^2 and Γ_0 is the decay rate at Born level. The sum runs over all the possible partons that can fragment in the heavy quark.
Logarithms of the mass can be resummed to all orders up to a given logarithmic accuracy thanks to DGLAP evolution equation,
μ^2 / μ^2𝒟_i (N,μ^2,m^2) = ∑_j γ_ij(N,(μ^2) ) 𝒟^j (N,μ^2,m^2),
𝒟_i (N,μ^2,m^2)= ℰ_ij(N,μ^2,μ_0^2) 𝒟_0(μ_0^2,m^2),
where 𝒟^j denotes the Mellin transform of the j-th fragmetation function defined as:
f(N)=∫^1_0 x x^N-1 f(x).
for some function f. γ_ij are the Mellin transform of the Altarelli Parisi splitting function, while ℰ is the so called DGLAP evolution operator.
The initial condition of the fragmentation functions 𝒟_0 at a given scale μ_0^2 is needed to completely solve the differential equation and for the b quark we can naturally set this scale such that μ_0^2≃ m^2. This means that the initial condition can be computed perturbatively <cit.>.
Merging these two different frameworks gives us a better prediction for the different regions of q^2
Γ(N,ξ)= Γ_k^(4)(N,ξ)+Γ^(5)_ℓ(N,ξ)-double counting, ξ=m^2/q^2.
with Γ^(4)(N,ξ) the Mellin transform of the massive calculation at perturbative order k and Γ^(5)_ℓ(N,ξ) is the Mellin transform of the massless decay rate at logarithmic accuracy ℓ.
Finally the “double counting" is the perturbative expansion of Γ^(5)_ℓ(N,ξ) to order k. We will restrict ourselves to the case ℓ=1 (FONLL) <cit.>.
Our main task is to generalize FONLL scheme including also the threshold resummation. In the next section we now explain the main difficulties in the merging of the two approaches.
§ LOGARITHMIC STRUCTURE AT LARGE N
Both the quantities appearing in the rhs of Eq. (<ref>) display a logarithmically divergent behaviour as N→∞ due to the presence in the coefficients of the physical spectrum Γ/ x of distributions:
d_k(x)=[log^k-1(1-x)/1-x]_+
which diverge at the threshold x→ 1.
The limit x→ 1 is mapped into N→∞ by the Mellin transform.
These logarithmic contributions can be resummed to all orders up to a given logarithmic accuracy but the merging of the two resummed formula is far from trivial
due to the different structures at large N.
Specifically, in the large N limit Γ^(4)_k(N,ξ) is a polynomial of degree n in log N whose coefficients c^(4)_n(ξ) carry the full ξ dependence:
Γ^(4)_k(N,ξ)=∑^k_n=0 c^(4)_n(ξ) (/π)^n log^nN +𝒪(1/N)
On the other hand expanding Γ^(5)_ℓ(N,ξ) at ^k we obtain a polynomial of degree 2n in log N, whose coefficients depend only on the log of the mass:
Γ^(5)_k(N,ξ)=∑^k_n=0 c^(5)_n(ξ) (/π)^n log^2nN +𝒪(1/N)
At expanding the threshold-resummed formulas in <cit.> we find:
Γ^(5)_k=1(N,ξ)= 1+/π(-1/2log^2N̅ +logξlogN̅+7/4logN̅-3/4logξ)+𝒪(N^0)
Γ_k=1^(4)(N,ξ)= 1+/π(1/2log^2 ξ+2logN̅logξ+2logN̅-1/2logξ)+ 𝒪(N^0,ξ^0)
with N̅= N e^γ_E, γ_E the Euler-Mascheroni constant.
Equation (<ref>) presents two main problems: the first, as already outline, is the fact that the five flavour formalism contains double log of N whereas the four flavor do not. This is a consequence of the fact that soft and small mass limit does not commute <cit.>.
Another important problem that cause a mismatch between the two formulas is that also the mass logs have different structures. At within the massless framework only at most single logs of the mass are present, whereas in Eq (<ref>) also double logs of the mass appear <cit.>.
Due to the non-commutativity of the limits it is impossible to define a matching scheme like FONLL: the main problem is that we do not know how to identify an all order double counting term.
In the following, we will present a solution to this problem, valid to next-to-leading log accuracy.
§ 4 VS 5 FLAVOR-SCHEME
The explicit calculation of Γ^(4,res)_ℓ_1(N,ξ), where ℓ_1 denotes the logarithmic accuracy of the threshold resummation, was performed in Ref. <cit.>. In particular for the case ℓ_1=0:
Γ^(4,res)_ℓ_1=0(N,ξ)=(1+/π𝒦^(1)(ξ,)) e^-2∫^1_1/N̅ z/z(z^2 μ_0^2) (ξ),
with 𝒦^(1)(ξ,) a process-dependent factor that exhibits the double mass log in the massless limit, and , the so-called first order massive soft anomalous dimension:
(ξ)=(1+β^2/2βlog1+β/1-β-1), β=√(1-4ξ).
On the other hand in the five flavor approach the complete calculation was performed in <cit.> and in <cit.> for ℓ_2=1, where ℓ_2 denotes the logarithmic accuracy of the threshold resummation in the 5 flavor scheme: it was shown that the resummed decay rate in Mellin space in can be seen as the product of two independent jet functions:
Γ_ℓ=1, ℓ_2=1^(5,res)(N,ξ) =
(1+ (μ^2) /π𝒞_0^(1)) (1+ (μ_0^2) /π𝒟_0^(1)) ℰ̃(N,μ_0^2,μ^2,(^2))
exp[J(N,μ^2/q^2,(μ^2),μ_0^2/m^2,(μ_0^2))+J̅(N,μ^2/q^2,(μ^2)) ].
The factor exp(J) in Eq. (<ref>) describes soft radiation emitted collinearly to the tagged b-quark and has the following form:
J=D_0 + E+Δ,
where
E=-∫^μ^2_μ_0^2 k^2/k^2{A((k^2)) logN̅ + 1/2B((k^2)) },
is the logarithmically enhanced contribution in the large N limit to the DGLAP evolution kernel and
Δ= ∫_1/N̅^1 z/z∫^μ^2_z^2q^2 k^2/k^2 A((k^2)),
D_0= -∫_1/N̅^1 z/z{∫^μ_0^2_z^2 m^2 k^2/k^2 A((k^2))+H((z^2 m^2))}.
On the other hand J̅ describe, collinear radiation with respect to the anti-quark.
J̅= -∫_1/N̅^1 z/z{∫^zq^2_z^2q^2 k^2/k^2 A((k^2))+1/2B((zq^2))}.
The remaining terms in Eq. (<ref>) are costant or vanish by construction in the large N limit.
For sake of simplicity in the following we will perform a fixed coupling analysis.
We see from Eqs. (<ref>) and (<ref>) that the jet function J does not contain any double threshold log of N (see Fig. <ref>). Therefore the double logs of N that appears in Eq. (<ref>) has to be addressed only to the recoiling jet function J̅.
We modify it in such a way that when 1/N>ξ we recover <cit.> and in the opposite case we recover the massive expression <cit.> in the small mass limit.
This is achieved including finite mass effects in the computation of J̅ which means that the recoil jet function has to be computed in the quasi-collinear limit.
We outline the fact that in the 5-flavor scheme the jet function J is already computed in the quasi-collinear limit (b mass effects are taken into account by the initial condition D_0) but the b̅ is assumed to be massless, therefore double logs of N appear at .
We can visualize the meaning of this claim in the Lund plane in Fig. (<ref>)
By definition in the quasi collinear limit the ratio ξ is kept of the same order of the emission angle off the b̅ quark θ̅^2:
J̅(N,ξ) =-∫_0^1 z̅∫_0^q^2 k_t^2/k_t^2+z̅^2 m^2^CMW(k_t^2)/2π P_𝒬g (z̅,k_t^2)Θ(1-θ̅^2) Θ( z̅(θ̅^2 +ξ) -1/N̅).
where P_𝒬g denotes the massive splitting function:
P_𝒬g(z̅,k_t^2)= (1+(1-z̅)^2/z̅-2 z̅(1-z̅)m^2/k_t^2+z̅^2 m^2),
and z̅ the fraction of energy taken away by the gluon from the heavy quark.
Solving the integral at fixed coupling we find:
if 1/N̅> ξ, J̅(N,ξ) =/π(-1/2log^2N̅+3/4logN̅),
if 1/N̅<ξ, J̅(N,ξ) =/π(1/2log^2ξ+logN̅logξ+logN̅+1/4logξ).
We explicitly see that taking into account mass effects in J̅ we obtain two different regimes: when 1/N̅>ξ we get the double log of N as expected. On the other hand when 1/N̅<ξ we recover the double mass log in Eq (<ref>), meaning that the mass in the log squared is the one of the b̅.
§ FINAL RESUMMED EXPRESSION
Once we have included the running coupling corrections we are able to obtain an all order resummed differential decay rate that interpolates consistently between the 5-flavor resummed expression and the 4-flavor one.
1/Γ_0Γ/ x= ∫_c-i ∞^c+ i ∞ N/2 π i x^-NΓ^(1)(N,ξ), if 1-x >√(ξ),
Γ^(2)(N,ξ), if ξ<1-x <√(ξ),
Γ^(3)(N,ξ), if 1-x <ξ,
with
Γ^(1)(N,ξ) =Γ_ℓ=1, ℓ_2=1^(5,res,sub)exp[J^(1)+J̅^(1)],
Γ^(2)(N,ξ) = Γ^(match)exp[J^(2)+J̅^(2)],
Γ^(3)(N,ξ) = Γ^(4,res,sub)_ℓ_1=0exp[J^(2)+J̅^(3)].
Γ_ℓ=1, ℓ_2=1^(5,res,sub),Γ^(4,res,sub)_ℓ_1=0, are the subtracted version of the 5 and 4 flavour resummation:
the subtracted 5-flavour result is defined starting from Eq. (<ref>):
Γ_ℓ=1, ℓ_2=1^(5,res-sub)(N,ξ) =
(1+ (μ^2) /π𝒞_0^(1)) (1+ (μ_0^2) /π𝒟_0^(1)) ℰ̃(N,μ_0^2,μ^2,(μ^2)),
By construction Γ^(1) coincides with the 5-flavour result of <cit.>.
Similarly, Γ^(4,res,sub)_ℓ_1=0 is built from Eq .(<ref>): it is the resummed 4-flavor calculation subtracted by all the logs of N and ξ which are not power suppressed since they are already taken into account in the jet functions:
Γ^(4,res-sub)_ℓ_1=0(N,ξ) = (1+ (μ_0^2)/π𝒦^sub_1(ξ)) exp[-2 (β) ∫^1_1/N̅ z/z(z^2 μ_0^2)/π].
Γ^(3) coincide with the four flavor calculation in <cit.> with the difference that also mass logs have been resummed.
Finally Γ^(match) is a matching function that interpolates between the two subtracted expressions.
The expression of the jet functions J^(i) and J̅^(i) are derived including running coupling corrections in the decoupling scheme.
§ CONCLUSIONS
We have developed a resummed expression that effectively bridges the gap between the 5-flavor scheme and the 4-flavor scheme. Specifically, when 1/N̅>ξ, our approach aligns with the results obtained in <cit.>, whereas when 1/N̅<ξ, we reproduce the findings of <cit.>. This implies that the double threshold logarithms present in the massless scheme are inherently linked to the double logarithms of the mass observed in the massive framework.
Our derivation heavily relies on the NLL approximation, which allows for the separation of the resummed expression into the computation of two distinct jet functions. An interesting avenue for future research would involve extending this framework to NLL accuracy, which would require accounting for gluon interference between the hard particles.
It is important to note that these results are currently being prepared for publication and are the outcome of collaborative work with S. Marzani and G. Ridolfi <cit.>.
We thank Simone Marzani and Giovanni Ridolfi for the aid in the drafting of this proceeding.
We thank Simone Caletti, Matteo Cardi, Samuele Grossi for useful discussions on this topic.
unsrt
|
http://arxiv.org/abs/2307.05300v2 | 20230711144519 | Unleashing Cognitive Synergy in Large Language Models: A Task-Solving Agent through Multi-Persona Self-Collaboration | [
"Zhenhailong Wang",
"Shaoguang Mao",
"Wenshan Wu",
"Tao Ge",
"Furu Wei",
"Heng Ji"
] | cs.AI | [
"cs.AI",
"cs.CL"
] |
Signal-background separation and energy reconstruction of gamma rays using pattern spectra and convolutional neural networks for the Small-Sized Telescopes of the Cherenkov Telescope Array
[
August 12, 2023
============================================================================================================================================================================================
Human intelligence thrives on the concept of cognitive synergy, where collaboration and information integration among different cognitive processes yield superior outcomes compared to individual cognitive processes in isolation. Although Large Language Models (LLMs) have demonstrated promising performance as general task-solving agents, they still struggle with tasks that require intensive domain knowledge and complex reasoning.
In this work, we propose (), which transforms a single LLM into a cognitive synergist by engaging in multi-turn self-collaboration with multiple personas. A cognitive synergist refers to an intelligent agent that collaborates with multiple minds, combining their individual strengths and knowledge, to enhance problem-solving and overall performance in complex tasks.
By dynamically identifying and simulating different personas based on task inputs, unleashes the potential of cognitive synergy in LLMs. We have discovered that assigning multiple, fine-grained personas in LLMs elicits better problem-solving abilities compared to using a single or fixed number of personas.
We evaluate on three challenging tasks: Trivia Creative Writing, Codenames Collaborative, and Logic Grid Puzzle, encompassing both knowledge-intensive and reasoning-intensive types. Unlike previous works, such as Chain-of-Thought, that solely enhance the reasoning abilities in LLMs, effectively elicits internal knowledge acquisition abilities, reduces hallucination, and maintains strong reasoning capabilities.
Code, data, and prompts can be found at: <https://github.com/MikeWangWZHL/Solo-Performance-Prompting.git>.
§ INTRODUCTION
Although large language models (LLMs) have demonstrated impressive performance as general task-solving agents, they still encounter challenges <cit.> in various knowledge-intensive and reasoning-intensive tasks due to hallucination <cit.> and a lack of slow-thinking <cit.> capabilities.
Unlike humans, who can leverage the power of collaboration and information integration among different cognitive processes and individuals (referred to as cognitive synergy <cit.>), current LLMs are akin to "jack-of-all-trades" with a vast mixture of knowledge and characteristics. Recent advancements, such as Chain-of-Thought (CoT) prompting <cit.> and Self-refinement <cit.>, have successfully enhanced the reasoning abilities of LLMs by simulating slow-thinking through the generation of intermediate steps or iterative revision. However, hallucination and factual errors in internal knowledge acquisition continue to pose major challenges in state-of-the-art LLMs.
A cognitive synergist denotes an intelligent agent that works in conjunction with several minds, merging their unique abilities and expertise to improve problem-solving and overall efficacy in intricate tasks.
In this work, we aim to develop a cognitive synergist based on a single LLM that can "split into" multiple personas and engage in multi-persona self-collaboration to address both knowledge-intensive and reasoning-intensive tasks. The underlying biological intuition stems from the significance of pretend play and role-playing <cit.> in a child's cognitive development. According to Piaget's developmental theory <cit.>, engaging in pretend play and taking on different roles allows children to cultivate essential skills such as problem-solving, critical thinking, empathy, and cooperation.
The main inspiration for this work originates from recent findings <cit.> suggesting that assigning personas to an LLM can elicit specific behaviors. For instance, <cit.> demonstrates that when conditioned on a task-specific expert identity, an LLM can generate superior answers compared to having no assigned persona. Another closely related line of work <cit.> hints at the possibility of constructing an AI society with multiple LLM agents collaborating in different roles. However, some lingering limitations of these previous works include: (1) personas are typically fixed or task-specific, necessitating human supervision; (2) such collaboration often requires multiple individual LLM instances, resulting in a doubling or tripling of inference costs.
To unleash the potential of cognitive synergy in LLMs, we propose (), which prompts a single LLM to identify, simulate, and collaborate with multiple personas to solve challenging tasks. Figure <ref> provides a high-level overview of . Here, a persona can represent either a domain expert, such as a movie enthusiast, or a target audience, such as a ten-year-old child.
Through the dynamic identification of various personas, we empower a single LLM to acquire diverse domain knowledge accurately without additional retrieval systems. By facilitating multi-turn self-collaboration, we enable self-revision and self-feedback from various perspectives without requiring additional agents.
In real-world scenarios, particularly in creative industries, there is often a need to incorporate diverse information from different domains. Figure <ref> presents a concrete example of how operates on a challenging task that necessitates creative integration of information from various domains, such as the Legend of Zelda game, Harry Potter movies, and Jay Chou's albums. Standard prompting fails to generate satisfactory output due to missing essential information and factual errors. In contrast, correctly provides all the necessary information by automatically identifying participants with special personas, such as Harry Potter Fan and Jay Chou Fan. A leader persona, AI Assistant, then initiates a multi-turn dialogue with all participants, where it iteratively writes drafts of the story, solicits feedback, and revises. Once all participants provide positive feedback, the collaboration concludes, and a final answer is provided.
To summarize, the key contributions of this paper are as follows:
* We present (), a novel approach that leverages a single LLM as a cognitive synergist to solve tasks by dynamically identifying personas and engaging in multi-turn self-collaboration.
* We evaluate on three challenging tasks, Trivia Creative Writing, Codenames Collaborative and Logic Grid Puzzle, spanning both knowledge- and reasoning-intensive domains. significantly enhances both knowledge acquisition and reasoning abilities in LLMs, without the need for external resources.
* We conduct an in-depth analysis of the impact of identified personas and provide insights into why dynamic, fine-grained personas are necessary, as opposed to fixed, coarse-grained personas.
§
§.§ Task-Solving Procedure
To unleash the power of synergizing different personas to tackle complex problems within a single LLM, we propose () which instructs a model to perform the following the procedure for solving general tasks: (1) Persona Identification: Identify multiple participants with special personas (including a leader persona: AI Assistant) that are essential for solving the particular task. (2) Beginning Remarks: Each of the participants delivers a beginning remarks providing suggestions or information on how to approach the task based on their own expertise. (3) Multi-Persona Iterative Collaboration: The leader persona, AI Assistant, proposes initial solutions, consults the other participants for feedback, and revise the answer iteratively. Figure<ref> shows a walking example of during inference. Next, we formally describe the procedure in detail.
Given an input sequence x and a model ℳ, let a prompt (including demonstration examples) prepended to the input to be p and the final output to be y. Denote an intermediate generation before generating the final y as z. Under this formulation, Standard Prompting and Chain-of-Thought (CoT) Prompting can be described as:
Standard Prompting: y = ℳ(x)
CoT Prompting: y = ℳ(p_cot‖ x ‖{z_1,z_2,...,z_n})
where p_cot is the CoT prompt, e.g., and {z_1,z_2...,z_n} are the intermediate steps. In contrast, our proposed can be described as follows:
: y = ℳ(p_spp‖ x ‖ z_p ‖{z^1_b, z^2_b,...,z^m_b}‖{z^0_s, z^1_f,...,z^m_f}_j=1..n)
where the prompt (p_spp) includes a high-level instruction and two carefully crafted demonstration examples[The tasks we use in the demonstration examples do not overlap with the evaluation tasks.] that showcase the expected task-solving procedure of . We describe the design details of the prompt in <ref>.
The corresponding intermediate generations (z) of are detailed below.
Persona Identification (z_p). Given an input task, first generates a list of participants with different personas that can potentially contribute to the task solving. The personas can be either domain experts or targeted audiences whose feedback is important. For example in Figure <ref>, the model identified a Jay Chou Fan persona for helping retrieving the knowledge of "the last song in the second album by Jay Chou".
And for some tasks involving special audiences, e.g., "Explain quantum computing to a ten-year-old kid", including a ten-year-old kid as a participant can provide valuable feedback from the audience's perspective.
We let the language model identify the personas dynamically instead of manually defining them. Given only two demonstration examples, we observe that a state-of-the-art large language model, e.g., GPT-4 <cit.>, can identify accurate and meaningful personas for diverse tasks. We denote this part of intermediate generation as z_p.
Beginning Remarks (z^i_b).
Among the identified participants, "AI Assistant (you)" is treated as a leader persona that initiates the collaboration and generates initial solutions. Before generating the initial answer, each of the personas gives a beginning remark on how to approach the task from their own perspectives. For the example in Figure <ref>, the Jay Chou Fan gives a beginning remark pointing out that the last song in Jay Chou's second album is "An Jing" ("Silence").
We find that this effectively improves the quality of the initial solution generated by the AI Assistant.
We use i=0 to denote the "AI Assistant" persona, and i>1 for other dynamically identified personas.
Thus the beginning remarks can be denoted as {z^1_b, z^2_b,...,z^m_b} where m is the number of personas excluding the "AI Assistant".
Multi-Persona Iterative Collaboration (z^0_s, z^i_f).
Based on the beginning remarks, the AI Assistant persona generates an initial solution denoted as z^0_s, then it consults each of the other participants for feedback {z^i_f}. For example in Figure <ref>, the Jay Chou Fan persona checks whether the song "An Jing" ("Silence") is nicely included in the story. The participants are also encouraged to critique the current generation and give revision suggestions.
This process can be repeated for multiple times until every participant is satisfied with the current solution.
We denote the intermediate generations of the multi-turn dialogue as {z^0_s, z^1_f,...,z^m_f}_j=1...n where n is the number of iterations before reaching the final answer. The collaboration is marked to be complete by "Finish collaboration!" And then the final solution is generated afterwards.
Based on only a single large language model, enables multi-persona self-collaboration which effectively elicits domain knowledge and reduces hallucination. Meanwhile, the iterative procedure inherits the benefit of CoT prompting for eliciting reasoning ability. The main advantage over CoT is that at each step we can receive feedback from diverse perspectives due to the dynamically assigned personas. A comprehensive comparison with previous prompting methods can be found in Table <ref>.
§.§ Prompt Design
To prompt an LLM to behave as a cognitive synergist that follows the expected task-solving procedure as mentioned in <ref>, we carefully designed the structure of the prompt as follows. The full prompt can be found in Appendix <ref>.[We use the same prompt for any arbitrary tasks.]
System Principle. The first part of the prompt contains a high-level instruction:
Demonstration Examples. Then, we include two manually crafted demonstration examples to showcase the expected task-solving behavior. The first example describes a Game of 24 task, where we only include two personas: an AI Assistant and a Math Expert. This task aims to provide an example of a reasoning-intensive task, where the AI Assistant needs to propose multiple proposals, and the other participants need to give fine-grained feedback on where the current solution went wrong and how to improve it. The second example describes a poem-writing task with diverse requirements, including lexical constraints, semantic constraints, and audience awareness. This task aims to provide an example of a knowledge-intensive task, where diverse personas are required to collaboratively solve the task. This example also demonstrates a case where it is important to assign a dedicated persona to the audience, e.g., a ten-year-old child.
Task Prefix. The last part of the prompt reminds the model to followed by task-specific format instructions and inputs.
§ EXPERIMENTS
We explore the effectiveness of for versatile task-solving by examining three challenging tasks that encompass both knowledge-intensive and reasoning-intensive domains. We introduce the task, which requires the model to internally acquire and integrate diverse information from various fields. We observe that even the most advanced LLMs, such as GPT-4 <cit.>, frequently exhibit hallucination and factuality errors in the task.
We also propose the task, an extension of the Codenames task from the BigBench <cit.> that features a two-role collaboration setup.
demands creative reasoning across a broad range of related knowledge and challenges the model's theory-of-mind skills. Lastly, we include a challenging pure-reasoning task, Logic Grid Puzzle, from the BigBench <cit.> which necessitates complex multi-step reasoning.
Methods. We primarily compare our approach with Standard Prompting and Chain-of-Thought (CoT) prompting methods (outlined in <ref>).
In CoT, a similar prompt design to <cit.> is employed, where the model is prompted to generate a plan or a series of steps before producing the final output.
We examine two variants of , and . Inspired by <cit.> that suggested a detailed expert description may help elicit distinguished abilities, we include , which involves generating profiles for each persona during the Persona Identification phase.
Full prompts for the methods can be found in Appendix <ref>.
Inference Configurations. All experiments are conducted using the GPT-4-32k API[The specific model version we employ is "2023-3-15-preview". There are some rare cases when a generation triggers the content filter of the API. We exclude those instances from our results.]. The temperature is set to 1.0 and top_p to 1.0 for all generations to maximize reproducibility. To evaluate the potential impact of initial persona assignment through a system message, we consider two inference settings: with or without the default system message, . We observe divergent patterns across various tasks and methods regarding the use of the system message, and report
the average metric scores across both inference settings in the Tables <ref>, <ref>, and <ref>. Full results for each setting can be found in Appendix <ref>.
§.§ : A Knowledge-Intensive Task
Task Description.
The task aims to push the limits of large language models in retrieving internal self-compressed knowledge and incorporating diverse information.
As a scalable extension of the example task shown in Figure <ref>, asks a model to write a coherent story around a topic while incorporating answers to N trivia questions. We consider two evaluation settings, N=5 and N=10, where a larger N involves more trivia questions and thus requires the model to elicit more diverse domain knowledge. We built a benchmark with 100 instances for each N, covering a total of 1000 trivia questions[To select difficult question instances that can pose challenges to GPT-4, we use a smaller open-source LLM, fastchat_t5_3b <cit.>, to obtain preliminary performance on the validation set, and then choose the failure cases as our question selection.] extracted from the TriviaQA <cit.> dataset.
The topic list is automatically generated by prompting GPT-4 to provide 100 nouns from pop culture that are PG or PG-13 rated[The full prompt for generating the topic list can be found in Figure <ref>. We performed further human curation to avoid potential harmful content.]. Figure <ref> shows an example instance in .
Evaluation Metrics. Instead of focusing on evaluating the coherence of the generation, which can be highly subjective, we employ an automatic metric to detect factual errors and quantify a model’s ability to incorporate diverse domain knowledge.
As shown in Figure <ref>, we perform string matching with the ground truth target answers for each question on the output generation. The target answers are provided by the TriviaQA dataset, and each question can have a list of answer aliases. A match to any of the answer aliases of a question is considered as a correct mention. The metric score is computed as follows.
Metric Score = # correct answer mentions/# trivia questions
Results.
Table <ref> shows the results of the four methods on the task. We have the following main observations: (1) Chain-of-Thought (CoT) does not outperform Standard prompting. This indicates that CoT may not be effective in eliciting an LLM's knowledge abilities. As shown in Figure <ref>, we find that although CoT generates reasonable plans for solving the task, the final generation still suffers from factual errors and hallucination. (2) Our proposed and significantly outperform both Standard and CoT. The improvement is more noticeable in the N=10 setting compared with N=5 (10% vs. 7%). This indicates that when the task requires incorporating knowledge from a large number of different domains, can be particularly helpful by identifying different personas for eliciting different expertise.
§.§ : A Knowledge+Reasoning Task
Task Description. is a challenging task that requires the model to reason over a wide range of knowledge while considering collaboration with another agent. We aim to use this task to investigate the effectiveness of on collaborative tasks that require knowledge, reasoning, and theory of mind abilities. involves two player roles: a Spymaster and a Guesser. The Spymaster is given a set of target words along with some other distractor words. The Guesser does not have the information about which words are the target words. The goal of the Spymaster is to come up with a single hint word that is closely related to the target words while being remotely related to the distractor words. The goal of the Guesser is to find the target subset of words from the entire word set based on the hint given by the Spymaster.
Finding a good hint word or guessing the target subset of words both require a strong capability of selecting, composing, and reasoning over various knowledge related to a certain word. For example, "director, popcorn" can be linked by the word "movie" because movies are created by a director and people often eat popcorn when watching movies in a cinema.
We use the same LLM (GPT-4 <cit.>) to play the Spymaster and the Guesser sequentially. That is, each game instance involves one inference as the Spymaster and then another inference as the Guesser, where the Guesser's input is dependent on the Spymaster's output. We construct a dataset with 50 instances based on the data from the Codenames task in the BigBench <cit.>. Figure <ref> shows an example of the task.
Evaluation Metrics. As illustrated in Figure <ref>, we compute the overlapping ratio between the predicted words from the Guesser and the target words given to the Spymaster as the metric. A major limitation of the original Codenames task in the BigBench dataset is that it only considers the Spymaster role and provides a ground truth answer to the hint word, which can be highly subjective and exclude many potentially good alternatives. Our task addresses this issue by making the evaluation setting self-contained, which can faithfully reflect the model's capability without the need for human annotation.
Results.
Table <ref> shows the results on the task. Similar to the task, we find that CoT does not bring positive gains compared with the Standard prompting. In contrast, brings significant improvements (~5%), which indicates the effectiveness of the proposed on collaborative tasks that require knowledge, reasoning, and theory of mind skills. Figure <ref> provides a qualitative example illustrating that generates detailed and interpretable intermediate dialogues, contributing to superior performance when compared with CoT.
§.§ : A Reasoning-Intensive Task
Task Description.
We leverage the task from the Bigbench <cit.> dataset, which contains 200 instances. Each instance describes a logic puzzle typically involving 2 - 5 houses, where each house is inhabited by a person with certain characteristics, e.g., having a vase of tulips or being a pianist. Given some partial clues, such as "the flutist lives in the second house," the goal is to answer the final question that queries the house number of the person with a specific characteristic. To obtain the final answer, the model is required to perform multi-step reasoning and select the most relevant clue to use at each step. Challenging instances may involve considering multiple clues simultaneously for deducing the next useful piece of information. Figure <ref> shows an example input and output of the task.
Evaluation Metrics. We compute the accuracy of the predicted house numbers by comparing them with the ground truth targets provided by the dataset.
Results.
Table <ref> presents the results on . In contrast to the previous two tasks, as expected, we find that CoT brings significant improvements compared to Standard prompting, verifying the observation from previous work that CoT elicits better reasoning abilities on reasoning-intensive tasks. Furthermore, we discover that also outperforms CoT on this task, indicating competitive reasoning capabilities on pure-reasoning tasks. This result demonstrates that the increased number of personas does not deteriorate the models' reasoning abilities.
§ ANALYSIS
effectively improves internal knowledge acquisition and reasoning in LLMs. As demonstrated by the results in <ref>, () not only brings significant improvements to knowledge-intensive tasks such as and without relying on external knowledge bases, but also achieves strong performance on reasoning-intensive tasks like . This indicates the potential of using LLM-based cognitive synergists as a default paradigm for general task solving by .
LLMs can effectively identify useful personas without additional fine-tuning. We visualize the personas[The visualization excludes the default persona, AI Assistant.] automatically identified by using a word cloud for each task in Figure <ref>, where a larger font indicates a higher frequency.
The identified personas are closely correlated with the particular task; for example, on , even though "logic puzzle" is not mentioned in the input, the LLM frequently assigns the persona "Logic Puzzle Expert" to a participant. It indicates that current LLMs are inherently capable of identifying useful expert personas for diverse tasks. We also find that on knowledge-intensive tasks, such as , identifies more diverse and specific personas, while on reasoning-intensive tasks, such as , the personas are more homogeneous.
Moreover, the fact that does not outperform in two of the three tasks suggests that a fine-grained name of the persona without a detailed description may already be sufficient for eliciting certain domain knowledge.
Dynamic personas vs. fixed personas. To further investigate the importance of dynamically identifying personas (synergizing dynamic cognitive processes) for each task instance instead of fixing a general persona (synergizing fixed cognitive processes), an ablated variant of , , is introduced. For , we modify the prompt of to force the personas to be fixed as an "AI Assistant" and an "Expert", while keeping all the information in the demonstration examples intact. The full prompt of can be found in Figure <ref>.
Figure <ref> shows the comparison between and . We have the following main insights: (1) consistently outperforms across all tasks, suggesting that dynamic, fine-grained personas are more effective than fixed, general personas. Figure <ref> shows qualitative examples from , where fine-grained personas such as "Film Expert" and "Sports Enthusiast" correctly find the answers, while the fixed persona "Expert" fails.
(2) suffers from a unique problem we refer to as early-termination, where the LLM stops the generation after the Expert persona gives the beginning remarks. The model behaves as if it were waiting for input from a user instead of simulating the response by itself. An example of the early-termination problem can be found in Figure <ref>. The problem is particularly severe on certain tasks, e.g., , resulting in unexpectedly low performance. The problem can be largely alleviated by removing the system message, , but cannot be entirely eliminated. Table <ref> shows the number of early-termination instances for each task and method. In contrast, we did not observe early-termination on , , Standard, or CoT prompting.
§ RELATED WORK
LLMs as role-playing agents.
Recent work <cit.> has shown that assigning personas or roles to LLMs can significantly influence their generation behavior. <cit.> demonstrated that assigning specific personas, such as the boxer Muhammad Ali, to an LLM can increase the toxicity of its generated content. Inspired by how humans form societies to effectively collaborate on complex tasks, recent work <cit.> has explored the possibility of creating an AI society where different model agents with distinct personas or occupations collaborate with each other. Generative Agents <cit.> prototyped a small AI neighborhood where generative models can simulate believable human behavior and collaborate on performing complex tasks, such as throwing a Valentine's Day party. However, current studies on enabling LLMs as role-playing agents have several limitations. Previous work on persona assignment is either limited to a single persona per agent <cit.> or a fixed number of personas <cit.> defined by humans. Additionally, current research on multi-agent collaboration often requires multiple LLM instances, which significantly increases the inference cost.
In this work, we investigate the possibility of using a single LLM to simulate multi-persona collaboration. Instead of fixing the personas, we allow the LLM to dynamically identify useful personas for each task instance. Our approach, , effectively outperforms the fixed persona variant (as shown in <ref>) without additional computational overhead.
Improving reasoning and knowledge acquisition abilities in LLMs.
Although LLMs have demonstrated impressive performance in a wide range of natural language understanding and generation tasks, they still face challenges when dealing with complex knowledge-intensive tasks due to hallucination <cit.> and reasoning-intensive tasks due to the lack of human-like slow thinking <cit.>.
Representative works aimed at enhancing LLMs' reasoning abilities include Chain-of-Thought (CoT) and Self-Refinement. CoT prompting <cit.> and its variants <cit.> encourage LLMs to solve tasks step by step instead of directly generating the final answer. By generating intermediate steps, the model effectively "slows down" its thinking process, resulting in improved reasoning ability. <cit.> recently extended the linear thought process in CoT to a tree-like structure, which demonstrated enhanced performance on complex reasoning tasks requiring trial-and-error. Self-Refinement <cit.> focuses on enabling LLMs to "talk" to themselves, provide feedback on their own generation, and iteratively revise their answers. <cit.> proposed a three-step framework in which a single LLM plays the roles of a generator, a feedback provider, and a refiner iteratively, showing consistent improvements on seven diverse tasks. <cit.> further incorporated an episodic memory for self-feedback, demonstrating promising results on decision-making and reasoning tasks. Despite their impressive improvements on reasoning-intensive tasks, CoT and Self-Refinement do not necessarily reduce hallucination or improve factuality in generated content, as shown in our results in Tables <ref> and <ref>.
On the other hand, retrieval augmented LLMs <cit.> have shown promising results in enhancing LLMs's knowledge acquisition based on external knowledge resources. However, retrieving from external sources does not improve a model's reasoning abilities, posing challenges for tasks that require both intensive knowledge and multi-step reasoning.
To elicit both internal knowledge acquisition and reasoning abilities in LLMs, we propose (), which significantly improves factuality while maintaining strong performance on pure-reasoning tasks. The key difference compared to previous prompting methods is that dynamically identifies multiple personas instead of one and simulates iterative collaboration to generate intermediate "thoughts".
§ DISCUSSION
Limitations and future work.
Although exhibits promising improvements in acquiring factually correct knowledge compared to Standard prompting, it has some limitations. For instance, even when a fine-grained persona is assigned, the answer may still be incorrect. It remains unclear to what extent assigning a persona can help enhance domain knowledge in a specific area. Dedicated diagnostic experiments and theoretical efforts are needed to quantify the impact of having a persona or not.
Furthermore, we currently adopt an identical prompt with the same two demonstration examples for any given task inputs, which may be suboptimal.
Future work investigating how to find better demonstration examples conditioned on each input could further improve the effectiveness of .
Last but not least, if given sufficient computational budget, a natural variant of could extend to a multi-agent cognitive synergist setup where a leader persona identifies several expert agents and forms a cabinet to collaboratively solve a task. The multi-agent setup allows for leveraging richer computation power, larger local memory, and more flexible human-computer interaction, which could be essential for deploying to real-world applications.
Conclusion.
In this work, we have made an initial attempt to mimic the cognitive synergy in human intelligence using a single large language model (LLM). We introduced an LLM-based cognitive synergist using , which effectively improves both internal knowledge acquisition and reasoning abilities compared to the native LLM. With , a single LLM can dynamically identify, engage, and collaborate with multiple personas to solve general tasks. To assess the performance of LLMs in terms of factuality, knowledge integration, and theory-of-mind reasoning, we have created novel and challenging tasks, namely and . Our results demonstrate superior performance compared to Standard and CoT prompting on both knowledge-intensive and reasoning-intensive tasks, indicating the promising potential of unleashing the power of cognitive synergy in LLMs with .
iclr2023_conference
§ PROMPTS
Figures <ref>, <ref> and <ref> show the full prompts for , and respectively. Figure <ref> shows the full prompts for Chain-of-Thought (CoT) prompting.
§ FULL RESULTS
Full results of the three tasks: , and can be found in Tables <ref>, <ref> and <ref>, respectively.
§ EARLY-TERMINATION WITH
Figure <ref> shows an example of the early-termination problem where the generation stops before reaching the final solution as if the models is waiting input from an external user.
Table <ref> shows the number of instances that suffer from the early-termination (defined in <ref>) with for each task. We find that removing the system message can largely reduce the problem but not be able to eliminate it.
|
http://arxiv.org/abs/2307.04830v2 | 20230710180826 | Double-Fourier engineering of Josephson energy-phase relationships applied to diodes | [
"A. Mert Bozkurt",
"Jasper Brookman",
"Valla Fatemi",
"Anton R. Akhmerov"
] | cond-mat.supr-con | [
"cond-mat.supr-con",
"cond-mat.mes-hall"
] |
Double-Fourier engineering of Josephson energy-phase relationships applied to diodes
A. Mert Bozkurt,1,2,*
Jasper Brookman,1
Valla Fatemi,3†
and Anton R. Akhmerov1
1 Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 4056, 2600 GA Delft, The Netherlands
2 QuTech, Delft University of Technology, P.O. Box 4056, Delft 2600 GA, The Netherlands
3 School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853 USA
[email protected]
†[email protected]
[email protected]
August 12, 2023
§ ABSTRACT
We present a systematic method to design arbitrary energy-phase relations using parallel arms of two series Josephson tunnel junctions each.
Our approach employs Fourier engineering in the energy-phase relation of each arm and the position of the arms in real space.
We demonstrate our method by engineering the energy-phase relation of a near-ideal superconducting diode, which we find to be robust against the imperfections in the design parameters.
Finally, we show the versatility of our approach by designing various other energy-phase relations.
§ INTRODUCTION
Josephson junction circuits allow to create many functional devices (such as SNAILs, quartons etc).
The Josephson tunnel junction is the fundamental building block of superconducting circuits <cit.>.
These junctions have enabled the development of a wide range of functional devices such as superconducting quantum interference devices (SQUIDs), superconducting low-inductance undulatory galvanometers (SLUGs) <cit.>, superconducting nonlinear asymmetric inductive elements (SNAILs) <cit.>, quantum-limited amplifiers <cit.>, and a bevy of superconducting qubits <cit.>.
One such device is a superconducting diode, which also exists in SNS junctions under magnetic field.
An example device that can be realized using Josephson junctions is a superconducting diode: a junction with unequal critical currents in different directions.
Superconducting diode effect manifests generically in inhomogeneous Josephson junctions subject to a magnetic field <cit.>.
Recently, however, there has been renewed interest in studying different physical mechanisms for the creation of superconducting diodes.
While superconducting diodes require breaking both time-reversal and inversion symmetries—otherwise the current-phase relationship (CPR) is anti-symmetric in phase—the way in which these symmetries are broken reveals information about the underlying physical systems.
To name several examples, recent studies reported superconducting diode effect in spin-orbit coupled in 2d-electron gases under external magnetic field <cit.>, superconducting thin films <cit.>, topological insulators <cit.>, finite-momentum superconductors <cit.>.
An alternative to controlling the junction CPR for creating a supercurrent diode is to combine multiple junctions in a supercurrent interferometer either consisting of multiple high transparency junctions <cit.> or arrays of Josephson tunnel junctions <cit.>.
We propose an approach to design arbitrary energy-phase relationships using Josephson junction arrays.
We propose a systematic approach to engineer arbitrary energy-phase relationships (EPRs) of a two-terminal device using parallel arrays of Josephson tunnel junctions.
We draw inspiration in the observation that circuits of conventional tunnel Josephson junctions implement a variety of Hamiltonians <cit.>, originally proposed for difficult-to-engineer microscopic structures.
We show that the EPR of a Josephson junction array can be engineered by combining Fourier engineering of the EPRs of each arm of the array, variation of the arm strengths in real space, and phase offsets created by an external magnetic field.
Our design relies on using standard fabrication techniques and is resilient against fabrication imperfections.
We promote that the schemes presented here may be useful in designing sophisticated energy-phase landscapes for decoherence-protected qubit designs <cit.>.
Our recipe consists of several steps: creation of higher Fourier components, FT, and adding zero trick.
§ THE ARBITRARY EPR ALGORITHM
The elementary unit (or building block) of our design consists of two tunnel junctions in series, which behaves as a short classical junction.
Our conceptual algorithm relies on the following realizations:
* The current-phase relation of two Josephson junctions in series matches the functional form of that of a short Josephson junction with a finite transparency. This allows single-parameter control over the Fourier components in the energy-phase relation of the arm.
* The energy-phase relation of multiple parallel junctions is a convolution of the individual energy-phase relations with the vector of junction strengths when each arm has equal phase offsets and transparency.
* Shifting the total Josephson energy of all arms by the same amount does not change the lowest Fourier components, and therefore the overall shape of the current-phase relation stays the same.
The elementary unit of the design used to generate higher harmonics of a CPR, of an arm of the Josephson junction array, consists of two Josephson tunnel junctions connected in series, with Josephson energies E_J1 and E_J2 [see Fig. <ref>(a)].
The EPR of each Josephson junction is U_i(φ_i) = - E_Jicos(φ_i), where φ_i is the phase drop across the junction. [We ignore the weak higher harmonic terms that have recently been reported in single Josephson tunnel junctions <cit.>.
Their influence can be easily incorporated and does not substantially alter the claims of our work.]
Current conservation and the additivity of phase differences yields:
E_J1sin(φ_1) = E_J2sin(φ - φ_1),
where φ = φ_1 + φ_2, with φ the total phase difference across the arm (see Figure <ref>).
Solving for φ, we obtain the CPR of an arm:
I_▸◂(φ) = E_J τ/4 Φ_0sin(φ)/√(1 - τsin^2(φ/2)),
with Φ_0 = ħ/2e the superconducting flux quantum.
The corresponding EPR is
E_▸◂(φ) = -E_J√(1 - τsin^2(φ/2)),
where E_J ≡ E_J1 + E_J2 is an overall Josephson energy of an arm and τ≡ 4 E_J1E_J2 / (E_J1 + E_J2)^2 controls the relative strength of the higher harmonics of the EPR.[We note that Eq. (<ref>) is consistent with the microscopic model for supercurrent in double barrier Josephson junctions <cit.>. For the case τ=1, Eq. (<ref>) is also consistent with the semiclassical model for CPR in a superconducting trilayer system <cit.>.]
This EPR has the same functional form as that of a short, single-channel finite transparency junction with transparency τ and gap E_J—a remarkable coincidence, for which we have no explanation.
The EPR and CPR of an arm become highly nonsinusoidal at τ≈ 1 or E_J1≈ E_J2, see Fig <ref>.
We introduce the Fourier transform of the normalized EPR of an arm:
𝒰(τ, φ) = √(1 - τsin^2(φ/2))≡∑_m=-∞^∞𝒰_m (τ) e^i mφ,
where 𝒰_m are the Fourier coefficients of 𝒰(τ,φ).
In the high transparency limit, τ≈ 1, 𝒰_m ∼ 1/m^2 for m ≲ 1/(1-τ).
We plot τ-dependence of several lowest Fourier coefficients of a single arm EPR in Fig. <ref>(c).
Connecting these elementary units in parallel and threading flux through them enables us to design the overall energy-phase relationship.
With this way to create higher order harmonics of a single arm EPR, we utilize a Josephson junction array shown in Fig. <ref>(b) to engineer arbitrary EPRs.
In addition to varying the strengths of each Josephson junction, and therefore E_J,n and τ_n of n-th arm, we utilize phase offsets by adding magnetic flux between the arms.
Magnetic flux gives rise to phase differences δφ_n between arms n and n-1.
In this way, we shift the phase offset of each arm by an amount ϕ_n=∑_n'=1^nδφ_n' with respect to a reference arm n=0.
For the rest of the discussion, we define an arm strength distribution by assigning a position to each arm, namely E_J,n≡ E_J(x_n), and correspondingly distributions of the effective transparency τ_n ≡τ(x_n) and phase offsets ϕ_n ≡ϕ(x_n).
The EPR of the Josephson junction array is
U(φ) = -∑_n=0^N-1 E_J(x_n)𝒰(τ(x_n), φ + ϕ(x_n)),
where N is the total number of arms.
This EPR is highly nonlinear in τ_n and x_n, and linear in E_J.
Our goal is to find U(φ) that approximates a target EPR, U_target(φ), by optimizing the design parameters E_J, τ and ϕ.
Because the role of τ is to introduce higher harmonics, and the role of x_n is to break time-reversal symmetry, we choose to make τ_n and x_n uniform to simplify the problem.
Specifically, we use ϕ(x_n) = 2π n/N and τ(x_n) = τ≈ 1, which makes the right hand side of Eq. (<ref>) a convolution of E_J(x_n) and 𝒰(τ, φ).
We then find an approximate solution of the optimization problem by requiring that two EPRs agree at a set of discrete points U(2 π m/N) = U_target(2 π m/N), with integer 0 ≤ m < N.
In other words, the Josephson junction strengths E_J(x_n) are obtained by Fourier transforming U_target, dividing the coefficients by the Fourier components of 𝒰(φ) and applying an inverse Fourier transform:
E_J(x_n) = -ℱ^-1{ℱ{U_target(φ_m)}/𝒰_m}_n.
We find that adding the most negative junction strength makes all E_J positive, while only minimally changing the current-phase relationship.
In general, the set of Josephson energies E_J(x_n) found by inverse discrete Fourier transform includes negative values, whereas the stable state of a single arm has a positive E_J.
We resolve this obstacle by adding the most negative E_J,min to all the Josephson energies E_J(x_n).
Because ∑_n 𝒰(ϕ - 2π n/N) has a period of 2π/N, N of its lowest Fourier components are absent, and therefore adding it to the EPR only changes it minimally, as shown in Fig. <ref>.
This concludes the design of a Josephson junction array with a target EPR.
§ OPTIMIZING THE SUPERCONDUCTING DIODE EFFICIENCY
The ideal SC diode EPR is special case of interest because it features a discontinuity, and the target metric is highly nonlinear.
We now apply our approach to design a superconducting diode.
This device has an asymmetric CPR with unequal critical currents in opposite directions.
The diode efficiency η is the degree of asymmetry of its two critical currents:
η = | I_c+ - I_c-|/I_c+ + I_c-,
where I_c± are the maximum critical currents for current flow in opposite directions.
An ideal superconducting diode with η=1 has a sawtooth-shaped EPR:
U_sawtooth(φ) = φ/2π - ⌊φ/2π⌋,
where ⌊φ⌋ is the floor function.
To optimize a superconducting diode, we apply the algorithm of the previous section with U_target = U_sawtooth, with the results shown in Fig. <ref>.
Because U_sawtooth is discontinuous, its Fourier approximation exhibits oscillatory behavior near the discontinuity, known as the Gibbs phenomenon.
This reduces the superconducting diode efficiency by allowing small side peaks of the opposite sign next to the main peak in the CPR.
To attenuate the Gibbs phenomenon, we modify the Fourier coefficients of E_J using the σ-approximation <cit.>.
In Fig. <ref>(a), we demonstrate the effect of the σ-approximation on CPR of a superconducting diode.
With increasing degree of regularization the efficiency of the superconducting diode increases and eventually peaks at η=0.92 (for N=78 arms).
We then choose a degree of regularization that maximizes the efficiency for a given number of arms and τ.
In Fig. <ref>(b), we show N dependency of the EPR and CPR of a Josephson junction array for a fixed τ=0.95.
As N increases, the main peak in the CPR gets higher and narrower, resulting in a larger efficiency.
§ GENERALIZATION OF THE ALGORITHM
We then relax the evenly spaced arms condition and perform stochastic optimization to test the robustness of our diode against imperfections that may arise from fabrication.
The discrete Fourier transform approach yields a closed form solution, it applies to any target EPR using the setup of Fig. <ref>.
On the other hand, it relies on several simplifications:
* It makes U(φ) agree with U_target(φ) at N points, instead of minimizing an error norm.
* It requires that all τ_n are equal and x_n are equidistant.
* It does not take into account the random variation of junction strengths.
To relax the first limitation we observe that as long as the error norm is quadratic in U(φ) - U_target(φ), the optimization problem stays a least squares problem (LS), implemented in the SciPy library <cit.>.
Relaxing the second and third limitations makes the problem nonlinear, but keeps it solvable using stochastic global optimization techniques.
We use LS to minimize off-current curvature.
To apply LS to the superconducting diode design, we use the error norm
E_J(x_n)min∑_i[U^''(φ_i)]^2,
which makes the negative current as constant as possible in the range φ_i ∈[ φ_min, φ_max].
To make the solution nonzero we fix E_J(x_0)=1 and solve for the Josephson junction strengths of the remaining N-1 arms.
After finding a solution to the LS problem, we add the most negative junction strength, similar to the Fourier method.
We then apply a brute force optimization to determine the phase region [ φ_min, φ_max] that yields highest η.
We implement stochastic optimization using differential evolution.
We solve the nonlinear problem by applying the SciPy's <cit.> implementation of the differential evolution method <cit.> to the problem of finding max_{x_n}, {E_Jn}η for a given N.
This procedure yields the results shown in Fig. <ref>.
Because differential evolution allows the presence of noise, we allow the junction strengths to vary by ± 2%, similar to the experimental state of the art <cit.>.
We find mean diode efficiency of η≈ 0.71 for N=5 arms, much larger than the result of the Fourier method.
We compare the three optimization methods.
In Fig. <ref>, we compare the diode efficiencies produced by the three optimization methods in perfect conditions and in presence of noise.
All three methods show improvement with increasing N.
The differential evolution method yields highest efficiencies for low N and converges the fastest, while showing only limited degradation in presence of disorder.
The superior performance of this method is expected, however the computational costs become prohibitively high for large N.
The discrete Fourier transform method is the most constrained, and therefore it performs worst, albeit the difference with LS vanishes at high N.
The LS approach is the least resilient to disorder once N becomes large due to overfitting.
§ OTHER EXAMPLE EPRS
Finally, we demonstrate the generality of our approach through other energy-phase relationship examples: A square barrier, a triangular barrier and a double well.
To demonstrate the generality of our approach, we apply it to other example EPRs: a square wave, a triangular wave, and a double well potential.
For square and triangular wave potentials, we employ the discrete Fourier transform approach.
Similar to the superconducting diode EPR case, we choose a constant τ and solve for the Josephson energy distribution.
The convergence of this method with N, shown in Fig. <ref> confirms that it allows to generate arbitrary EPRs.
Through the double well example, we show that our method is not only effective in designing overall energy-phase relationships, but also for a specific phase range.
The double well EPR example demonstrates how to apply the same device to design an EPR that is only defined within a limited phase range.
Specifically, we consider a double well potential of the form:
U_dw(φ) = φ^4 - 1/2φ^2.
By discretizing Eq. (<ref>) and eliminating equations outside the region of interest, we obtain an overdetermined set of equations, which we solve using LS and shift the Josephson energies by the most negative one when necessary.
Due to absence of sharp features in double well potential, we choose a low value of τ=0.1.
The resulting EPR of the Josephson junction array with N=4 arms, shown in Fig. <ref>, agrees with target EPR given in Eq. (<ref>) in the phase region of interest, depicted by the yellow dashed line.
§ CONCLUSION AND OUTLOOK
We proposed and investigated an approach to design arbitrary energy-phase relationships using Josephson tunnel junction arrays.
In particular, our approach allows to design a superconducting diode with a desired efficiency and the resulting design is robust against variation in device parameters.
The main building block of our approach is possibly the simplest source of a non-sinusoidal CPR: two Josephson tunnel junctions in series.
While our method does not rely on a specific arm EPR, this choice offers practical advantages.
For example, more than two junctions in series generally have a multi-valued CPR <cit.> and does not allow for a simple parametrization.
An alternative way of generating higher harmonics is a Josephson junction in series with an inductor <cit.>, however it has a non-periodic CPR, and is therefore more complicated to use.
We have focused on the DC properties of the circuit, and we envision engineering the RF characteristic as the next logical step.
For example, we expect that diode effects are correlated with odd-order RF nonlinearities, which we could explore <cit.>.
Furthermore, so far, we have ignored the role of junction capacitance E_C, which sets the plasma frequency of the superconducting junctions, and consequently the islands.
This plasma frequency limits the range of operation frequencies, therefore incorporating the dynamics of the superconducting islands into the picture would be relevant for designing quantum coherent devices.
Finally, we can extend our scheme to 2- or 3-dimensional energy-phase landscapes and include sensitivity to parametric knobs as optimization inputs for design of protected qubits <cit.>.
Hence, to make our design usable in quantum circuits, we need to have high E_C compared to the range of operation frequencies.
This step would be more relevant for designing quantum coherent devices such as qubits, parametric amplifiers.
§ ACKNOWLEDGEMENTS
We acknowledge useful discussions with Alessandro Miano, Nicholas E. Frattini, Pavel D. Kurilovich, Vladislav D. Kurilovich, and Lukas Splitthoff.
Data availability
The code used to generate the figures is available on Zenodo <cit.>.
Author contributions
A.R.A. and V.F. defined the research question.
A.R.A oversaw the project.
J.B. implemented the initial version of the optimization as a part of his bachelor project.
A.M.B. implemented the final version of the optimization and performed the numerical simulations in the manuscript.
All authors contributed to identifying the final algorithm.
A.M.B., A.R.A. and V.F. wrote the manuscript.
Funding information
This work was supported by the Netherlands Organization for Scientific Research (NWO/OCW) as part of the Frontiers of Nanoscience program, an Starting Grant 638760, a subsidy for top consortia for knowledge and innovation (TKl toeslag) and a NWO VIDI Grant (016.Vidi.189.180). AMB acknowledges NWO (HOTNANO) for the research funding.
|
http://arxiv.org/abs/2307.05266v1 | 20230711135906 | On the efficient preconditioning of the Stokes equations in tight geometries | [
"Vladislav Pimanov",
"Oleg Iliev",
"Ivan Oseledets",
"Ekaterina Muravleva"
] | math.NA | [
"math.NA",
"cs.NA"
] |
Assessing Peer Award Diversification on Reddit
Amaury Trujillo
==============================================
If the Stokes equations are properly discretized, it is well-known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa and Krylov-Uzawa algorithms.
However, in the case of complex geometries, the Schur complement matrix can become arbitrarily ill-conditioned having a significant portion of non-unit eigenvalues, which makes the established Uzawa preconditioner inefficient. In this article, we study the Schur complement formulation for the staggered finite-difference discretization of the Stokes problem in 3D CT images and synthetic 2D geometries. We numerically investigate the performance of the CG iterative method with the Uzawa and SIMPLE preconditioners and draw several conclusions. First, we show that in the case of low porosity, CG with the SIMPLE preconditioner converges faster to the discrete pressure and provides a more accurate calculation of sample permeability. Second, we show that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix, while the dependence is inverse for the Schur complement matrix preconditioned with the SIMPLE. As an explanation, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement.
Keywords: Stokes problem, tight geometry, preconditioned Krylov subspace methods
§ INTRODUCTION
The steady-state flow of a slow incompressible fluid in the domain Ω is governed by the Stokes equations:
-Δ𝐮 + ∇ p = 𝐟, in Ω
-∇·𝐮 = 0, in Ω
+ b.c. on ∂Ω,
equipped with proper boundary conditions on ∂Ω (see Section <ref> for the boundary conditions considered in this paper).
Here p denotes the fluid pressure, 𝐮 is the fluid velocity, and 𝐟 is the source.
The discretization of the BVP
(<ref>) leads to a block system of linear equations of the following form:
𝔸[ 𝐮_h; p_h ]
=
[ 𝐟_h; g_h ], 𝔸 =
[ 𝐀 𝐁^T; 𝐁 ],
where 𝐀 and 𝐁 are discrete counterparts of the negative velocity Laplacian operator and the negative divergence operator which, under proper discretization is an adjoint of the discrete pressure gradient operator 𝐁^T. The boundary conditions are incorporated in the discretization matrices.
We consider the Pressure Schur Complement formulation (according to the terminology from <cit.>) which reduces the system (<ref>) to the following equivalent equation on the discretized pressure:
S p_h = g_h^S,
where S is the Schur complement of the (0,0)-th block of the matrix 𝔸 and g_h^S is the right-hand-side in the reduced equation, which are defined as follows:
S = 𝐁𝐀^-1𝐁^T, g_h^S = 𝐁𝐀^-1𝐟_h - g_h.
Once the pressure is computed, the velocity can be recovered by solving the following equation:
𝐀𝐮_h = 𝐟_h - 𝐁^Tp_h.
If the Stokes equations (<ref>) are properly discretized such that the discrete operators preserve important properties of the continuous ones, then (up to a constant in its nullspace) the Schur complement matrix S is known to be spectrally equivalent to the identity operator acting on the discrete pressure space. For example, in the context of the finite element method, the equivalence to the pressure mass matrix takes place with the right choice of LBB-stable elements (see e.g. <cit.>), and in the context of the finite difference discretization, this equivalence can be observed for mimetic discretization of the divergence and gradient operators, see, e.g., <cit.>. For earlier introduction of the mimetic approach see, e.g., <cit.>.
In particular, spectral equivalence to the identity means that the condition number of S does not increase under the mesh refinement process.
Moreover, it is often observed that most of the eigenvalues of S are equal to one. In fact, when solving the boundary value problem (<ref>), the spectrum of S strongly depends on the boundary conditions imposed. The most common boundary conditions for the Stokes flow on solid walls are the no-slip boundary conditions, and
this is due to the presence of the no-slip boundaries the spectrum of the Schur complement contains eigenvalues which are not equal to one. When one solves problems with small surface-to-volume ratio, which is the case most commonly considered in papers analyzing iterative solvers for Stokes problems, only a small part of eigenvalues of the Schur complement are not equal to one, which justifies using a diagonal matrix or even the identity matrix as a preconditioner. These facts are the basis for the Uzawa and Uzawa-like algorithms <cit.>, which are classical algorithms for solving the steady Stokes problem (<ref>).
Note, that Uzawa algorithm can be written in two equivalent formulations (see, e.g., <cit.> p.44). In the first one, as a stationary iteration for the coupled system (<ref>) preconditioned with a block triangular preconditioner (see, e.g., <cit.>), while in the second one, the Uzawa algorithm is written as an iterative method for the Schur complement equation (<ref>) (see, e.g., <cit.>).
Numerous computational studies, using either of the formulations, demonstrate the efficiency of the Uzawa algorithms in the case of simple geometries. A number of reviews and theoretical studies are dedicated to this subject advancing the knowledge in the area, see, e.g. <cit.>.
Even a superlinear convergence of the Krylov-Uzawa algorithm can be established for general smooth geometries, see <cit.>.
On the other hand, the Schur complement matrix S can become arbitrarily ill-conditioned for very complex geometries, like ones representing the pore space in rock samples, filter media, membranes, etc.. In particular, we demonstrate the later issue in Section <ref>, where we present specific rock samples from tight reservoirs for which the condition number of S is greater than 10^5.
Recently, it was demonstrated in <cit.> that adding discrete diffusion to the established preconditioner significantly reduces the number of iterations in the case of channel-dominated domains. Similar results were demonstrated in <cit.> for the diffusion-like SIMPLE preconditioner in the case of complex geometries from tight porous media.
The SIMPLE and the Uzawa preconditioners for the Schur complement S, Ŝ_simple and Ŝ_uzawa, can be written as follows:
Ŝ_simple = 𝐁𝐀̂_simple^-1𝐁^T, 𝐀̂_simple = diag(𝐀), Ŝ_uzawa = I.
In fact, the preconditioner Ŝ_simple is widely-known in the CFD community since the same approximation is used in the SIMPLE iterative method (Semi-Implicit Method for Pressure Linked Equations), which is one of the classical methods for solving the stationary Navier-Stokes equations <cit.>.
It should be noted, that in the Stokes case, the spectrum of Ŝ_simple is qualitatively different from the spectrum of S. Namely, the matrix Ŝ_simple behaves essentially as the pressure Laplacian matrix 𝐁𝐁^T, so its condition number increases quadratically as the grid resolution decreases.
However, such spectral behavior turns out to be justified in the case of tight geometries in the presence of narrow channels where the nature of flow is predominantly diffusive.
In the present article, we numerically investigate the behavior of the two preconditioners for the staggered finite-difference discretization of Stokes equations describing the flow in thight rock samples and in synthetic 2D geometries, and identify certain domain characteristics which are indicators for the performance of the respective Stokes solvers.
The reminder of the paper is organized as follows. Next section is dedicated to the problem statement. The description of the considered iterative methods, namely the CG-Uzawa and CG-SIMPLE methods, is provided in the third section. The main results are presented in the fourth section where the computational experiments are performed and can be summarized as follows.
* In Section <ref>, we compare the CG-SIMPLE and CG-Uzawa algorithms for 3D samples from real rocks with low porosity and confirm by the numerical experiments that the preconditioner Ŝ_simple provides orders of magnitude lower condition numbers than Ŝ_uzawa and demonstrates robust convergence while the CG-Uzawa method stagnates. Further on, we show that the CG-SIMPLE algorithm calculates the permeability much more accurately, which means it calculates the pressure gradient more accurately.
* We show that there is a correlation between the condition number of the Schur complement matrix and the surface-to-volume ratio of the samples. In Section <ref>, we perform a systematic study for the synthetic 2D geometries - random packings of squares, which demonstrates a clear dependence between the condition number of both unpreconditioned and preconditioned with the SIMPLE Schur complement matrices and the surface-to-volume ratio. The results are valid for 3D samples as well (see Section <ref>).
* Further on, we compute the full spectra of the Schur complement matrix and observe that the number of its non-unit eigenvalues is determined by the surface area of the boundary where the no-slip conditions are posed, and by the connectivity of the flow domain.
Finally, conclusions are drawn.
§ PROBLEM STATEMENT
The considered here Stokes problem is presented in the Introduction, along with the general algebraic form of its discretization. In this section, the way of representing the geometries, the grid, and the boundary conditions are discussed.
§.§ Voxel-based geometries
In voxel-based geometry, the domain is discretized into small cubic volumes called voxels. For the three-dimensional case, let the computational domain Ω be a cube with side length L, and let it be decomposed into n^3 voxels, where n is the number of voxels in each dimension:
Ω = ⋃_(i,j,k) ∈𝕀^nω(i,j,k), 𝕀^n = (i,j,k): i,j,k ∈1, …, n,
where 𝕀^n denotes a three-dimensional index set. The voxels ω(i,j,k) are defined as cubic regions of length h=L/n:
ω(i,j,k) = [x_i-1, x_i] × [y_j-1, y_j] × [z_k-1, z_k],
where x_i = (i-1)h, y_j = (j-1)h, and z_k = (k-1)h.
Each voxel can either represent fluid or solid material. The domain Ω is partitioned into two disjoint parts, the pure fluid domain Ω_f and the solid domain Ω_s:
Ω = Ω_f ∪Ω_s.
which corresponds to disjoint decomposition:
𝕀^n = 𝕀^n_f ⊔𝕀^n_s.
Example of voxel-based geometry for the case d=2 can be seen in Fig. <ref>, and several samples for the case d=3 are shown in Fig. <ref>. The classical finite difference method is used to discretize the Stokes problem in Ω_f on fully-staggered grids. The described voxel-based geometry serves as a grid for the pressure, the momentum equations are discretized on staggered grids. For detailed description see, e.g., <cit.>.
§.§ Boundary conditions
Recall, in the fluid domain Ω_f = Ω∖Ω_s, we consider the Stokes equations:
-Δ𝐮 + ∇ p = 𝐟0 in Ω_f,
-∇·𝐮 = 00 in Ω_f,
𝐮 = 0f on Γ_0.
No-slip boundary conditions are imposed on the interior solid boundary Γ_0, defined as follows:
Γ_0 = Ω_f ∩Ω_s.
Periodic boundary conditions on the velocity and pressure are imposed on the outer boundaries of the computational domain:
𝐮|_Γ_x^0∩Ω_f = 𝐮|_Γ_x^1∩Ω_f, 𝐮|_Γ_y^0∩Ω_f = 𝐮|_Γ_y^1∩Ω_f,
p|_Γ_x^0∩Ω_f = p|_Γ_x^1∩Ω_f,
p|_Γ_y^0∩Ω_f = p|_Γ_y^1∩Ω_f,
where Γ_x^0 = {(x,y) ∈∂Ω | x = 0}, and Γ_x^1, Γ_y^0, Γ_y^1 are defined similarly.
In the case of periodic boundary conditions, the periodicity of the geometry is usually also assumed.
It should be noted, that the Stokes problem (<ref>)-(<ref>) is well-posed if and only if the fluid domain Ω_f is connected and ∫_Ω_f p ∂ω = 0.
§ ITERATIVE METHODS UNDER CONSIDERATION
A variety of methods for solving (<ref>) can be roughly divided into two categories: coupled methods, in which the system is solved in the full form, and reduced methods, in which one of the variables is first excluded from the equations and then restored.
In what follows, we consider the (reduced) preconditioned Schur complement equation (<ref>), which can be written as follows:
Ŝ^-1S p_h = Ŝ^-1 g_h^S,
with the preconditioners either Ŝ = Ŝ_uzawa or Ŝ = Ŝ_simple defined in (<ref>).
As it was previously mentioned, the classical SIMPLE and Uzawa algorithms can be equivalently written as stationary iterations in either coupled or reduced form.
For the coupled form, usually the GMRES Krylov subspace method is used to accelerate stationary iterations (see, e.g., <cit.>).
Following <cit.>, we use here the Conjugate Gradient (CG) method as the Krylov accelerator for solving the Schur complement problem (<ref>). In fact, different Krylov accelerators of the SIMPLE or Uzawa algorithms may result in non-equivalent iterative methods. In the present section we describe the respective CG-SIMPLE and CG-Uzawa algorithms.
§.§ Description of the methods
A wide class of preconditioners for the stationary Stokes equations (see, e.g., <cit.>), including the classical SIMPLE and Uzawa algorithms in their coupled forms, can be constructed by discarding one or more factors in the following approximation of the matrix 𝔸:
𝔸̂ = 𝕃̂𝔻̂𝕌̂ =
[ 𝐈 ; 𝐁𝐀̂_1^-1 I ][ 𝐀̂_2 ; -Ŝ ][ 𝐈 𝐀̂_3^-1𝐁^T; I ]
,
where 𝐀̂_1, 𝐀̂_2, 𝐀̂_3 are (possibly different) approximations of 𝐀, and Ŝ is an approximation of S.
Popular options are block lower triangular, block diagonal, and block upper triangular preconditioners:
𝕃̂𝔻̂ =
[ 𝐀̂ ; 𝐁 -Ŝ ]
, 𝔻̂ =
[ 𝐀̂ ; -Ŝ ]
, 𝔻̂𝕌̂ =
[ 𝐀̂ 𝐁^T; -Ŝ ]
,
where 𝐀̂ is an approximations of 𝐀.
Let us consider a stationary iteration corresponding to the operator splitting 𝔸 = 𝔸̂ - (𝔸̂ - 𝔸), given as follows:
[ 𝐮_h^k+1; p_h^k+1 ]
=
[ 𝐮_h^k; p_h^k ]
+ 𝔸̂^-1(
[ 𝐟_h; g_h ]
- 𝔸[ 𝐮_h^k; p_h^k ]).
For certain choices of the preconditioner 𝔸̂ (see below), the coupled iterations (<ref>) can be equivalently rewritten as a preconditioned Richardson iteration for the reduced system (<ref>):
p_h^k+1 = p_h^k + αŜ^-1 (g^S_h - Sp_h^k),
where α > 0 is a stationary parameter of the Richardson iteration.
§.§.§ Uzawa and CG-Uzawa algorithms
Let us consider the classical Uzawa algorithm in the following form (see., e.g., <cit.>):
𝐀𝐮_h^k+1 = 𝐟_h - 𝐁^T p_h^k,
Ŝ_uzawaδ p_h^k = α_uzawa(𝐁𝐮_h^k+1 - g_h),
p_h^k+1 = p_h^k + δ p_h^k,
where α_uzawa is a relaxation parameter of the algorithm.
Firstly, as it is shown in <cit.>, the Uzawa iteration (<ref>) can be regarded as a stationary iteration of the form (<ref>) with a preconditioner 𝔸̂ given as follows:
𝔸̂ = 𝔸̂_uzawa =
[ 𝐀 ; 𝐁 -(1/α_uzawa)Ŝ_uzawa ]
,
which corresponds to the block lower triangular preconditioner 𝕃̂𝔻̂ defined in (<ref>) if we take:
Ŝ =
(1/α_uzawa) Ŝ_uzawa, 𝐀̂ = 𝐀.
Secondly, the Uzawa iteration (<ref>) is equivalent to the Richardson iteration (<ref>) (see, e.g. <cit.>), if we take:
Ŝ =
Ŝ_uzawa, α = α_uzawa.
The CG-Uzawa algorithm is obtained by replacing the stationary iteration (<ref>) by the Conjugate Gradient iteration applied to S with the identity preconditioner Ŝ_uzawa, see, e.g., <cit.> for details.
§.§.§ SIMPLE and CG-SIMPLE algorithms
Let us now consider the classical SIMPLE algorithm in the following form <cit.>:
𝐀𝐮_h^k + 1/2 = 𝐟_h - 𝐁^T p_h^k,
Ŝ_simpleδ p_h^k = α_simple (𝐁𝐮_h^k + 1/2 - g_h),
𝐀̂_simpleδ𝐮_h^k = -𝐁^T δ p_h^k,
p_h^k+1 = p_h^k + δ p_h^k,
𝐮_h^k+1 = 𝐮_h^k + 1/2 + δ𝐮_h^k,
where α_simple is a pressure damping parameter of the algorithm and Ŝ_simple is defined in (<ref>).
Firstly, as it is shown in <cit.>, the iterative process (<ref>) can be regarded as a coupled iteration of the form (<ref>) with the preconditioner 𝔸̂ given as follows:
𝔸̂ = 𝔸̂_simple
=
[ 𝐀 ; 𝐁 -(1/α_simple)Ŝ_simple ][ 𝐈 𝐀̂_simple^-1𝐁^T; I ]
,
which corresponds to a preconditioner 𝔸̂ of the block structure (<ref>) if we take:
𝐀̂_1 = 𝐀̂_2 = 𝐀̂, 𝐀̂_3 = 𝐀̂_simple, Ŝ = (1/α_simple)Ŝ_simple.
Secondly, the SIMPLE iteration (<ref>) is equivalent to the Richardson iteration (<ref>) (see, e.g. <cit.>), if we take:
Ŝ = Ŝ_simple, α = α_simple.
Similarly to the CG-Uzawa algorithm, the CG-SIMPLE algorithm is defined by applying the Preconditioned Conjugate Gradient method to S with the preconditioner Ŝ_simple.
It should be noted, that different splittings of 𝔸 in the coupled iterations (<ref>) may result in the same reduced iteration (<ref>).
For example, replacing Ŝ_uzawa, α_uzawa by Ŝ_simple, α_simple in (<ref>) results in the following preconditioner 𝔸̂:
𝔸̂ = 𝔸̂_simple^* [ 𝐀 ; 𝐁 -(1/α_simple)Ŝ_simple ]
,
which corresponds to the block lower triangular preconditioner 𝕃̂𝔻̂ defined in (<ref>) if we take:
Ŝ = (1/α_simple)Ŝ_simple, 𝐀̂ = 𝐀.
Then, the coupled iteration (<ref>) with the preconditioner 𝔸̂ = 𝔸̂_simple^* determines the following iterative process:
𝐀 u_h^k+1 = 𝐟_h - 𝐁^T p_h^k,
Ŝ_simpleδ p_h^k = α_simple(𝐁𝐮_h^k+1 - g_h),
p_h^k+1 = p_h^k + δ p_h^k,
which can be regarded as the preconditioned (with Ŝ_simple) Uzawa algorithm.
In the same time, the iterative process (<ref>) is also equivalent to the Richardson iteration (<ref>) for the choice (<ref>), hence it is also equivalent to the classical SIMPLE algorithm (<ref>). So, following the terminology from <cit.>, the CG-SIMPLE algorithm can be also identified as the preconditioned CG-accelerated Uzawa algorithm.
§.§ Inner iterations and stopping criteria
For both the CG-Uzawa and CG-SIMPLE algorithms, in our numerical experiments we use unpreconditioned relative residual norm as a stopping criteria for the outer iterative process. Namely, given an input tolerance ε_S, the outer CG iterations stop as soon as:
r_S^# / r_S^0 < ε_S,
where the superscript # denotes the final iteration number, and r_S^k denotes the residual on the k^th outer iteration, given as follows:
r_S^k = Sp_h^k - g^S_h.
On each step of the outer iteration, applying matrix S requires inversion of the velocity Laplacian matrix 𝐀 to recover intermediate velocity from the intermediate pressure. We use the Preconditioned Conjugate Gradient for solving with the velocity Laplacian matrix 𝐀, so formally we deal with inexact versions <cit.> of the Uzawa and SIMPLE algorithms. Thus, we have a two-level inner-outer iterative process: at each step of the outer CG iteration for S, inner CG iterations for 𝐀 are performed.
In our numerical experiments, we use preconditioned relative residual norm as the stopping criteria for inner iterations with matrix 𝐀. For example, given an input tolerance ε_𝐀, the inner CG iteration for computing 𝐮_h = 𝐀^-1𝐟^h stops as soon as:
(𝐀̂)^-1r_𝐀^# / (𝐀̂)^-1 r_𝐀^0 < ε_𝐀,
where r_𝐀^k denotes the residual on the k^th inner iteration, given as follows:
r_𝐀^k = 𝐀𝐮_h^k - 𝐟_h.
Additionally, at each step of the outer CG-SIMPLE iteration, we have to solve the system with the preconditioner matrix Ŝ_simple. We use preconditioned relative residual norm as the stopping criteria for inner iterations with the matrix Ŝ_simple. For example, given an input tolerance ε_Ŝ, the inner CG iteration for computing p_h = (Ŝ_simple)^-1g^h stops as soon as
(Ŝ̂_simple)^-1 r_Ŝ^# / (Ŝ̂_simple)^-1 r_Ŝ^0 < ε_Ŝ,
where r_Ŝ^k denotes the residual on the k^th inner iteration, given as follows:
r_Ŝ^k = (Ŝ_simple) p_h^k - g_h.
For building preconditioners 𝐀̂ and Ŝ̂_simple for the Laplacian matrices 𝐀 and Ŝ_simple, we use Algebraic Multigrid method (AMG) <cit.>.
In practice, we use the implementation BoomerAMG <cit.> from HYPRE library. It is also worth mentioning recent advances in developing monolithic AMG methods for the Stokes problem <cit.>.
§ COMPUTATIONAL EXPERIMENTS
Computational experiments are performed in order to study numerically:
* The performance of the two preconditioners in solving the Schur complement problem (<ref>);
* The performance of the two preconditioners in computing the permeability of the samples according to (<ref>);
* Possible correlation between the surface-to-volume ratio and the condition number of the preconditioned\unpreconditioned Schur complement matrix;
* Possible correlation between the number of boundary nodes where the no-slip boundary conditions are imposed and the number of non-unit eigenvalues of the Schur complement matrix.
Two sets of experiments are performed.
In the first one, 3D CT images of samples of rocks from tight reservoirs (low porosity) are considered. The performance of the inexact Uzawa and SIMPLE preconditioners in conjunction with the conjugate gradient method for the Schur complement matrix is compared. It is observed that the SIMPLE preconditioner behaves much better for this class of problems. A correlation between the surface-to-volume ratio and the condition number of the Schur complement matrix is observed. Furthermore, the permeabilities computed with the two preconditioners are compared, and it is shown that the usage of the CG-SIMPLE leads to a much more robust and accurate calculation of the permeability. This implies that the SIMPLE preconditioner not only ensures more robust convergence in solving the Schur complement problem with respect to the pressure, but also provides a more accurate computation of the gradient of the pressure.
In the second set of computational experiments, a more rigorous study of the two preconditioners is conducted by considering the synthetic 2D geometries and the exact inner iterations. In addition to comparing the performance of the preconditioners, a systematic study of the spectra of the Schur complement matrix is carried out, and a conjecture that the no-slip boundary conditions are the main reason for the non-unit eigenvalues of the Schur complement is formulated.
Moreover, for the considered synthetic samples, we clearly observe that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix, while the dependence is inverse for the Schur complement matrix preconditioned with the SIMPLE.
§.§
3D rock samples from tight reservoirs
§.§.§ Preliminaries: samples and notations
In this section, we study the performance of the CG-Uzawa and CG-SIMPLE algorithms for the pore space images of real rock samples from tight reservoirs. We consider six 3D images: five ultra-tight samples A-E considered in the article <cit.>, and one image of the classical Berea's sandstone with medium porosity. The corresponding pore space images are depicted on Fig. <ref> using the Geodict visualization tool. The samples A-E are scanned with the resolution 1.2 mkm and have the size n = 600, and the sample S is scanned with the resolution 4 mkm and has the size n = 300. Detailed information about the samples can be found in Table <ref>, which includes the reference permeability κ_zz^ref defined in (<ref>), the number of fluid voxels 𝕍^f = |𝕀^n_f|, and the porosity ν, defined as follow:
ν = (𝕍^f / 𝕍) · 100 %,
where 𝕍 = |𝕀^n| = n^d is the total number of voxels.
Additionally, for each sample we compute the surface-to-volume (s-t-v) ratio which is defined as follows:
σ_s = (𝕍_surf^s / 𝕍^f) · 100%,
where the surface area 𝕍_surf^s of the no-slip boundary is determined as the number of near-boundary solid voxels, i.e., solid voxels face-adjacent with the fluid domain:
𝕍_surf^s = |𝕀^n_surf|, 𝕀^n_surf = {(i,j,k) ∈𝕀^n_s: ω_(i,j,k)∩Ω_f ≠∅}.
Computing permeability.
Computing permeability of 3D CT images of rocks or of synthetic 2D geometries is a basic task in the Digital Rock Physics.
To compute permeability of a sample in x direction, we solve the Stokes equations with the forcing term 𝐟 = (1,0,0)^T.
For Ω having a unit length, such a unit volume force corresponds to a unit pressure drop <cit.>, so in this case permeability equals the Darcy (averaged) velocity, which is given as follows:
(κ_xx, κ_xy, κ_xz)^T = (⟨ u ⟩, ⟨ v ⟩, ⟨ w ⟩)^T = ⟨𝐮⟩ = |Ω|^-1∫_Ω_f𝐮.
Note, in order to compute permeabilities in y and z directions, similar computations are required with 𝐟 = (0,1,0)^T and 𝐟 = (0,0,1)^T.
For non-unit domains, the non-dimensional permeability κ_xx is scaled by the size of the physical domain in the flow direction:
κ_xx = L^2 κ_xx.
§.§.§ Performance of the preconditioners in solving the Schur complenment problem and in computing the permeability
In the present subsection, we study the performance of the SIMPLE and Uzawa preconditioners in solving the Schur complement problem (<ref>) as well as in computing the permeability for the samples A-S described in Table <ref>.
As a stopping criteria for the outer CG iterations, we use ε_S = 10^-3 for the CG-Uzawa algorithm and ε_S = 10^-2 for the CG-SIMPLE algorithm. It is worth noting that such a smaller stopping tolerance for the CG-Uzawa was used in an attempt, albeit unsuccessful, to achieve convergence in permeability to its reference values κ_zz^ref. For the inner iterations, we use a higher precision ε_𝐀 = 10^-5 in both cases. Also, for the CG-SIMPLE algorithm we use ε_Ŝ = 10^-5 for solving with the SIMPLE preconditioner Ŝ_simple. The reference permeabilities were computed using the CG-SIMPLE algorithm with higher precision ε_S = 10^-4, ε_𝐀 = 10^-7, ε_Ŝ = 10^-7. Such inexact solves for 𝐀 and Ŝ_simple make it difficult to rigorously analyze underlying numerical methods. However, the purpose of this section is to demonstrate convergence problems with the established Uzawa preconditioner that occur in practical permeability calculations when computing flows in tight porous media. So, we compare methods for the settings that we usually use in our practical calculations.
The convergence history for selected tolerances is presented in Fig. <ref>, where the relative unpreconditioned residual norm r_S^k / r_S^0 defined in (<ref>) is shown on the left.
Furthermore, the scaled permeability error, defined as:
e_κ^k = |κ_zz^k / κ_zz^ref - 1|,
is shown in Fig. <ref> on the right. Recall, that the permeability is computed according to (<ref>), and the velocity is computed according to (<ref>). Having in mind that the left hand side of (<ref>) is identical for both the Uzawa and SIMPLE preconditioned algorithms, the difference in the accuracy with which the permeability is computed can be explained only by the difference in the accuracy with which 𝐁^Tp is computed in both cases.
Recall that in <cit.> the computation of the permeability with the current CG-SIMPLE algorithm was compared to computations with other commercial and academic codes, and it was shown that all the values are in the same range.
The summary of the results with CG-Uzawa and CG-SIMPLE algorithms for the 3D real rock samples A-S can be found in Table <ref>, which includes the number of iterations, the required computational time, and the scaled permeability error e_κ^# computed on the final iteration. Additionally, we provide estimations for the condition numbers of the preconditioned and unpreconditoned Schur complement matrices, computed for free during the outer CG iterations using Lanczos algorithm.
The following hardware was used in our numerical experiments: 48x MPI compute node (Dell PowerEdge M640), dual Intel Xeon Gold 6132 ("Skylake") @ 2.6 GHz, i.e. 28 CPU cores per node. The computational times shown in Table <ref> were obtained using 8 CPU nodes.
Several observations can be drawn from the results presented in Table <ref>. For the considered low porosity images, the SIMPLE preconditioner appears to perform better than the Uzawa preconditioner, as it converges robustly while the CG-Uzawa stagnates. Firstly, despite the CG-SIMPLE algorithm being more expensive (approximately x1.5) per iteration, its total computational time is smaller compared to the CG-Uzawa because significantly fewer number of iterations is required. This observation holds true for low porosity samples with high surface-to-volume ratios, whereas for higher porosity, the CG-Uzawa shows better performance. Secondly, the estimated condition number for the preconditioned Schur complement matrix is about three orders of magnitude smaller for the SIMPLE preconditioner than for the Uzawa (identity) preconditioner in the case of low porosity. For moderate porosity (sample S), the condition number for both preconditioners is comparable. Thirdly, the CG-SIMPLE computes the permeability much more accurately for the considered samples. As mentioned earlier, this means that it calculates the pressure gradient more accurately for such problems. Actually, achieving even 10% accuracy in computing permeability is not always possible with the CG-Uzawa method.
§.§.§ Correlation between the surface-to-volume ratio and the condition number of the Schur complement matrix
In Fig. <ref>, we show that for the considered samples A-S there is a strong correlation between the surface-to-volume ratio σ_s (<ref>) and the estimated condition number of the unpreconditioned Schur complement matrix.
However, the dependence in the case of SIMPLE preconditioned Schur complement matrix is not so distinctive here, so we perform a more rigorous study in the subsequent Section <ref>.
§.§
Two-dimensional synthetic geometries
To identify consistent patterns that affect the performance of the methods under consideration in complex pore space domains, we consider synthetic 2D geometries with a transparent generation process and directly available geometric information (such as porosity, no-slip surface area, etc). As in the 3D case, we present and discuss the convergence of the two algorithms, as well as the accuracy with which the permeability is computed. Additionally, for the considered synthetic samples we compute and analyse the full spectra of the preconditioned and unpreconditioned Schur complement matrices.
§.§.§ Generation of synthetic 2D geometries.
We study flows passing around arrays of solid square obstacles randomly placed in a fluid bed. First of all, a uniform voxel (pixel in 2D) grid is generated in Ω as described in Section <ref>. For ease of generation, we consider square obstacles of the same size, and each obstacle is located in the center of the square cell, or is slightly shifted, so that a cell contains the obstacle and a part of the flow domain around it. The obstacles do not touch the boundary of the cell. Each generated geometry is defined by four integer parameters (N,n_c,n_avg,n_min), where N and n_c determine the number of cells in one direction and their size measured in voxels, while the parameters n_avg and n_min control the average and minimal thicknesses (in voxels) of the fluid channels between two adjacent solid squares. Note, that the cells and the obstacles in all cases are adjusted to the introduced computational grid, so that each voxel is fully occupied either by fluid or by solid.
An example for N = 7, n_c = 50, n_avg = 10, n_min = 2 is presented in Fig. <ref>. Formally, we have a domain of the total size n×n, n = Nn_c which represents an N × N array of cells of the size n_c× n_c; each cell contains a solid square of the size (n_c - n_avg) × (n_c - n_avg) with the origin (n_c/2 + r_1, n_c/2 + r_2), where r_1, r_2 ∈ [-(n_avg-n_min)/2, (n_avg-n_min)/2] are (integer) random shifts. It should be noted, that randomness is necessary to observe non-trivial solutions which take place in the case of fully periodic arrays.
§.§.§ Performance of the preconditioners in solving the Schur complement problem and in computing permeability
In the present subsection, we study the performance of the SIMPLE and Uzawa preconditioners in solving the Schur complement problem (<ref>) as well as in computing the permeability for the synthetic 2D geometries with variable average channel thicknesses n_avg. Namely, we randomly generated five geometries according to the procedure described in the previous subsection for N = 7, n_c = 50, n_min = 2, and n_avg = {4,6,8,10,12}.
Detailed information about the samples can be found in Table <ref>, which includes the reference permeability κ_zz^ref defined in (<ref>), the number of fluid voxels 𝕍^f, the porosity ν (<ref>), and the surface-to-volume (s-t-v) ratio σ_s (<ref>).
It should be noted, that the average channel thickness for synthetic 2D geometries is straightforwardly related to the surface-to-volume ratio for general porous media. Therefore, by varying the average thickness of the synthetic geometries, we can investigate the effect of surface-to-volume ratio on the convergence of the algorithms.
As a stopping criteria for the outer CG iterations, we use ε_S = 10^-3 for both preconditioners. However, in this experiment we use machine epsilon ε_𝐀 = ε_Ŝ = 10^-13 for inner iterations, which means that exact Uzawa is used here.
The convergence history for selected tolerances is presented in Fig. <ref>, where the relative unpreconditioned residual norm r_S^k / r_S^0 defined in (<ref>) is shown on the left and the scaled permeability error e_κ^k defined in (<ref>) is shown on the right.
The summary of the results obtained with CG-Uzawa and CG-SIMPLE algorithms for the synthetic 2D geometries with variable channel thicknesses can be found in Table <ref>, which includes the number of iterations, the required computational time, the scaled permeability error e_κ^# computed on the final iteration, and the condition numbers of the preconditioned and unpreconditoned Schur complement matrices.
Similar observations to those made for the 3D simulations can be made from the results presented in Table <ref>. Again, the CG-SIMPLE algorithm requires less total computational time compared to the CG-Uzawa for low-porosity (and large surface-to-volume ratio) samples. For higher porosity, CG-Uzawa tends to show better performance. The condition number for the preconditioned Schur complement matrix (here it is computed exactly, see below) is about two orders of magnitude smaller for the SIMPLE preconditioner than for the Uzawa (identity) preconditioner in the case of low porosity. As in the 3D case, CG-SIMPLE more accurately computes permeability for low-porosity samples, indicating that it computes pressure gradient more accurately for such problems, as we discussed earlier in Section <ref>.
§.§.§ Correlation between the surface-to-volume ratio and the condition number of the Schur complement matrix
In Fig. <ref>, for the considered 2D geometries with variable channel thicknesses, we show the dependence between the surface-to-volume ratio σ_s (<ref>) and the condition number of both the unpreconditioned and preconditioned Schur complement matrices.
The main conclusion that we draw is that an increase in the surface-to-volume ratio and a decrease in porosity lead to an increase in the condition number of the Schur complement matrix. However, unlike the 3D case, for the considered here synthetic geometries the inverse dependence is clearly observed for the Schur complement matrix preconditioned with the SIMPLE.
We conjecture that for low porosity samples the condition number of the Schur complement matrix is influenced by the number of non-unit eigenvalues in its spectrum. In the next subsection, we show that the large number of non-unit eigenvalues in the spectrum of the Schur complement matrix is directly related to low porosity and large surface-to-volume ratio.
§.§.§ Number of non-unit eigenvalues and no-slip surface area.
In the present section, for the class of geometries, we show the relation between surface-to-volume ratio and the number of non-unit eigenvalues of the Schur complement matrix.
We consider various configurations of synthetic 2D geometries. First, we vary the number of squares N, which determines the degree of connectivity of the flow domain Ω_f. Second, for each N we vary the surface area 𝕍_surf^s defined in (<ref>).
For selected configurations, we compute the full spectrum of the Schur complement matrix S and calculate the number of eigenvalues, denoted N_ev, which are not equal to one, including zero eigenvalue. The results are presented in Figure <ref>, where we show the dependence between the surface area 𝕍_surf^s and the number of non-unit eigenvalues N_ev.
For the considered here class of geometries, we observe, that the following empirical formula for the number of non-unit eigenvalues holds:
N_ev = 𝕍_surf^s + 3N^2 - 1.
In particular, the formula (<ref>) reveals that the surface-to-volume ratio is related to the ratio of non-unit to unit eigenvalues of the Schur complement matrix, i.e., the greater is the ratio the further the Schur complement is from the identity, and the worse is the performance of the Uzawa algorithm.
§ CONCLUSIONS
In conclusion, the article presents a comparative study of the CG-SIMPLE and CG-Uzawa algorithms for samples with low porosity. The results show that the CG-SIMPLE algorithm applied to solve Stokes problem for low porosity 3D rock samples provides significantly lower condition numbers of the Schur complement matrix and ensures robust convergence to high accuracy both in solving the Schur complement problem and in computing the permeability. The latter indicates that the pressure gradient is more accurately computed in this case. This behavior is further explained through a systematic study of synthetic 2D geometries. We demonstrate that the number of non-unit eigenvalues of the Schur complement matrix is determined by the surface area of the boundary where the no-slip conditions are posed, and the connectivity of the flow domain. Moreover, we show that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix. However, the inverse correlation holds for the Schur complement matrix preconditioned with the SIMPLE. These findings provide important insights into the behavior of the solvers for the Schur complement matrix and suggest effectiveness of the SIMPLE preconditioner for solving the Stokes problem in tight geometries.
§ ACKNOWLEDGEMENTS
Ivan Oseledets was supported by Alexander von Humboldt Research Award. Oleg Iliev was supported by BMBF under grant 05M20AMD ML-MORE.
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http://arxiv.org/abs/2307.04026v1 | 20230708182039 | Dowker-type theorems for disk-polygons in normed planes | [
"Bushra Basit",
"Zsolt Lángi"
] | math.MG | [
"math.MG",
"52A40, 52A21, 52A30"
] |
Dowker-type theorems]Dowker-type theorems for disk-polygons in normed planes
B. Basit]Bushra Basit
Z. Lángi]Zsolt Lángi
Bushra Basit, Department of Algebra and Geometry, Budapest University of Technology and Economics,
Műegyetem rkp. 3., H-1111 Budapest, Hungary
[email protected]
Zsolt Lángi, Department of Algebra and Geometry, Budapest University of Technology and Economics, and MTA-BME Morphodynamics Research Group,
Műegyetem rkp. 3., H-1111 Budapest, Hungary
[email protected]
Partially supported by the National Research, Development and Innovation Office, NKFI, K-147544 grant.
[2020]52A40, 52A21, 52A30
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body K in the Euclidean plane, the areas of the maximum (resp. minimum) area convex n-gons inscribed (resp. circumscribed) in K is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, the Euclidean plane by an arbitrary normed plane, or convex n-gons by disk-n-gons, obtained as the intersection of n closed Euclidean unit disks. The aim of our paper is to investigate these problems for C-n-gons, defined as intersections of n translates of the unit disk C of a normed plane. In particular, we show that Dowker's theorem remains true for the areas and the perimeters of circumscribed C-n-gons, and the perimeters of inscribed C-n-gons. We also show that in the family of origin-symmetric plane convex bodies, for a typical element C with respect to Hausdorff distance, Dowker's theorem for the areas of inscribed C-n-gons fails.
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§ INTRODUCTION
For any integer n ≥ 3 and plane convex body K, let A_n(K) (resp. a_n(K)) denote the the infimum (resp. supremum) of the areas of the convex n-gons circumscribed about (resp. inscribed in) K. Verifying a conjecture of Kerschner, Dowker <cit.> proved that for any plane convex body K, the sequences { A_n(K) } and { a_n(K) } are convex and concave, respectively. It was proved independently by L. Fejes Tóth <cit.>, Molnár <cit.> and Eggleston <cit.> that the same statements remain true if we replace area by perimeter, where the last author also showed that these statements are false if we replace area by Hausdorff distance. These results are known to be true also in any normed plane <cit.>. Dowker's theorems have became important in many areas of discrete geometry, in particular in the theory of packing and covering <cit.> and are often used even today (see e.g. <cit.>).
Among many variants of Dowker's theorems that have appeared in the literature, we mention only one, which is related to the notion of spindle convexity. This concept goes back to a paper of Mayer <cit.> who, for any given convex body C in Euclidean space, considered sets X with the property that for any points p,q ∈ X, X contains the intersection of all translates of C containing p,q. He called these sets hyperconvex. His paper led to several papers in this topic in the 1930s and 40s, which, however, seems to have been forgotten by the end of the century. In modern times, a systematic investigation of hyperconvex sets was started in the paper <cit.> in 2007 for the special case that C is a closed Euclidean ball, and a similar paper <cit.> appeared in 2013, dealing with any convex body C (see also <cit.>).
Hyperconvex sets have appeared in the literature under several different names: spindle convex, strongly convex or superconvex sets (see e.g. <cit.>), and appear in different areas of mathematics <cit.>. In this paper, we follow the terminology in <cit.>, and call a set satisfying the property in Mayer's paper C-spindle convex, or shortly C-convex, and if C is a closed Euclidean unit ball, we call it spindle convex (see Definition <ref>).
One of the results related to spindle convex sets is due to G. Fejes Tóth and Fodor <cit.> who extended Dowker's theorems, together with their variants for perimeter, for spindle convex sets; in these theorems the role of inscribed or circumscribed convex n-gons is played by the so-called disk-n-gons, obtained as the intersections of n closed Euclidean unit disks. They also proved similar theorems in hyperbolic or spherical plane.
Our main goal is to investigate a normed version of the problem in <cit.>. To state our results, recall that the unit ball of any finite dimensional normed space is a convex body symmetric to the origin o, and any such body is the unit ball of a finite dimensional normed space. Thus, in the paper we choose an arbitrary o-symmetric convex disk C in the real normed space ^2, and work in the normed plane with unit disk C, which we regard as ^2 equipped with the norm ||·||_C of C.
In the paper, by a convex disk we mean a compact, convex planar set with nonempty interior. We denote the family of convex disks by , and the family of o-symmetric convex disks by _o. In the paper we regard and _o as topological spaces with the topology induced by Hausdorff distance.
Before presenting our results, recall the well-known fact that any finite dimensional real normed space can be equipped with a Haar measure, and that this measure is unique up to multiplication of the standard Lebesgue measure by a scalar (cf. e.g. <cit.>). This scalar does not play a role in our investigation and in the paper (·) denotes 2-dimensional Lebesgue measure.
For any C ∈_o and convex polygon Q, we define the C-perimeter of Q as the sum of the lengths of the sides of Q, measured in the norm generated by C. The C-perimeter of a convex disk K ⊂^2, denoted by _C(K), is the supremum of the C-perimeters of all convex polygons inscribed in K.
We note that, moving its vertices one by one to the boundary of K in a suitable direction, for any convex polygon Q contained in K one can find a convex polygon Q' inscribed in K with _C(Q) ≤_C(Q'). This shows, in particular, that for any two plane convex bodies K ⊆ L ⊂^2, we have _C(K) ≤_C(L), with equality if and only if K=L (see also <cit.>).
Furthermore, it is worth observing that a straightforward modification of Definition <ref> can be used to define the C-length of a rectifiable curve Γ⊂^2, denoted by _C(Γ).
Our next definition can be found in <cit.> and its origin goes back to <cit.>.
Let C ∈_o and consider two (not necessarily distinct) points p, q ∈^2 such that a translate of C contains both p and q.
Then the C-spindle (denoted as [p,q]_C) of p and q is the intersection of all translates of C
that contain p and q. If no translate of C contains p and q, we set [p,q]_C = ^2.
We call a set K ⊂^2 C-spindle convex (or shortly C-convex), if for any p,q ∈ K, we have [p,q]_C ⊆ K.
We recall from <cit.> that a closed set in ^2 different from ^2 is C-convex if and only if it is the intersection of some translates of C.
The intersection of n translates of C is called a C-n-gon for n ≥ 3.
In our next definition and throughout the paper, (·) denotes standard Lebesgue measure.
Let n ≥ 3 and let K be a C-convex disk in ^2, where C ∈_o. We set
Â_n^C(K) = inf{(Q) : Q is a C-n-gon circumscribed about K };
â_n^C(K) = sup{(Q) : Q is a C-n-gon inscribed in K };
P̂_n^C(K) = inf{_C(Q) : Q is a C-n-gon circumscribed about K };
p̂_n^C(K) = sup{_C(Q) : Q is a C-n-gon inscribed in K }.
For any C ∈_o and C-convex disk K, the sequences {Â_n^C(K) }, {P̂_n^C(K) } are convex, and the sequence {p̂_n^C(K) } is concave. That is, for any n ≥ 4, we have
Â_n-1^C(K)+Â_n+1^C(K) ≥ 2 Â_n^C(K), P̂_n-1^C(K)+P̂_n+1^C(K) ≥ 2 P̂_n^C(K), and
p̂_n-1^C(K)+p̂_n+1^C(K) ≤ 2 p̂_n^C(K).
As a consequence of Theorem <ref>, we prove Theorem <ref>, and recall that similar statements have been derived in <cit.> for the Euclidean areas of inscribed and circumscribed polygons from the classical results of Dowker in <cit.> (for their spindle convex variants, see <cit.>).
Let n ≥ 3 and k ≥ 2. Assume that k is a divisor of n and both K and C have k-fold rotational symmetry. Then there are C-n-gons Q^A, Q^P circumscribed about K which have k-fold rotational symmetry, and (Q^A)= Â_n^C(K) and _C(Q^P)= P̂_n^C(K). Similarly, there is a C-n-gon Q^p inscribed in K which has k-fold rotational symmetry, and _C(Q^p)= p̂_n^C(K).
Before our next theorem, we remark that in a topological space ℱ, a subset is called residual if it is a countable intersection of sets each of which has dense interior in ℱ. The elements of a residual subset of ℱ are called typical.
Our next result shows that Dowker's theorem for the sequence { A_n^C(K) } fails in a strong sense.
A typical element C of _o satisfies the property that for every n ≥ 4, there is a C-convex disk K with
â_n-1^C(K) + â_n+1^C(K) > 2 â_n^C(K).
The structure of the paper is as follows. In Section <ref>, we present the necessary notation and prove some lemmas. Then in Sections <ref> and <ref> we prove Theorems <ref> and <ref>, and Theorem <ref>, respectively. Finally, in Section <ref>, we collect our additional remarks and propose some open problems.
§ PRELIMINARIES
In the paper, for simplicity, for any x,y ∈^2, we denote by [x,y] the closed segment with endpoints x,y. We equip ^2 also with a Euclidean norm, which we denote by ||·||, and use the notation B^2 for the Euclidean closed unit disk centered at o. Recall that the Euclidean diameter of a compact set X ⊂^2 is the Euclidean distance of a farthest pair of points in X. If we replace Euclidean distance by distance measured in the norm of C, we obtain the C-diameter of X.
Recall that for any set X ⊆^2, the C-convex hull, or shortly C-hull is the intersection of all C-convex sets that contain C. We denote it by _C(X), and note that it is C-convex, and if X is closed, then it coincides with the intersection of all translates of C containing X <cit.>.
In the following list we collect some elementary properties of C-spindles and C-n-gons that we are going to use frequently in the paper.
We have the following.
(a) For any x,y ∈^2 with ||x-y||_C ≤ 2, [x,y]_C is the intersection of at most two translates of C, and if [x,y]_C is a translate of C, then ||x-y||_C=2.
(b) Conversely, a nonempty intersection of at most two translates of C is the C-spindle of two (not necessarily distinct) points.
(c) For any x, y ∈^2, [x,y]_C=[x,y] if and only if a translate of C contains [x,y] in its boundary.
(d) If [x,y]_C ≠ [x,y], then [x,y]_C is a centrally symmetric convex disk whose boundary consists of two arcs, connecting x and y, that are contained in the boundary of some translates of C.
(e) Any C-n-gon is the C-hull of at most n points contained in a translate of C, and vice versa.
Let x,y ∈ C ∈_o, with ||x-y||_C < 2. Then, for any sequences x_m → x, y_m → y, C_m → C with x_m,y_m ∈^2 and C_m ∈_o, we have [x_m,y_m]_C_m→ [x,y]_C.
We observe that the statement in Remark <ref> does not necessarily hold if ||x-y||_C = 2. As an example, we can choose C as a parallelogram, x_m=x and y_m=y as the midpoints of two opposite sides S_1, S_2 of C, and { C_m } as a sequence of o-symmetric hexagons inscribed in C whose elements intersect S_1 and S_2 only in x and y, respectively.
For any n ≥ 4, let ^n_a denote the subfamily of the elements C of _0 satisfying the Dowker-type inequality â_n-1^C(K) + â_n+1^C(K) ≤ 2 â_n^C(K) for any C-convex disk K. We define ^n_A, ^n_p and ^n_P similarly.
Our first lemma describes the topological properties of these families.
For any n ≥ 4, ^n_a, ^n_A, ^n_p and ^n_P are closed.
We prove the assertion only for ^n_a, as for the other quantities the proof is analogous.
Let C ∉^n_a, and suppose for contradiction that there is a sequence C_m ∈_a^n with C_m → C.
Since C ∉^n_a, there is a C-convex disk K satisfying â_n-1^C(K) + â_n+1^C(K) > 2 â_n^C(K). By Remark <ref>, if K contains points at C-distance equal to 2, then K is a C-spindle, which yields that â_j(K) = (K) for any j ≥ 3. Thus, according to our assumptions, K does not contain points at C-distance equal to 2, i.e its C-diameter is strictly less than 2. On the other hand, since K is C-convex, K is the intersection of the translates of C that contain it. Thus, there is a set X ⊂^2 such that K = ⋂_x ∈ X (x+C).
Let K_m = ⋂_x ∈ X (x+C_m). Then, clearly, K_m is C_m-convex, and K_m → K. For j=n-1,n+1, let Q_j be a C-j-gon inscribed in K such that (Q_j)=â_j^C(K). Then, as K_m → K and C_m → C, there are sequences { Q_n-1^m } and { Q_n+1^m } such that for j=n-1,n+1, Q_j^m is a C_m-j-gon inscribed in K_m, and Q_j^m → Q_j. By the properties of Hausdorff distance, the C_m-diameter of K_m is strictly less than 2 if m is sufficiently large. Then we can apply Remark <ref>, and obtain that (Q_j^m) →(Q_j) for j=n-1,n+1. From this, we have (Q_n-1^m)+(Q_n+1^m) →â_n-1^C(K) + â_n+1^C(K). On the other hand, since C_m ∈^n_a, there is a sequence { Q_n^m } such that Q_n^m is a C_m-n-gon inscribed in K_m, and 2 (Q_n^m) ≥(Q_n-1^m)+(Q_n+1^m). By compactness, we may assume that { Q_n^m } converges to a C-n-gon Q_n. Clearly, Q_n is contained in K, and by Remark <ref>, (Q_n^m) →(Q_n). Thus, â_n-1^C(K) + â_n+1^C(K) ≤ 2 (Q_n) ≤ 2 â_n^C(K); a contradiction.
Lemma <ref> readily yields Corollary <ref>, since the intersection of arbitrarily many closed sets is closed.
The family ⋃_n=4^∞_a^n of the elements C of _0 satisfying â_n-1^C(K) + â_n+1^C(K) ≤ 2 â_n^C(K) for all n ≥ 4 and all C-convex disks K is closed in _0. Similar statements hold for the families ⋃_n=4^∞_p^n, ⋃_n=4^∞_A^n and ⋃_n=4^∞_P^n.
Let C ∈_o, and let x,y be points with ||x-y||_C ≤ 2.
Then the arc-distance ρ_C (x,y) of x,y with respect to C (or shortly, C-arc-distance of x and y) is the minimum of the C-length of the arcs, with endpoints x,y, that are contained in z+(C) for some y ∈^2.
For any x,y ∈^2 with ||x-y||_C ≤ 2, if [x,y]_C ≠ [x,y], then ρ_C (x,y) = 1/2_C ([p,q]_C). Furthermore, if [x,y]_C = [x,y], then ρ_C(x,y)=||x-y||_C.
We recall the following version of the triangle inequality from <cit.>.
[Lángi, Naszódi, Talata]
Let C ∈_0, and let x,y,z be points such that each pair has a C-arc-distance.
(a) If y ∈ [x,z]_C, then ρ_C(x,y)+ρ_C(y,z) ≤ρ_C(x,z).
(b) If y ∈ [x,z]_C, then ρ_C(x,y)+ρ_C(y,z) = ρ_C(x,z).
(c) If y ∉ [x,z]_C and C is smooth, then ρ_C(x,y)+ρ_C(y,z) ≥ρ_C(x,z).
We start with a consequence of this inequality.
Let p,q,r,s ∈^2 be distinct points contained in a translate of the smooth o-symmetric convex disk C, and assume that _C {p,q,r,s} contains all of them and in this counterclockwise order. Then
ρ_C(p,q)+ρ_C(r,s) ≤ρ_C(p,r)+ρ_C(q,s).
Note that according to our conditions, the two C-arcs in the boundary of [p,r]_C intersect both C-arcs consisting of the boundary of [q,s]_C. Let s' denote the intersection point of one of the C-arcs in [p,r]_C and one of the C-arcs in [q,s], where the arcs are chosen to satisfy s' ∈_C { p,q,r } and s' ∈_C {p,r,s}. Then s' ∉ [p,q]_C and s' ∉ [r,s]_C. Since [s,s']_C, [q,s']_C ⊂ [q,s]_C, it is easy to see that p,q,s', and also r,s,s' are in C-convex position.
Thus, by Lemma <ref>, we have ρ_C(p,q) ≤ρ_C(p,s')+ρ_C(q,s') and ρ_C(r,s) ≤ρ_C(r,s') + ρ_C(s,s'), implying the assertion.
In the following lemma, let ^1 denote the Euclidean unit circle centered at the origin. For simplicity, if x,y ∈^1, we denote by xy the Euclidean closed circle arc obtained as the orbit of x when it is rotated around o in counterclockwise direction until it reaches y. Let 𝒮 denote the family of closed circle arcs xy of S. Furthermore, we say that a function f : 𝒮→ has a k-fold rotational symmetry for some positive integer k, if for any S,S' ∈𝒮, where S' is a rotated copy of S in counterclockwise direction with angle 2π/k, we have f(S)=f(S'). Lemma <ref> can be regarded as a functional form of Dowker's theorems.
Let f : 𝒮→ be a bounded function with f(xx)=0 for all x ∈^1. For any integer n ≥ 3, let
M_n = sup{∑_S ∈ X f( S ) : X ⊂𝒮 is a tiling of ^1 with |X| = n }.
If for any x_2x_3⊂x_1x_4, we have
f(x_1x_3)+f(x_2x_4) ≥ f(x_1x_4)+f(x_2x_3),
then the sequence { M_n } is concave.
Furthermore, if in addition, there is some positive integer k such that k | n and f has k-fold rotational symmetry, and there is an n-element tiling X of ^1 such that M_n = ∑_S ∈ X f(S) then there is an n-element tiling X' of ^1 with k-fold rotational symmetry such that M_n = ∑_S ∈ X' f(S).
Before the proof, we remark that X ⊂𝒮 is called an m-tiling of ^1 for some positive integer m if every point of ^1 belongs to at least m members of X, and to the interiors of at most m members of X.
To prove the assertion for { M_n }, we need to show that M_n-1+M_n+1≤ 2M_n is satisfied for any n ≥ 4. In other words, we need to show that for any tilings X={x_0x_1, …x_n-2x_n-1}, Y={y_0y_1, …y_n y_n+1} of ^1, there are tilings Z={z_0z_1, …z_n-1z_n} and W={w_0w_1, …w_n-1w_n} of ^1 such that
∑_i=1^n-1 f(x_i-1x_i) + ∑_i=1^n+1 f(y_i-1y_i) ≤∑_i=1^n f(z_i-1z_i) + ∑_i=1^n f(w_i-1w_i).
Note that the union A_0 of the two tilings is a 2-tiling of ^1.
Assume that x_1, x_2, …, x_n-1, and y_1,y_2, …, y_n+1 are in this counterclockwise order in ^1, and that y_1 ∈x_1x_2.
Due to the possible existence of coinciding points in the above two sequences, we unite these sequences as a single sequence v_1, v_2, …, v_2n in such a way that the points are in this counterclockwise order in ^1, v_1=x_1, and removing the x_i (resp. y_j) from this sequence we obtain the sequence y_1, …, y_n+1 (resp. x_1, …, x_n-1). In the proof we regard this sequence as a cyclic sequence, where the indices are determined mod 2n, and, with a little abuse of notation, we say that v_iv_j covers v_kv_l only if v_kv_l⊆v_iv_j and i < k < l < j < i+2n.
Our main goal will be to modify the 2-tiling A_0 in such a way that the value of f does not decrease but the number of covering pairs strictly decreases.
Note that since A_0 is the union of two tilings consisting of (n-1) and (n-1) arcs, respectively, A_0 contains covering pairs.
Assume that v_iv_j covers v_kv_l. Then let A_1 denote the 2-tiling of ^1 in which v_iv_j and v_kv_l
are replaced by v_iv_l and v_kv_j. According to our conditions, ∑_S ∈ A_0 f(S) ≤∑_S ∈ A_1 f(S), and the number of covering pairs in A_1 is strictly less than in A_0. Repeating this procedure we obtain a 2-tiling A_t of ^1 for which ∑_S ∈ A_0 f(S) ≤∑_S ∈ A_t f(S) and which does not contain covering pairs. Then, A_t decomposes into the two tilings {v_1,v_3, v_3v_5, …, v_2n-1v_1} and {v_2,v_4, v_4v_6, …, v_2nv_2}, each of which contains exactly n arcs. This proves the assertion for { M_n }.
Now we prove the second part. Let X be an n-element tiling of ^1 such that M_n = ∑_S ∈ X f(S). Assume that X does not have k-fold rotational symmetries. For i=1,2,…, k, let X_i denote the rotated copy of X by 2iπ/k in counterclockwise direction. Then Y= ⋃_i=1^k X_i is a k-fold tiling of ^1 with k-fold rotational symmetry, and ∑_S ∈ Y f(S) = k ∑_S ∈ X f(S). Since X has no k-fold rotational symmetry, Y contains covering pairs, and we may apply the argument in the previous paragraph.
We remark that an analogous proof yields Lemma <ref>, the proof of which we leave to the reader.
Let f : 𝒮→ be a bounded function with f(pp)=0 for all p ∈^1. For any integer n ≥ 3, let
m_n = inf{∑_S ∈ X f( S ) : X ⊂𝒮 is a tiling of ^1 with |X| = n }.
If for any x_2x_3⊂x_1x_4, we have
f(x_1x_3)+f(x_2x_4) ≤ f(x_1x_4)+f(x_2x_3),
then the sequence { m_n } is convex.
Furthermore, if in addition, there is some positive integer k such that k | n, and f has k-fold rotational symmetry, and there is an n-element tiling X of ^1 such that m_n = ∑_S ∈ X f(S) then there is a tiling X' of ^1 with k-fold rotational symmetry such that m_n = ∑_S ∈ X' f(S).
In the next lemma, by the partial derivatives (∂_p f) (p_0q_0) (resp. (∂_q f) (p_0q_0)) of the function f(pq) at p_0q_0, we mean the derivative of the function f(p(t)q_0) (resp. f(q(t)p_0)) at t=0, where p(t) (resp. q(t)) is the rotated copy of p_0 (resp. q_0) around o by angle t in counterclockwise direction.
Let f : 𝒮→ be a bounded function with f(pp) = 0 for all p ∈^1. Assume that for any p_0q_0∈^1, where p_0 ≠ q_0, (∂_p ∂_q f)(p_0q_0) is a continuous function of p_0q_0 in both variables.
Then, for any x_1, x_2, x_3, x_4 ∈^1 in this counterclockwise order, we have
f(x_1x_3)+f(x_2x_4) ≥ f(x_1x_4)+f(x_2x_3)
if and only if (∂_p ∂_q f)(p_0q_0) ≥ 0 for all p_0 ≠ q_0.
Similarly, for any x_1, x_2, x_3, x_4 ∈^1 in this counterclockwise order, we have
f(x_1x_3)+f(x_2x_4) ≤ f(x_1x_4)+f(x_2x_3)
if and only if (∂_p ∂_q f)(p_0q_0) ≤ 0 for all p_0 ≠ q_0.
We prove only the first part. Assume that (∂_p ∂_q f)(p_0q_0) ≥ 0 for all p_0 ≠ q_0. Let x_2x_3⊂x_1x_4.
Then, by the Newton-Leibniz Theorem we have
0 ≤∫_x_3^x_4∫_x_1^x_2 (∂_p ∂_q f)(p_0q_0) d p_0 d q_0 = f(x_2x_4)-f(x_2x_3)-f(x_1x_4)+f(x_1x_3).
Furthermore, if we have (∂_p ∂_q f)(p_0q_0) < 0 for some p_0 ≠ q_0, then, by continuity and the same argument, there are some points x_1,x_2 and x_3,x_4 sufficiently close to p_0 and q_0, respectively, such that x_2x_3⊂x_1x_4, and 0 > f(x_2x_4)-f(x_2x_3)-f(x_1x_4)+f(x_1x_3).
§ PROOF OF THEOREMS <REF> AND <REF>
Note that by Lemma <ref> and Corollary <ref>, it is sufficient to prove
Theorem <ref> for any everywhere dense subset of _o, and applying a similar consideration, we have the same for Theorem <ref>. Thus, we may assume that C has C^∞-class boundary and strictly positive curvature. Under this condition, the quantities defined in Definition <ref> are continuous functions of K for any fixed value of n, and thus, we may assume that K has C^∞-class boundary, and the curvature of (K) at any point p is strictly greater than the curvature of (C) at the point q with the same outer unit normal as p.
Under the above conditions, for any points p,q ∈ (K), [p,q]_C ∖{ p,q }⊂ (K).
In the proof we identify ^1 with the set / { 2kπ : k ∈ℤ}. Let us parametrize (K) as the curve Γ : ^1 →^2, where the outer unit normal vector at Γ(φ) is (cosφ, sinφ).
Then, for any two points Γ(φ_1), Γ(φ_2) with φ_1 < φ_2 < φ_1+2π, let us denote the arc of Γ connecting them in counterclockwise direction by Γ|_[φ_1,φ_2]. Furthermore, recall <cit.>, stating that K is the intersection of the translates of C containing it. Thus, for any φ∈ [0,2π], there is a unique translate x+C of C containing K with Γ(φ) ∈ (x+C).
We denote this translate by C(φ)=x(φ)+C, and call it the supporting C-disk of K at Γ(φ) (see Figure <ref>).
We define the following regions:
(i) r(φ_1,φ_2) is the closure of the connected component of K ∖ [Γ(φ_1), Γ(φ_2)]_C containing Γ|_[φ_1,φ_2];
(ii) R(φ_1,φ_2) is the closure of the connected component of (C(φ_1) ∩ C(φ_2) ∖ K) containing Γ|_[φ_1,φ_2];
(1) p(φ_1,φ_2) = _C(r(φ_1,φ_2) - _C(Γ|_[φ_1,φ_2]);
(2) A(φ_1,φ_2) = (R(φ_1,φ_2));
(3) P(φ_1,φ_2) = _C(R(φ_1,φ_2) - _C(Γ|_[φ_1,φ_2]).
§.§ The proof of Theorems <ref> and <ref> for Â_n^C(K)
Let I[X] : ^2 → denote the indicator function of X ⊂^2. Then it can be seen directly that for any φ_1 < φ_2 < φ_3 < φ_4 < φ_1+2π, the function
I[R(φ_1,φ_4)] + I[R(φ_2,φ_3)] - I[R(φ_1,φ_3)]- I[R(φ_2,φ_4)]
has nonnegative values at every point. Thus, the conditions of Lemma <ref> are satisfied, implying the statement.
§.§ The proof of Theorems <ref> and <ref> for p̂_n^C(K)
Let φ_1 < φ_2 < φ_3 < φ_4 < φ_1+2π. Then, by Lemma <ref>,
ρ_C(Γ(φ_1),Γ(φ_4))+ρ_C(Γ(φ_2),Γ(φ_3)) ≤ρ_C(Γ(φ_1),Γ(φ_3))+ρ_C(Γ(φ_2),Γ(φ_4)).
Thus, the conditions of Lemma <ref> are satisfied, implying our statement.
§.§ The proof of Theorems <ref> and <ref> for P̂_n^C(K)
By Lemmas <ref> and <ref>, it is sufficient to prove that for any φ_1 < φ_2 < φ_1+π, the function ∂_φ_1∂_φ_2 P is a continuous nonpositive function.
In the remaining part of the subsection we prove this property.
For brevity, for any α < β < α +2π, we define z(α,β) as the intersection point of (C(α)) and (C(β)) contained in the boundary of R(α,β).
First, observe that P(φ_1,φ_2) = ρ_C(Γ(φ_1),z(φ_1,φ_2))+ ρ_C(z(φ_1,φ_2),Γ(φ_2)). Clearly, since C has C^∞-class boundary, ρ_C(·,·) is a C^∞-class function, implying that P(φ_1,φ_2) is C^∞-class, and ∂_φ_1∂_φ_2 P is continuous.
Now, let 0 < | Δ_1| , |Δ_2 | ≤ε for some sufficiently small ε > 0, and set p=z(φ_1,φ_2), q_1=z(φ_1,φ_2+Δ_2), q_2 = z(φ_1 + Δ_1,φ_2) and q=z(φ_1+Δ_1,φ_2+Δ_2).
To prove the assertion, it is sufficient to prove that
0 ≥1/Δ_1( P(φ_1+Δ_1,φ_2+Δ_2)-P(φ_1+Δ_1,φ_2)/Δ_2 - P(φ_1,φ_2+Δ_2)-P(φ_1,φ_2)/Δ_2) =
= 1/Δ_1 Δ_2( P(φ_1+Δ_1,φ_2+Δ_2) - P(φ_1+Δ_1,φ_2) - P(φ_1,φ_2+Δ_2) + P(φ_1,φ_2) ).
We do it in the case that Δ_1 < 0 and Δ_2 > 0, in the other cases a straightforward modification yields the assertion.
Note that in this case it is sufficient to show that
ρ_C(p,q_1)+ρ_C(p,q_2) ≤ρ_C(q,q_1)+ρ_C(q,q_2).
For i=1,2, let v_i denote the tangent vector of C(φ_i) at p pointing `towards' q_i in its boundary, and let w_i denote the tangent vector of K at Γ(φ_i) pointing towards p in (C(φ_i)).
Let C(φ)= x(φ)+C. Then lim_Δ→ 0 ± 0x(φ+Δ)-x(φ)/|x(φ+Δ)-x(φ)| = ± v for any value of φ, where v is the unit tangent vector of (K) at Γ(φ) pointing in the positive direction.
Let Θ(φ) denote the point of (C) with outer unit normal vector (cosφ, sinφ). Then x(φ)=Γ(φ)-Θ(φ) and
more generally,
x(φ+Δ)-x(φ) = ( Γ(φ+Δ)- Γ(φ) ) - ( Θ(φ+Δ)- Θ(φ) ).
Note that lim_Δ→ 0 ± 0Γ(φ+Δ)- Γ(φ)/|Γ(φ+Δ)- Γ(φ)| = lim_Δ→ 0 ± 0Θ(φ+Δ)- Θ(φ)/|Θ(φ+Δ)- Θ(φ)| = ± v, and, by the choice of the parametrization of Γ and Θ, lim_Δ→ 0|Θ(φ+Δ)- Θ(φ)|/|Γ(φ+Δ)- Γ(φ)| = κ_Γ(φ)/κ_Θ(φ), where κ_Γ(φ) and κ_Θ(φ) denote, the curvature of Γ and Θ at Γ(φ) and Θ(φ),respectively. Thus, the assertion follows from our assumption that κ_Θ(φ) ≠κ_Γ(φ).
By Remark <ref>, C(φ_1) ∩ C(φ_2) is the C-spindle of p and another point, which we denote by p'.
By convexity, the tangent vectors of (C(φ_1)) pointing in counterclockwise direction, turn in counterclockwise direction from p to p'. Thus, the directions of the vectors v_2, w_1, v_1 are in this order in counterclockwise orientation, and the same holds for the vectors v_2, w_2, v_1.
For i=1,2, let C(φ_i+Δ_i)=y_i + C(φ_i). Then, by Lemma <ref>, if Δ_i is sufficiently small, we have that the vectors y_1,y_2 are between v_1 and v_2 according to counterclockwise orientation.
Consider the translate C_i' of C(φ_i) by q_i-p. The boundary of this translate contains q_i, and v_i is a tangent vector of C_i' at q_i. Thus, if q' = q_1+q_2-p (i.e. q' is the unique point for which p,q_1,q',q_2 are the vertices of a parallelogram in this counterclockwise order), then q' lies in the boundary of both C_1' and C_2'. On the other hand, by our observation about the tangent lines, if Δ_i are sufficiently small, then q' is contain in Q. By symmetry, ρ_C(p,q_1) = ρ_C(q',q_1) and ρ_C(p,q_2) =ρ_C(q',q_2), and thus, the required inequality follows from the remark after Definition <ref>.
§ PROOF OF THEOREM <REF>
We prove the statement in several steps. For brevity, for any points z_1,z_2, …, z_k ∈^2, we set [z_1,z_2,…,z_k] = { z_1,z_2,…, z_k } and
[z_1,z_2,…,z_k]_C = _C { z_1,z_2,…, z_k }.
Step 1.
Let us fix a Cartesian coordinate system, and consider the points p_1=(0,-1-t), p_2=(2.1,-0.9-t), p_3=(t+2,-1), p_4=(t+2,1), p_5=(2.1, 0.9+t), p_6=(0,1+t), q_1=(t,-1), q_2=(t,1), q_3=(-t,1) and q_4=(-t,-1) (see Figure <ref>). In the construction we assume that t is a sufficiently large positive value.
We define the hexagon H= [p_1,q_1,q_2,p_6,q_3,q_4] and the octagon K_1 = [p_1,p_2,…,p_6,q_3,q_4]. Note that H ⊂ K_1, and set G = (K_1) ∖(H), and G'=(K_1) ∩(H). In the following, D_1 denotes the Euclidean diameter of K_1.
We define C_1 as an o-symmetric convex 14-gon with vertices x_1,x_2,…,x_14 in counterclockwise order such that
(a) x_1 and x_8 are on the negative and the positive half of the y-axis, respectively;
(b) C_1 is symmetric to both coordinate axes;
(c) the sides [x_1,x_2], [x_2,x_3], [x_3,x_4], [x_4,x_5] are parallel to [p_1,p_2], [p_1,p_3], [p_2,p_3] and [p_3,p_4], respectively;
(d) we have ||x_2-x_1||, ||x_3-x_2||, ||x_4-x_3|| > D_1, and ||x_5-x_4||=2, i.e. [x_4,x_5] is a translate of [p_3,p_4].
Note that by our conditions, for any two point u,v ∈ G, each of the two C_1-arcs in the boundary of [u,v]_C_1 consists of translates of subsets of at most two consecutive sides of C_1, or they contain translates of [x_4,x_5] and possibly translates of subsets of the sides [x_3,x_4] and [x_5,x_6].
In particular, [p_1,p_6]_C_1 = H.
We estimate ([p_1,q,p_6]_C_1) for any q ∈ G with nonnegative y-coordinate. In the following p̅=(0,t+2) denotes the midpoint of [p_3,p_4].
Case 1: q ∈ [p̅,p_4]. Then ([p_1,q,p_6]_C_1) consists of G', parts of the segments [p_1,p_3] and [p_4,p_6], and two segments with q as an endpoint, parallel to [p_2,p_3] and [p_4,p_5], respectively. Thus, ([p_1,q,p_6]_C_1) is maximal if q=p̅, implying that
([p_1,q,p_6]_C_1) ≤([p_1,p̅,p_6]_C_1) = (H)+3/2t + 3
Case 2: q ∈ [p_4,p_5]. Assume that the x-coordinate of q is at least t+1. Then the curve ([p_1,q,p_6]_C_1) consists of G', a segment containing [p_1,q_1], a segment parallel to [p_3,p_4] and ending at q, and segment parallel to [p_4,p_6] and ending at q, and a subset of [p_5,p_6]. Observe that if t is sufficiently large, in this case ([p_1,q,p_6]_C_1) is maximal if the x-coordinate of q is equal to t+1. A similar consideration shows that if the x-coordinate of q is at most t+1, then ([p_1,q,p_6]_C_1) is maximal if q=p_5. Thus, in Case 2 we have
([p_1,q,p_6]_C_1) ≤([p_1,p_5,p_6]_C_1) =
= (H)+ ([q_2,p_4,p_5,p_6)=1/2( (H)+(K_1) ) - 2
Case 3: q ∈ [p_5,p_6]. Then ([p_1,q,p_6]_C_1) consists of G', a segment parallel to [q_2,p_6] and ending at q, a segment containing [p_1,q_1] as a subset, and a translate of [p_3,p_4]. Thus, in this case ([p_1,q,p_6]_C_1) is maximal if q=p_5, and we have
([p_1,q,p_6]_C_1) ≤([p_1,p_5,p_6]_C_1) = 1/2( (H)+(K_1) ) - 2.
Combining our results, if t is sufficiently large, for any q,q' ∈ G
([p_1,q,p_6]_C_1) + ([p_1,q',p_6]_C_1) ≤(H)+(K_1) - 4 <
< (H)+(K_1) = ([p_1,p_6]_C_1)+([p_1,p_2,p_5,p_6]_C_1),
where we used the observation that [p_1,p_2,p_5,p_6]_C_1 = K_1.
In the remaining part of the construction, we fix t in such a way that (<ref>) is satisfied.
Step 2.
In the next step, based on Step 1, we construct some C_2 ∈_o and a C_2-convex disk K_2 such that
â_3^C_2(K_2) + â_5^C_2(K_2) > 2 â_4^C_2(K_2).
Let p_7 = (-s,0), where s is sufficiently large, and set K_2 = (K_1 ∪{ p_7 }) (see Figure <ref>). Let D_2 denote the Euclidean diameter of K_2, and let C^+_1 (resp. C^-_1) denotes the set of the points of (C_1) with nonnegative (resp. nonpositive) x-coordinates.
We define C_2 as follows:
(a) C_2 is symmetric to both coordinate axes.
(b) (C_2) contains some translates u+ C^+_1 and -u+C^-_1, where u points in the direction of the positive half of the x-axis. We set w_3=u+x_1.
(c) In addition to the above two translates, (C_2) consists of segments [w_1,w_2], [w_2,w_3] and their reflections about one or both of the coordinate axes, such that [w_1,w_2], [w_2,w_3] are parallel to [p_6,p_7] and [p_5,p_7], respectively, and |w_1-w_2|, |w_2-w_3| > D_2.
We remark that if s is sufficiently large, then there is some C_2 ∈_o satisfying the above conditions, and K_2 is C_2-convex.
In the following, let Q_4 = [z_1,z_2,z_3,z_4]_C_2 denote a maximal area C-4-gon inscribed in K_2.
Let H'= (H ∪{ p_7 }) =[p_1,p_6,p_6]_C_2 and observe that K_2 = [p_1,p_2,p_5,p_6,p_7]_C_2. Then, to show the inequality in (<ref>), it is sufficient to show that (H')+(K_2) > 2 (Q_4).
Let Q = [p_1,p_5,p_6,p_7]_C_2. By the consideration in Step 1, we have that (Q) = 1/2 ((H')+(K_2))-2. Thus, we have (Q_4) ≥1/2 ((H')+(K_2))-2.
Let us define the points v_1 and v_6 as the images of p_1 and p_6, respectively, under the homothety with center p_7 and homothety ratio 1/√(s). An elementary computation shows that then v_1 = ( -(1-1/√(s))s, -1+t/√(s)) ∈ [p_1,p_7] and v_6 = ( -(1-1/√(s))s, 1+t/√(s)) ∈ [p_6,p_7]. Note that since |v_2-v_1| = 2(1+t)/√(s) < 2 if s is sufficiently large, and (C_2) contains two vertical segments of length 2, we may assume that [v_1,v_6]_C_2 = [v_1,v_6]. In other words, we may assume that there is a translate of C that contains K_2 ∖ [v_1,p_7,v_6] and does not overlap [v_1,p_7,v_6]. Thus, if z_i ∉ [v_1,p_7,v_6] for any 1 ≤ i ≤ 4, then Q_4 ⊆ K_2 ∖ [v_1,p_7,v_6], implying that in this case (Q_4) ≤(K_2) - ([v_1,p_7,v_6]) = (K_2) - 2 √(s)(1+t) < 1/2 ((H')+(K_2))-2; a contradiction. Consequently, in the following we may assume that z_4 ∈ [v_1,p_7,v_6].
Let v'_5 and v'_7 be the images of p_5 and p_7, respectively, under the homothety with center p_6 and ratio 1/√(s). Note that since there is a side of C parallel to [v_5',v_7'], we have [v_5',v_7']_C_2= [v_5',v_7'], and, as in the previous paragraph, if z_i ∉ [v_1,p_7,v_6] for any 1 ≤ i ≤ 4, then (P_4) ≤(K_2) - ([v_5',v_7',p_6]). On the other hand, we have |p_6-p_7| > s and that the length of the corresponding height of [p_5,p_6,p_7] is greater than 0.1 by the definition of p_5. Thus, ([v_5',v_7',p_6])=([p_5,p_6,p_7])/√(s^2) > 0.1 √(s), implying that since (Q_4) ≥(Q), which otherwise by our inequalities does not hold if s is sufficiently large, we may assume that some z_i, say z_3, is an element of [v_1,p_7,v_6].
We obtain similarly that if s is sufficiently large, some z_i, say z_1, is contained in the triangle [v_7”,p_1,v_2”], where v_7” and v_2” are the images of p_7 and p_2, respectively, under the homothety with center p_1 and ratio 1/√(s). These observations, the consideration in Step 1, and the inequality (Q_4) ≥(Q) yield that as s →∞, we have z_1 → p_1, z_3 → p_6 and z_4 ∈ [v_1,p_7,v_6], and min{ | z_2 - p_2|, |z_2-p_5| }→ 0, implying that in this case (Q_4) →(Q). This shows that if s is sufficiently large, then (H')+(K_2) > 2 (Q_4).
Before proceeding to the final step, we make two important observations that we are going to use. Here, by C^+_2 and C^-_2, we denote the parts of (C_2) contained in the closed half planes { x ≥ 0} and { x ≤ 0}, respectively.
(1) A straightforward modification of the construction in Step 2 yields, for any n ≥ 4, the existence of some C_n ∈_0 and a C_n-convex disk K_n such that â_n-1^C_n(K_n) + â_n+1^C_n(K_n) > 2 â_n^C_n(K_n).
(2) To guarantee the required inequalities in Steps 1 and 2, we used the properties of the arcs of C_2 entirely contained in C^+_2 or C^-_2. Thus, if C_2' is an o-symmetric plane convex body containing C^+_2 and C^-_2 in its boundary, then we have
â_3^C_2'(K_2) + â_5^C_2'(K_2) > 2 â_4^C_2'(K_2).
We combine these two observations in the following remark.
For any n ≥ 4, there is some C_n ∈_o and a C_n-convex disk K_n such that if any C_n' ∈_o contains C_n^+ and C_n^- in its boundary, where by C^+_n and C^-_n, we denote the parts of (C_n) contained in the closed half planes { x ≥ 0} and { x ≤ 0}, respectively, then K_n is C_n'-convex, and
â_n-1^C_n'(K_n) + â_n+1^C_n'(K_n) > 2 â_n^C_n'(K_n).
Step 3.
Now we prove Theorem <ref>. Let n ≥ 4. Recall that ^n_a denotes the elements C of _o such that for any C-convex disk K, we have â_n-1^C(K) + â_n+1^C(K) ≤ 2 â_n^C(K), and set ^n_a = _o ∖^n_a.
Observe that by Lemma <ref>, ^n_a is open. We show that it is everywhere dense in _o.
Let C be an arbitrary element of _o and let ε > 0. Note that for any nondegenerate linear transformation h : ^2 →^2, K is C-convex if and only if h(K) is h(C)-convex, and for any n ≥ 4, if K is C-convex, then â_n^C(K) = â_n^h(C)(h(K)).
Thus, without loss of generality, we may assume that there are vertical supporting lines of C meeting (C) at some points ± p of the x-axis. We choose our notation such that p is on the positive half of the axis.
Consider the convex disk C_n ∈_0 in Remark <ref>. Let us define the nondegenerate linear transformation h_λ, μ : ^2 →^2 by h_λ,μ(x,y)=(λ x, μ y). Then, if we choose suitable sufficiently small values μ, λ > 0, then there is a translate C^+ of h_λ,μ(C^+_n), and an o-symmetric convex disk C' containing C^+ in its boundary such that C^+ ⊂ (C+ ε B^2) ∖ C, and C ⊂ C'. Then C' ∩ (C+ ε B^2) ∈_o contains translates of h_λ,μ(C^+_n) and h_λ,μ(C^-_n) in its boundary, the Hausdorff distance of C and C' is at most ε, and, if we set K'=h_λ,μ(K_n), by Remark <ref> we have
â_n-1^C'(K') + â_n+1^C'(K') > 2 â_n^C'(K').
Thus, ^n_a is everywhere dense, which immediately yields that ⋂_n=4^∞^n_a is residual, implying Theorem <ref>.
§ REMARKS AND QUESTIONS
For C ∈_o, K ∈ and positive integer n ≥ 3, let
P̅_n^C(K) = inf{_C(Q) : Q is a convex n- gon circumscribed about K };
p̅_n^C(K) = sup{_C(Q) : Q is a convex n- gon inscribed in K }.
As we have observed in the introduction, it is known <cit.> that for any C ∈_o and K ∈, the sequences {P̅_n^C(K) } and {p̅_n^C(K) } are convex and concave, respectively. Our approach yields a new proof of these statements by applying Theorem <ref> for λ C, where λ→∞.
Applying Theorem <ref> for λ C with λ→∞, we obtain the following.
Let C ∈_o, K ∈ and n ≥ 3. If, for some positive integer k,
Let C ∈_o, K ∈, n ≥ 3 and k ≥ 2. Assume that k is a divisor of n and both K and C have k-fold rotational symmetry. Then there is a convex n-gon Q^P circumscribed about K with _C(Q^P)= P̅_n^C(K) such that Q^P has k-fold rotational symmetry. Similarly, there is a convex n-gon polygon Q^p inscribed in K which has k-fold rotational symmetry, and _C(Q^p)= p̅_n^C(K).
In the remaining part of the paper, we denote the set (1,∞) ∪{∞} by [1,∞].
Let p,q ∈ [1,∞] satisfy the equation 1/p + 1/q = 1. For any K, L ∈, G. Fejes Tóth <cit.> introduced
the weighted area deviation of K,L with weights p,q as the quantity ^p,q(K,L)=p (K ∖ L) + q (L ∖ K).
He proved that if for any K ∈, a̅_K^C(n,p,q) denotes the minimal weighted area deviation of K and an arbitrary convex n-gon, then the sequence {a̅_K^C(n,p,q) } is convex. Based on this idea, we introduce the following quantity.
Let p,q ∈ [1,∞] satisfy the equation 1/p + 1/q = 1, and let C ∈_0, K ∈_0.
We call the quantity
_C^p,q(K,L) = p ( _C((K) ∖(L))- _C((L) ∩ K) ) +
+ q ( _C((L) ∖(K)) - _C ((K) ∩ L) )
the weighted C-perimeter deviation of K,L with weights p,q. Here we note that by convexity,
_C((K) ∖(L)) ≥_C((L) ∩ K) and _C((L) ∖(K)) ≥_C ((K) ∩ L), with equality if and only if K ⊆ L and L ⊆ K, respectively. Let p̅_K^C(n,p,q) denote the minimal C-perimeter deviation of K and an arbitrary convex n-gon. We remark that if K is C-convex, by replacing the convex n-gons in the definitions of a̅_K^C(n,p,q) and p̅_K^C(n,p,q) with C-n-gons, we may analogously define the quantities â_K^C(n,p,q) and p̂_K^C(n,p,q), respectively.
This leads to the following problems.
Prove or disprove that for any p,q ∈ [1,∞ ] with 1/p + 1/q = 1, C ∈_o and K ∈, the sequence {p̅_K^C(n,p,q) } is convex.
Prove or disprove that for any p,q ∈ [1,∞ ] with 1/p + 1/q = 1, C ∈_o and C-convex disk K ∈, the sequence {p̂_K^C(n,p,q) } is convex. Does the same hold for {â_K^C(n,p,q) } if C is the Euclidean unit disk?
Before our last problem, we remark that â_K^C(n,1, ∞) = (K) - â_K^C(n) and â_K^C(n,∞,1) = Â_K^C(n)-(K).
Is there a value p_0 ∈ (1,∞) such that for any p with p_0 < p ≤∞ and q satisfying 1/p + 1/q = 1, for any C ∈_o and C-convex disk K ∈, the sequence {â_K^C(n,p,q) } is convex?
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|
http://arxiv.org/abs/2307.07227v1 | 20230714084706 | Secure Short-Packet Communications via UAV-Enabled Mobile Relaying: Joint Resource Optimization and 3D Trajectory Design | [
"Milad Tatar Mamaghani",
"Xiangyun Zhou",
"Nan Yang",
"A. Lee Swindlehurst"
] | cs.IT | [
"cs.IT",
"eess.SP",
"math.IT"
] |
IEEEexample:BSTcontrol
Secure Short-Packet Communications via UAV-Enabled Mobile Relaying: Joint Resource Optimization and 3D Trajectory Design
Milad Tatar Mamaghani0000-0002-3953-7230, Member, IEEE, Xiangyun Zhou0000-0001-8973-9079, Fellow, IEEE
Nan Yang0000-0002-9373-5289, Senior Member, IEEE, and A. Lee Swindlehurst0000-0002-0521-3107, Fellow, IEEE
This work was supported by the Australian Research Council’s Discovery Projects funding scheme (DP220101318).
M. Tatar Mamaghani, X. Zhou, and N. Yang are with the Research School of Engineering, Australian National University, Canberra, ACT 2601, Australia (email: mailto:[email protected]@anu.edu.au; mailto:[email protected]@anu.edu.au; mailto:[email protected]@anu.edu.au).
A. L. Swindlehurst is with the Center for Pervasive Communications and
Computing, Henry Samueli School of Engineering, University of California, Irvine, CA 92697, USA (email: mailto:[email protected]@uci.edu).
August 12, 2023
========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
Short-packet communication (SPC) and unmanned aerial vehicles (UAVs) are anticipated to play crucial roles in the development of 5G-and-beyond wireless networks and the Internet of Things (IoT). In this paper, we propose a secure SPC system, where a UAV serves as a mobile decode-and-forward (DF) relay, periodically receiving and relaying small data packets from a remote IoT device to its receiver in two hops with strict latency requirements, in the presence of an eavesdropper. This system requires careful optimization of important design parameters, such as the coding blocklengths of both hops, transmit powers, and UAV’s trajectory. While the overall optimization problem is nonconvex, we tackle it by applying a block successive convex approximation (BSCA) approach to divide the original problem into three subproblems and solve them separately. Then, an overall iterative algorithm is proposed to obtain the final design with guaranteed convergence. Our proposed low-complexity algorithm incorporates 3D trajectory design and resource management to optimize the effective average secrecy throughput of the communication system over the course of UAV-relay’s mission. Simulation results demonstrate significant performance improvements compared to various benchmark schemes and provide useful design insights on the coding blocklengths and transmit powers along the trajectory of the UAV.
Short-packet transmissions, machine-type communications, UAV-aided relaying, physical-layer security, 3D trajectory design, resource management, and convex optimization.
§ INTRODUCTION
Short-packet communication (SPC) has recently emerged as a critical communication paradigm for meeting the stringent demands of massive machine-type communications (mMTC) and ultra-reliable and low-latency communications (uRLLC) for the support of Internet of Things (IoT) in 5G-and-beyond wireless networks <cit.>. Indeed, unlike conventional wireless communications, SPC generally involves the transmission of small data packets whose size can potentially go down to tens of bytes, wherein the length of control information is comparable with that of the data payload. This approach is gaining popularity in the context of IoT, since there is a growing demand for real-time and efficient communication between devices. The transmission of short packets also plays an important role in mission-critical applications such as intelligent transportation, telemedicine, and industrial automation, where stringent latency requirements are in place <cit.>.
One of the critical challenges of SPC is guaranteeing communication reliability, considering the fact that utilizing short packets for transmissions is inherently accompanied by severely degraded channel coding gain <cit.>. In addition, SPC in IoT networks encounters severe security challenges, primarily due to the broadcast nature of wireless media as well as the mission-critical and private SPC data that IoT networks often need to share <cit.>. SPC-IoT communication systems are vulnerable to eavesdropping. For example, in the case of intelligent transportation, if a message is intercepted by an adversary, private information such as user identity or location may be exposed. Thus, ensuring both the stringent reliability and security of SPC-IoT systems presents significant challenges. While upper-layer encryption techniques have typically been used to strengthen the security of communication systems, they are challenging to implement in 5G-and-beyond IoT systems due to the deployment of a huge number of nodes in a dynamic heterogeneous environment, which makes cryptography key management difficult and unscalable. Also, resource-constrained IoT nodes have practical constraints such as limited energy, leading to the use of lightweight secrecy protocols
to reduce the demand for resources. To this end, various physical-layer security (PLS) techniques have been investigated to provide protection through wiretap coding and smart signaling by exploiting the physical-layer properties of wireless channels <cit.>.
Nonetheless, achieving the required secrecy by applying PLS techniques is nontrivial for mission-critical SPC-IoT applications. First, conventional PLS schemes and the adopted wiretap codes are primarily based on the assumption of infinite blocklength codes whose packet lengths are on the order of several kilobytes, as opposed to tens of bytes for SPC. Secondly, PLS-based designs for traditional large-packet communications are generally based on the so-called Secrecy Capacity <cit.>, from Shannon's classical information theory. However, secrecy capacity is not applicable for SPC scenarios due to the finite blocklength assumption, and therefore it is not appropriate to design classical PLS approaches for SPC scenarios since it could lead to suboptimal solutions. This calls for rethinking the analysis and design of PLS for the SPC-IoT systems.
Concurrently, unmanned aerial vehicles (UAVs) have recently become an increasingly popular technology
for a variety of wireless and IoT applications. Particularly, UAVs are envisioned to be integrated into IoT systems to provide a range of benefits such as aerial base station (BS) or mobile relaying, remote sensing and processing, real-time monitoring, etc., due to their relatively rapid on-demand deployment, inexpensive operation, low-cost maintenance, and flexible three-dimensional (3D) mobility <cit.>. One of the main advantages of UAVs in IoT systems is their ability to gather data from difficult-to-reach geographic areas or hazardous locations. Also, UAVs are particularly useful in applications where timely data transmission is critical, such as emergency response and disaster relief efforts. Since they operate above ground away from obstacles, UAVs are effective at reducing signal attenuation in wireless links arising from shadowing and blockage effects. Accordingly, battery-limited IoT devices, enjoying favorable air-ground (AG) channel conditions, need significantly lower power to transmit data to UAVs, leading to a significant increase in their lifespan. However, security challenges such as eavesdropping for UAV-aided IoT communications are of significant importance and need to be meticulously addressed <cit.>.
§.§ Prior Studies and Motivation
To safeguard different wireless networks, PLS techniques have been extensively studied in the literature for various applications such as cooperative non-orthogonal multiple access (NOMA) networks <cit.>, opportunistic relaying in IoT networks <cit.>, full-duplex relaying <cit.>, energy harvesting-based device-to-device communications <cit.>, and UAV-aided secure wireless communication <cit.>. However, previous research has often considered the design and optimization of infinite blocklength transmissions, which as mentioned earlier are not suitable for SPC-IoT scenarios due to their severely reduced channel coding gain and the impossibility of guaranteeing error-free transmissions. Recently, there has been significant research interest in exploring to what extent finite blocklength transmissions incur a loss in capacity <cit.>. Yang et al. in their seminal work <cit.> addressed the achievable secrecy rate (SR) for a general wiretap channel given fixed reliability and secrecy constraints in the finite blocklength regime, and analytically derived tight achievability and converse bounds on the maximum secret communication rate. Following these foundational results, the authors in <cit.> studied the design of PLS in SPC over fading channels by investigating secrecy throughput for both single and multiple antenna transmitters. In <cit.>, the authors investigated a setting similar to <cit.> from another perspective, where they defined a new outage probability metric relevant to the characteristics of SPC. In <cit.>, the authors studied secure SPC in a mission-critical IoT system in the presence of an external multi-antenna eavesdropper. Further, <cit.> explored secure transmission rate for downlink SPC with a queuing delay requirement under different assumptions on the channel state information (CSI), where the authors derived closed-form optimal power control policies for some special scenarios. We stress that the previous studies have considered system design with only static communication nodes and adopted a fixed number of information bits per short-packet transmission. Thus, the frameworks developed in <cit.> may not be applicable for highly dynamic UAV-IoT scenarios or when a variable amount of data is generated from IoT devices for transmission.
In light of this, some recent research efforts have considered UAV-IoT network designs with downlink SPC, e.g., <cit.>. Nevertheless, blocklength optimization is completely ignored in the aforementioned works due to the assumption of simple one-hop transmissions. As a result, further research is necessary to fully understand the performance of SPC in UAV-IoT networks.
§.§ Our Contributions
Inspired by the aforementioned research, in this work we propose a secure UAV-aided relaying scheme in which sensitive short packets need to be periodically transmitted from a remote IoT device to its receiver with a stringent latency requirement while combating a passive eavesdropper in an uncertain location. Our contributions are detailed below.
* We formulate a new optimization problem in terms of the effective average
secrecy throughput (EAST) for the considered UAV-aided SPC system under security, reliability, latency, and mobility constraints. The formulated problem is nonconvex with a nondeterministic objective function, and hence challenging to solve optimally.
* To tackle the challenging nonconvex problem, we derive an analytical and tractable lower-bound expression for the objective function, and then propose a computationally efficient algorithm based on the block successive convex approximation (BSCA) to iteratively solve a sequence of convex subproblems: joint power allocation, coding blocklength optimization, and 3D trajectory design. We then propose a low-complexity algorithm combining these solutions to optimize the EAST performance, and we discuss its complexity and convergence properties.
* We conduct extensive simulations to draw useful insights into the performance of the proposed joint resource allocation and trajectory design, and highlight its benefits compared to other benchmarks. We observe that the joint trajectory and resource optimization can significantly improve the EAST performance compared to designs with either fixed trajectory or predetermined communication power and coding blocklengths.
* We investigate the impact of key system parameters on the overall system performance. In particular, we find that in our joint design, both the transmit power and uplink coding blocklength adaptively increase when the mobile UAV-assisted relay approaches a location between the transmitter and receiver, while the relaying power maintains its maximum value with a decreasing downlink coding blocklength for the sake of EAST enhancement.
The remainder of this paper is organized as follows. Section <ref> introduces our proposed UAV-aided relaying system model with imperfect location information about the eavesdropper, followed by the formulation of an optimization problem to improve the overall system performance in Section <ref>. In Section <ref>, we present an efficient approach to tackle the optimization problem. Section <ref> discusses selected numerical results and the impact of key system parameters. Finally, conclusions are drawn in Section <ref>.
§ SYSTEM MODEL AND PROBLEM STATEMENT
We consider a UAV-assisted IoT communication system with secure SPC as illustrated in Fig. <ref>, wherein a transmitter (Alice) periodically sends short packets containing confidential information to a designated remote receiver (Bob) with the help of a trusted mobile UAV-Relay (UR) while a malicious passive eavesdropper (Eve), whose location is not perfectly known, attempts to overhear the ongoing confidential transmissions. Here, SPC is considered due to its importance for various delay-sensitive IoT applications. In practice, Alice can be considered as an IoT device that periodically generates a short packet of sensitive information from the environment and, if feasible, immediately transmits it to a desired remote receiver Bob with a stringent latency requirement for monitoring, controlling or sensing applications. We refer to a timeslot as a period of δ_t seconds, and assume that a packet is generated at the beginning of each timeslot. In this work, we allow Alice to generate and transmit a variable amount of information bits in each timeslot to accommodate different tasks, or to provide a suitable amount of information to Bob according to the communication channel conditions and other requirements.
§.§ Periodic Secure Short-packet Relaying
We assume that there exists no direct link between Alice and Bob due to the distance or blockages, and thus, a UAV-mounted relay, thanks to its flexible mobility and line-of-sight (LoS)-dominant AG channels, is employed to facilitate the end-to-end SPC. We assume that all the communication nodes are equipped with only a single antenna, as commonly done for resource-constrained IoT devices. In addition, the mobile relaying strategy adopted by the UR is assumed to be the decode-and-forward (DF) protocol with time division duplexing (TDD) on a shared bandwidth W for both reception and transmission. Compared with amplify-and-forward (AF) relaying <cit.>, DF relaying is beneficial for the considered UAV-aided scenario since Bob is not required to obtain CSI for the first hop, which in turn results in lower overhead, and DF relaying avoids noise amplification, leading to generally better signal reception quality at Bob.
We assume that the UR-aided DF relaying for SPC occurs at the beginning of each timeslot δ_t, as illustrated in Fig. <ref>. Recall that DF relaying consists of two phases. In the first phase (i.e., uplink transmission), Alice transmits one short packet containing sensitive information to the UR over l_u[n] channel uses, where n={1, 2, ⋯} denotes the index of each timeslot, and then the UR decodes the received signal to obtain the transmitted confidential message. In the second phase (i.e., downlink transmission), the UR encodes the message with a different codebook for security purposes, forwarding it to the desired destination Bob over l_d[n] channel uses and Bob retrieves the original confidential information. In the meantime, Eve wiretaps the ongoing transmissions to obtain the confidential data and thus poses a security threat.
Note that end-to-end SPC generally occupies much less time than one timeslot, i.e. δ_i[n] ≪δ_t ∀ n and i∈{u, d}, where δ_u[n] and δ_d[n] indicate the duration required by a finite blocklength SPC for uplink and downlink, respectively. Since the time taken for one channel use is inversely proportional to the available bandwidth, we have δ_i[n] = l_i[n]/W ∀ n.
§.§ System Assumptions and Constraints
Without loss of generality, we consider a 3D Cartesian coordinate system, where Alice, Bob, and Eve are respectively located on the ground at _a=[x_a,y_a,0]^T, _b=[x_b,y_b,0]^T, and _e=[x_e,y_e,0]^T, where [·]^T represents the transpose. While the location of the legitimate network nodes is perfectly known, Eve attempts to hide her location via passive eavesdropping. Nonetheless, we assume that an approximate estimate of Eve's location[Some information may be available about Eve's location due to geographical constraints, prior information about adversarial operations, or unintended emissions from Eve's RF electronics.]
within a given uncertainty region can be obtained and tracked such that
_e - _e≤Δ_e,
where _e is an estimate of Eve's 3D location within a sphere of radius Δ_e, and · represents the Euclidean norm. In practice, the uncertainty region is usually smaller than the distance between Alice-Eve and Bob-Eve pairs, i.e., _j - _e≥Δ_e, where j∈{a, b}.
§.§.§ UAV trajectory constraints
We assume that the UR's flight time horizon is set to T seconds, and is divided into
N timeslots such that T= N δ_t, and the timeslots are indexed by n∈={1, 2, ⋯, N}. Since the SPC duration is small, we can assume that the UR's location over the transmission phase in each timeslot remains approximately unchanged, but varies from one timeslot to another.
Therefore, the UR's 3D location in timeslot n can be denoted by _r[n]=[x[n], y[n], z[n]]^T ∀ n∈. With this setting, the UR's continuous trajectory can be approximated by the 3D (N+1)-waypoint sequence {_r[n]}^N_n=1. UAVs can generally control their horizontal and vertical speeds independently <cit.>. Thus, assuming that the UR's initial and final locations are denoted by _i=[x_i, y_i, z_i]^T and _f=[x_f, y_f, z_f]^T, respectively,
the following mobility constraints are imposed on the UR's 3D trajectory:
1: _r[1] = _i, _r[N] = _f,
2: √((x[n+1]-x[n])^2+(y[n+1]-y[n])^2)≤ v^max_xyδ_t, ∀ n∈∖ N
3: z[n+1] - z[n] ≤ v^max_zδ_t, ∀ n∈∖ N
4: H^min≤ z[n] ≤ H^max, ∀ n∈
where 2 and 3 limit horizontal and vertical displacements of the UR for consecutive timeslots, and v^max_xy and v^max_z denote the maximum velocity of the UR in the horizontal and vertical directions, respectively. Furthermore, 4 indicates that the altitude of the UR needs to be larger than a minimum required height H^min, to avoid collision with buildings or other obstacles <cit.>, and smaller than a maximum permitted height H^max.
§.§.§ Channel modeling
The AG channels are assumed to be dominated by path-loss with negligible fading <cit.>. Thus, for the Alice-UR link, the UR-Eve link, and the UR-Bob link, denoted as h_ra[n], h_re[n], and h_rb[n] ∀ n, respectively, we express their LoS-dominant channel power gains as
h_rj[n]=β_0/_r[n] - _j^2, ∀ n∈, j ∈{a, e, b}
where β_0 denotes the path-loss at a reference distance d_0 under omnidirectional propagation, and is given by β_0=(C/4π d_0 f_c)^2, with C being the speed of light and f_c denoting the carrier frequency. Furthermore, since both Alice and Eve are terrestrial nodes, the channel model for the Alice-Eve link constitutes both distance-dependent attenuation and small-scale fading <cit.>, the power gain of which can be represented as
h_ae[n]= β_0/_a - _e^αζ[n], ∀ n∈
where ζ[n] is a unit-mean exponential random variable, i.e., ζ[n] ∼𝐄𝐱𝐩(1), that accounts for independent and identically distributed (i.i.d) Rayleigh fading, and α is the corresponding environmental path-loss exponent, with a typical range between 2 < α≤ 4.
Remark: In this work, we assume that h_ra[n] and h_rb[n] ∀ n are perfectly known by Alice and the UR, respectively, through channel reciprocity and training. However, it is assumed that only statistical information about Eve's channel is available at the legitimate transmitters. We further assume that Eve has perfect knowledge of the CSI from Alice and the UR to herself, i.e., h_ae[n] and h_re[n] ∀ n, respectively, which can be considered as a worst-case scenario from the point of secrecy.
§.§.§ Constraints on radio resources and latency
Since the channel conditions and the UR's location are assumed to remain stable during the short-packet transmission in each timeslot but can change between timeslots,
we allow the transmit power to be chosen according to the channel condition of a timeslot. In other words, the transmit power of Alice and the UR can change from one timeslot to another. Furthermore, we impose a total power constraint over all timeslots for both Alice and the UR. Accordingly, denoting {p_a[n], ∀ n} and {p_r[n], ∀ n} as the transmit power per channel use in timeslot n for Alice and the UR, respectively, the total power budget constraints can be expressed as
5: ∑_n=1^Np_a[n]l_u[n] ≤P^tot_a,
6: ∑_n=1^Np_r[n]l_d[n] ≤P^tot_r,
where P^tot_a and P^tot_r represent the maximum available power budget for Alice and the UR over the mission time, respectively. The constraints (<ref>) and (<ref>) ensure that the amount of energy utilized to transmit short packets of confidential information by Alice and the UR will be sufficient for the duration of the mission. In addition, it is common practice to design SPC systems with appropriate power control mechanisms to limit the transmit power in each timeslot, i.e., within the peak power limit, which guarantees the reliable and safe operation of the communication system <cit.>. Accordingly, we further adopt the following peak power constraints:
7: 0 ≤ p_a[n] ≤ P^max_a, ∀ n∈
8: 0 ≤ p_r[n] ≤P^max_r, ∀ n∈
where P^max_a and P^max_r represent the maximum peak power for Alice and the UR, respectively.
In SPC, delay tolerance is often a critical factor since, to be useful, the communication system must deliver sensitive information within a short deadline. On the other hand, a larger blocklength results in a longer transmission duration and increased delay. So, for the considered short-packet delay-sensitive system, the requirement on delay tolerance can be imposed by constraining the number of total blocklengths per transmission to be less than a maximum allowable end-to-end delay, which can be expressed mathematically as
9: ∑_i l_i[n] ≤ L^max, i∈{u, d}, ∀ n∈
10: l_i[n] ∈ℤ^+, i∈{u, d}, ∀ n∈
where L^max denotes the maximum end-to-end latency.
§.§ Secrecy Metric for SPC
Human-centered secure communication systems usually assume channel coding with a sufficiently large (infinite) blocklength, and then consider the so-called secrecy capacity as the performance metric. With positive secrecy capacity, the legitimate source-destination pair can achieve perfectly reliable and secure communications based on the well-known wiretap coding scheme <cit.>. On the other hand, for machine-type SPC, perfect secrecy cannot be guaranteed due to the assumption of finite blocklength transmissions. In light of this, <cit.> addressed the achievable SR for SPC given the legitimate receiver's decoding error probability and tolerable information leakage to the illegitimate receiver, which we will exploit in the following analysis.
§.§.§ SR for uplink SPC
In the uplink transmission of timeslot n, Alice generates a short packet and transmits it over l_u[n] channel uses to the UR, while Eve attempts to passively overhear the transmission. According to <cit.>, given a reliability constraint on the UR's decoding error probability, denoted by ε_r, and a security constraint in terms of information leakage to Eve, denoted by η_e, we express the achievable average SR in bits per channel use for the finite blocklength uplink transmissions in timeslot n, denoted by R^sec_u[n], as
R^sec_u[n] = _ζ[n]{[C^sec_u[n]- √(V(γ_r[n])/l_u[n])( ε_r) - √(V(γ_ae[n])/l_u[n])( η_e) ]_+}, ∀ n∈
where C^sec_u[n]=log_2(1+γ_r[n]/1+γ_ae[n]) indicates the uplink secrecy capacity with infinite blocklength in timeslot n,
[x]_+=max{x,0}, _x{·} indicates expectation over the random variable x, and (x) is the inverse of the complementary Gaussian cumulative distribution function Q(x), defined as Q(x)=∫^∞_x1/√(2π)^-r^2/2dr. Further, γ_r[n] and γ_ae[n], denoting the received signal-to-noise ratios (SNRs) at the UR and Eve in timeslot n, are given respectively by
γ_r[n] = p_a[n]h_ra[n]/σ^2_r[n] = p_a[n]ρ_r[n]/_r[n] - _a^2, ∀ n∈
γ_ae[n] = p_a[n]h_ae[n]/σ^2_e[n] = p_a[n]ρ_e[n]/_a - _e^αζ[n], ∀ n∈
where σ^2_r[n] and σ^2_e[n] denote the additive white Gaussian noise (AWGN) power at the UR and Eve in timeslot n, respectively, and ρ_r[n] = β_0/σ^2_r[n] and ρ_e[n] = β_0/σ^2_e[n]. Furthermore, the function V(·) indicates the statistical variation of the channel (a.k.a the channel dispersion), which can be mathematically expressed, according to <cit.>, as
V(γ) = log^2_2()[1-(1+γ)^-2], ∀ n∈
As such, V(γ_r[n]) and V(γ_ae[n]) represent the corresponding channel dispersion in timeslot n for the Alice-UR and Alice-Eve's links, respectively.
Note that the channel dispersion is a monotonically increasing function of SNR.
§.§.§ SR for downlink SPC
In the downlink transmission of timeslot n, the UR decodes the confidential information bits transmitted by Alice, encodes them with a different codebook, and forwards the generated short packet towards Bob over l_d[n] channel uses. While Eve can still wiretap the transmissions from the UR, since the signals from both Alice and the UR have been encoded with different codebooks, she cannot take advantage of a diversity combining strategy by performing, for example, maximum ratio combining (MRC) to improve her reception quality. Accordingly, given Bob's decoding error probability ε_b, the achievable SR for short-packet downlink transmissions in timeslot n, denoted by R^sec_d[n], is given by
R^sec_d[n] = [ C^sec_d[n]- √( V(γ_b[n])/l_d[n])( ε_b) - √( V(γ_re[n])/l_d[n])( η_e) ]_+, ∀ n∈
where C^sec_d[n]=log_2(1+γ_b[n]/1+γ_re[n]) specifies the downlink secrecy capacity with infinite blocklength in timeslot n, and γ_b[n] and γ_re[n] denote the received SNRs at Bob and Eve in timeslot n, given respectively by
γ_b[n] = p_r[n]h_rb[n]/σ^2_b[n] = p_r[n]ρ_b[n]/_r[n] - _b^2, ∀ n∈
γ_re[n] = p_r[n]h_re[n]/σ^2_e[n] = p_r[n]ρ_e[n]/_r[n]- _e^2, ∀ n∈
where σ^2_b[n] denotes the power of the AWGN at Bob and ρ_b[n] = β_0/σ^2_b[n].
Additionally, V(γ_b[n]) and V(γ_re[n]) indicate the channel dispersion for the UR-Bob and UR-Eve links in timeslot n, respectively.
§ PROBLEM FORMULATION
In this section, we aim to optimize the secrecy performance of our proposed UAV-aided mobile relay with short-packet transmission by designing the transmit power for Alice and the UR ={_a={p_a[n], ∀ n}, _r={p_r[n], ∀ n}}, the transmission blocklengths Ł={l_u[n], l_d[n], ∀ n}, and the UR's 3D trajectory ={[n], ∀ n}. Assuming that Alice and the UR securely encode the short transmit packets in timeslot n to sustain the desired reliability and security requirements (ε_r, ε_b, η_e), we define the secrecy throughput metric as the rate of the effective number of securely transmitted information bits in bits per second (bps) as
B̅_s[n] =1/δ_tmin{R^sec_u[n]l_u[n] (1- ε_r), R^sec_d[n]l_d[n] (1- ε_b) }, ∀ n∈
The resulting optimization maximizes the Effective Average Secrecy Throughput () over the mission duration and can be formulated as
: {, , Ł}max = 1/N∑_n=1^NB̅_s[n]
s.t. 1-10.
Note that is a nonconvex optimization problem due to the nonconvex objective function with nonsmooth operator [·]_+, highly coupled optimization variables, and nonconvex constraints 5-6 and 9. Thus, it is too challenging to solve optimally. We note that the nonsmoothness of the objective function in can be handled since at the optimal point, B̅_s[n] should hold a nonnegative value. Otherwise, by setting p_a[n]=0 and/or p_r[n]=0 in the given timeslot, one obtains B̅_s[n]=0, which violates the optimality. In light of this, we can remove the nonsmoothness operator from the objective function without impacting the optimal solution.
Furthermore, some terms in the objective function are implicit due to _ζ[n]{·} in R^sec_u[n] as well as Eve's location uncertainty. In the following lemmas, we tackle the aforementioned problems to make more tractable.
A closed-form lower-bound expression on the uplink short-packet secure transmission (<ref>) can be obtained as
R^sec_u[n] ≥ A_0[n] +_ζ[n]{-log_2(1+γ_ae[n])} + A_1[n]_ζ[n]{-√(1-(1+γ_ae[n])^-2)},
≥log_2(1+γ_r[n]/1+γ̅_ae[n])- √(V(γ_r[n])/l_u[n])( ε_r) - √(V(γ̅_ae[n])/l_u[n])( η_e), ∀ n∈
where γ̅_ae[n] = p_a[n]ρ_e[n]/_a - _e^α ∀ n, and A_0[n] and A_1[n] are given by
A_0[n] = log_2(1+γ_r[n])- √(V(γ_r[n])/l_u[n])( ε_r), A_1[n] = log_2 /√(l_u[n])(η_e), ∀ n∈
The proof follows from Jensen's inequality, a fundamental theorem in mathematics which states that {f(x)}≥ f({x}) for a convex function f(x), and considering the convexity of the functions f_1(x)=max{x,0}, f_2(x)=-log_2(1+x), and f_3(x)=-√(1-(1+x)^-2).
Now, we deal with the uncertainty in Eve's location by applying a worst-case analysis to facilitate the optimization problem.
Lower bounds on the achievable short-packet SR in the downlink and uplink transmissions can be obtained as
R^sec_u[n] ≥log_2(1+γ_r[n]) - √(V(γ_r[n])/l_u[n])( ε_r)
-log_2(1+γ̃_ae[n]) - √(V(γ̃_ae[n])/l_u[n])( η_e) R̃^sec_u[n], ∀ n∈
R^sec_d[n] ≥log_2(1+γ_b[n]) - √(V(γ_b[n])/l_d[n])( ε_b)
-log_2(1+γ̃_re[n]) - √(V(γ̃_re[n])/l_d[n])( η_e) R̃^sec_d[n], ∀ n∈
where
γ̃_ae[n] = p_a[n]ρ_e[n]/(_a - _e - Δ_e)^α, γ̃_re[n] = p_r[n]ρ_e[n]/(_r[n] - _e - Δ_e)^2, ∀ n∈
The proof follows from Lemma <ref> and considering the fact that R^sec_u[n] and R^sec_d[n] are monotonically decreasing functions with respect to (w.r.t.) the terms _a - _e and _r[n] - _e, respectively. Then, the lower bounds given in (<ref>) and (<ref>) are obtained by using the following inequality
_i - _e ≥|_i - _e - _e - _e| ≥_i - _e - Δ_e,
where _i ∈{{_r[n], ∀ n}, _a}. Note that (<ref>) is obtained by applying the reverse triangle inequality and utilizing (<ref>).
Now, by introducing the slack variable vector τ={τ[n], ∀ n}, can be converted to a more tractable form, whose objective function is differentiable and serves as a lower bound to that of the original problem:
1: {, , Ł, τ}max 1/T∑_n=1^Nτ[n]
s.t. 1-10,
R̃^sec_u[n] l_u[n](1- ε_r) ≥τ[n], ∀ n∈
R̃^sec_d[n] l_d[[n](1-ε_b) ≥τ[n], ∀ n∈
In the sequel, we propose a low-complexity iterative solution to solve the above problem based on the BSCA algorithm, wherein we optimize each block of variables while keeping the others unchanged in an alternating manner. Such an algorithm generally approaches a sub-optimal solution while ensuring convergence.
§ PROPOSED SOLUTION
In this section, we divide 1 into three subproblems: i) joint power optimization for Alice and the UR, ii) short-packet blocklength optimization, and iii) joint optimization of the UAV's motion and altitude, which are tackled separately. Thereafter, we propose an efficient overall algorithm, considering the impact of the block optimization on the joint design. Note that in the sequel, we omit the fixed multiplicative term 1/T from the objective function, which does not impact the proposed solution.
§.§ Joint Power Optimization
In this subsection, we jointly optimize the transmit powers of Alice and the UR, i.e., ={_a, _r}, while keeping other optimization variables fixed. Introducing slack variable vectors ={s_a[n], s_r[n], ∀ n} and ν={ν_a[n], ν_r[n] ∀ n}, 1 can be equivalently reformulated as
2: {_a, _r, τ, , ν}max ∑_n=1^Nτ[n]
s.t. 5-8,
ln(1+k_1, j p_j[n]/1+k_2, j p_j[n]) ≥ k_3, js_j[n]+k_4, jν_j[n]+k_5, jτ[n], j∈{a, r}, ∀ n∈
s^2_j[n] ≥ 1-(1+k_1, jp_j[n])^-2, s_j[n] ∈^+, j∈{a, r}, ∀ n∈
ν^2_j[n] ≥ 1-(1+k_2, jp_j[n])^-2, ν_j[n] ∈^+, j∈{a, r}, ∀ n∈
where
k_1, a = ρ_r[n]/_r[n]-_a^2, k_2, a = ρ_e[n]/(_a-_e-Δ_e)^α,
k_3, a = (ε_r)/√(l_u[n]), k_4, a = (η_e)/√(l_u[n]), k_5, a = ln2/l_u[n](1- ε_r),
k_1, r = ρ_b[n]/_r[n]-_b^2, k_2, r = ρ_e[n]/(_r[n]-_e-Δ_e)^2,
k_3, r = (ε_b)/√(l_d[n]), k_4, r = (η_e)/√(l_d[n]), k_5, r = ln2/l_d[n](1- ε_b).
Note that (<ref>) and (<ref>) should hold with equality at the optimal point; otherwise, their values can be decreased to improve the objective function, which would violate the optimality of the solution. Nonetheless, subproblem 2 is still nonconvex due to nonconvex constraints (<ref>), (<ref>), (<ref>). Thus, to tackle 2, we replace these constraints with corresponding convex approximations by applying the first-order restrictive law of the Taylor expansion method at the given point <cit.>. Accordingly, at a local point ^lo_j={p^lo_j[n], j∈{a, r}, ∀ n}, we write 2 approximately as
2.1: {_a, _r, τ, , ν}max ∑_n=1^Nτ[n]
s.t. 5-8, s_j[n] ∈^+, ν_j[n] ∈^+, ∀ n∈
ln (1+k_1, j p_j[n]) ≥ k_3, js_j[n]+k_4, jν_j[n]+k_5, jτ[n]+k_6, jp_j[n]+k_7,j, ∀ n∈
ln s_j[n] + ln(1+ k_1, jp_j[n]) ≥ A_1(p^lo_j[n];k_1, j) (p_j[n]-p^lo_j[n]) + A_0(p^lo_j[n];k_1, j), ∀ n∈
lnν_j[n] + ln(1+ k_2, jp_j[n]) ≥ A_1(p^lo_j[n];k_2, j) (p_j[n]-p^lo_j[n]) + A_0(p^lo_j[n];k_2, j), ∀ n∈
where k_6, j = k_2, j/1+k_2, jp^lo_j[n], k_7,j = ln(1+k_2,jp^lo_j[n])-k_6, jp^lo_j[n], and the functions A_0(x; k) and A_1(x; k) are defined for k>0 respectively as
A_0(x;k) =1/2ln(kx[2+kx]) and A_1(x; k) = kx+1/x (kx+ 2).
Since 2.1 is convex w.r.t. the optimization variables, it can be efficiently solved using standard tools.
§.§ Short-packet Blocklength Optimization
In this subsection, we optimize the blocklength vectors of both the uplink and downlink SPC, i.e., Ł={l_u[n], l_d[n], ∀ n}. As such, the corresponding subproblem can be expressed as
3: Ł, τmax ∑_n=1^Nτ[n]
s.t. 5, 6, 9, 10
a_0,il_i[n] - a_1,i√(l_i[n])≥τ[n], i∈{u, d}, ∀ n∈
where
a_0,u = (1-ε_r)log_2(1+γ_r[n]/1+γ̃_ae[n]), a_1,u=(1-ε_r)[√(V(γ_r[n]))(ε_r) + √(V(γ̃_ae[n]))( η_e)],
a_0,d = (1-ε_b)log_2(1+γ_b[n]/1+γ̃_re[n]), a_1,d=(1-ε_b)[√(V(γ_b[n]))(ε_b) + √(V(γ̃_re[n]))( η_e)].
3 is a nonlinear integer programming problem due to 10. One possible approach to tackle this challenge is to relax 3 into a convex optimization problem by converting the integer-valued vector Ł to a positive continuous vector Ł={l̃_u[n], l̃_d[n], ∀ n} such that
3.1: Ł, τmax ∑_n=1^Nτ[n]
s.t. 5, 6, 9
l̃_i[n] ≥ 1, i∈{u, d}, ∀ n∈
a_0,il̃_i[n] - a_1,i√(l̃_i[n])≥τ[n], i∈{u, d}, ∀ n∈
Notice that (<ref>) in 3.1 is a nonconvex constraint, following the law of the second-order derivative <cit.>. Thus, the convex reformulation of 3.1 at a local point L^lo={l̃^lo_u[n], l̃^lo_d[n], ∀ n} can be given, using a first-order Taylor approximation, as
3.2: Ł, τmax ∑_n=1^Nτ[n]
s.t. (<ref>), (<ref>)
a_0,il̃_i[n] - τ[n] ≥ a_1,i√(l̃^lo_i[n]) + a_1,i/2√(l̃^lo_i[n])(l̃_i[n]-l̃^lo_i[n]), i∈{u, d}, ∀ n∈
3.2 is a convex optimization problem, and thus it can be solved efficiently. In order to obtain the integer solution to 3, denoted by Ł^⋆, one can simply round down the noninteger solution to 3.3 such that Ł^⋆= ⌊Ł^opt⌋, where ⌊ x⌋ indicates the largest integer less than or equal to x. We note that such a rounding approach provides a lower-bound solution to the original integer problem as the objective function is a monotonically increasing function of blocklength and none of the constraints is violated.
§.§ 3D UR Trajectory Optimization
This subsection explores the joint optimization of the UR's motion and altitude. In light of this, we recast 1 to optimize , while keeping the other variables fixed:
4: {_r, τ}max ∑_n=1^Nτ[n]
s.t. 1-4,
log_2(1+γ_r[n]) - b_0√(1-(1+γ_r[n])^-2)≥ b_2 τ[n] + b_1, ∀ n∈
log_2(1+γ_b[n]/1+γ̃_re[n]) - c_0√(1-(1+γ_b[n])^-2) - c_1√(1-(1+γ̃_re[n])^-2)≥ c_2 τ[n], ∀ n∈
where
b_0 = (ε_r)log_2 /√(l_u[n]), b_1 = log_2(1+γ̃_ae[n]) - √(V(γ̃_ae[n])/l_u[n])( η_e), b_2 = 1/l_u[n](1-ε_r),
c_0 = (ε_b)log_2 /√(l_d[n]), c_1 = (η_e)log_2 /√(l_d[n]), c_2 = 1/l_d[n](1-ε_b).
We stress that 4 is still a nonconvex optimization problem due to nonconvex constraints (<ref>) and (<ref>). In the following, we focus on transforming these constraints into convex approximations to make the problem tractable.
§.§.§ Convex reformulation of (<ref>)
First, we equivalently write (<ref>) in a more tractable way by introducing nonnegative slack variables λ={λ_1[n], λ_2[n], ∀ n} and β={β_1[n], ∀ n}:
log_2(1+λ_1[n]) - b_0β_1[n] ≥ b_2 τ[n] + b_1, ∀ n∈
λ_2[n] ≥_r[n] - _a ^2/p_a[n]ρ_r[n], ∀ n∈
λ_1[n] λ_2[n] ≤ 1, ∀ n∈
β^2_1[n] ≥ 1-(1+λ_1[n])^-2, ∀ n∈
We note that constraints (<ref>) and (<ref>) are convex, while (<ref>) and (<ref>), introduced to ensure the smoothness of 4, are nonconvex. We stress that (<ref>)-(<ref>) should hold with equality at the optimal point. Before proceeding further, we mention a fruitful lemma below.
Let f(x,y)=1/xy with x, y > 0. At any given point (x_0, y_0) in the domain of f, the following function serves as a global lower bound on f(x,y), i.e.,
f_lb(x,y;x_0,y_0) = -x y_0+x_0 y-3 x_0 y_0/x_0^2 y_0^2≤ f(x,y).
The Hessian of f(x,y)= 1/xy is given by
∇^2f(x,y) = [ ∂^2 f(x,y)∂ x^2 ∂^2 f(x,y)∂ x ∂ y; ∂^2 f(x,y)∂ y ∂ x ∂^2 f(x,y)∂ y^2 ]= [ 2/x^3 y 1/x^2 y^2; 1/x^2 y^2 2/x y^3 ].
We can easily see that ∇^2f(x,y) is a positive definite matrix, implying that f(x,y) is jointly convex w.r.t. the variables x and y. Thus, since the first-order Taylor expansion of a convex function provides a global lower bound at the given point (x_0,y_0), we reach the expression given in (<ref>), which completes the proof.
Thus, at a given point (^lo_r,λ^lo,β^lo), the following serve as convex approximations of the constraints (<ref>) and (<ref>).
1≤
f_lb(λ_1[n],λ_2[n];λ^lo_1[n],λ^lo_2[n]), ∀ n∈
ln(β_1[n]) + ln(1+λ_1[n]) ≥ A_0(λ^lo_1[n];1) + A_1(λ^lo_1[n];1)(λ_1[n]-λ^lo_1[n]), ∀ n∈
Note that (<ref>) follows from Lemma <ref> and (<ref>) follows from the concavity of the logarithm function.
§.§.§ Convex reformulation of (<ref>)
Introducing the nonnegative slack variables ω={ω_1[n], ω_2[n], ∀ n}, ψ={ψ_1[n], ∀ n}, 𝐮={u_1[n], ∀ n}, and 𝐯={v_1[n], v_2[n], ∀ n}, we can reformulate (<ref>) into the equivalent convex constraint
ℐlog_2(1+ω_1[n]) - c_0ψ_1[n] - ℐℐlog_2(1+1/u_1[n]) - c_1v_1[n]≥ c_2 τ[n], ∀ n∈
with the additional constraints for Part ℐ, given by
ω_2[n] ≥_r[n] - _b ^2/p_r[n]ρ_b[n], ∀ n∈
ω_1[n] ω_2[n] ≤ 1 , ∀ n∈
ψ^2_1[n] ≥ 1-(1+ω_1[n])^-2, ∀ n∈
and the extra constraints for Part ℐℐ
u_1[n] ≤(_r[n] - _e - Δ_e)^2/p_r[n]ρ_b[n], ∀ n∈
v^2_1[n] ≥ 1-(1+v_2[n])^-2, ∀ n∈
u_1[n]v_2[n] ≤ 1, ∀ n∈
We note that some constraints arising from the reformulation of (<ref>) are nonconvex. Thus, using Lemma <ref> and <cit.>, we obtain convex approximations of the constraints (<ref>) and (<ref>) at the given local point (^lo_r,ω^lo,ψ^lo, 𝐯^lo, 𝐮^lo) as
1≤ f_lb(ω_1[n],ω_2[n];ω^lo_1[n],ω^lo_2[n]), ∀ n∈
ln(ψ_1[n]) + ln(1+ω_1[n]) ≥ A_0(ω^lo_1[n];1) + A_1(ω^lo_1[n];1)(ω_1[n]-ω^lo_1[n])
, ∀ n∈
p_r[n]ρ_b[n]u_1[n]+2Δ_e_r[n] - _e≤ -^lo_r[n]^2 + 2(^lo_r[n] - _e)^T_r[n] +d_0, ∀ n∈
ln(v_1[n]) + ln(1+v_2[n]) ≥ A_0(v^lo_2[n];1) + A_1(v^lo_2[n];1)(v_2[n]-v^lo_2[n])
, ∀ n∈
u_1[n] ≥1/v_2[n], ∀ n∈
where d_0 = _e^2 + Δ^2_e. We now express the convex reformulation of subproblem 4 as
4.1: {_r, τ, λ, β, ω, ψ, 𝐮,𝐯}max ∑_n=1^Nτ[n]
s.t. 1-4, (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>)
Since 4.1 is convex, it can be efficiently solved.
§.§ Overall Algorithm and Complexity Analysis
In this subsection, we propose an overall iterative algorithm based on the sequential block optimization summarized in Algorithm <ref>. It can be proved that the proposed algorithm is guaranteed to converge to a local optimum commencing from a feasible point, since the objective function is non-decreasing over the iteration index and is upper bounded. Moreover, the time complexity of Algorithm <ref> depends on the complexity of each convex subproblem and the convergence accuracy parameter ϵ. The convex conic optimizations are typically solved using the interior-point method, whose complexity can be approximated based on the number of optimization variables and convex constraints. Accordingly, we provide Table <ref> to summarize the complexity of each subproblem and the overall algorithm using big-O notation. The overall complexity order of Algorithm <ref> is polynomial, and thus our proposed approach can be reasonably implemented for energy-constrained UAV-aided SPC-IoT scenarios.
§ NUMERICAL RESULTS AND DISCUSSION
In this section, we demonstrate the EAST enhancement achieved by our proposed optimization algorithm for the considered UAV-aided SPC-IoT scenario. To
exhibit the effectiveness of our joint trajectory and resource allocation design in Algorithm <ref>, labeled , we compare it with the following benchmark schemes:
* Benchmark 1: Resource Design with Fixed Trajectory (), where the UR's trajectory is fixed, and the joint transmit power and blocklength optimizations in 2.1 and 3.2 are solved sequentially until convergence, resulting in (^opt, Ł^opt) to improve EAST.
* Benchmark 2: Trajectory Design with Fixed Resources (), where the communication resources (, Ł) are fixed and only the UR's 3D trajectory is optimized using subproblem 4.1 in a sequential manner to achieve ^opt, maximizing the EAST performance.
Unless otherwise stated, the system parameters are set as in Table <ref>. Further, the initial feasible trajectory of the UR, ^(0), is taken to be on a direct line with fixed speed from the initial location to the final location. The uplink and downlink transmission blocklengths per timeslot are initialized as l^(0)_u[n]=l^(0)_d[n]=L^max/2 ∀ n, and the transmit power of Alice and the UR per timeslot are adapted such that the maximum and total power budget constraints are satisfied, i.e., p^(0)_a[n]=min{p^max_a, p^tot_a/Nl^(0)_u[n]} ∀ n and p^(0)_r[n]=min{p^max_r, p^tot_r/Nl^(0)_d[n]} ∀ n. The slack variables (λ^lo, β^lo, ω^lo, ψ^lo, 𝐮^lo, 𝐯^lo) are initialized such that the corresponding constraints (<ref>)-(<ref>), (<ref>), and (<ref>) are met with equality.
In Fig. <ref>, we plot the EAST against the iteration index for all schemes with different mission times T={100, 150} s and L^max={150, 400} to verify the quick convergence of Algorithm <ref> and the validity of our analysis, as well as to demonstrate the performance advantage of our joint design. We see that all algorithms converge quickly in just a few iterations. Our proposed approach achieves the best EAST performance in all cases.
For example, can reach up to 73 bps, approximately 15% more than , 43% better than , and nearly three times the EAST of the initial feasible setting for the case with T=100 s and L^max=400. Fig. <ref> shows that using the baseline trajectory and optimizing the resource allocation is more important for SPC than optimizing the trajectory for a fixed resource allocation, while the joint design of both is clearly preferable. Furthermore, outperforms for the considered system setup with L^max=400, indicating the significance of radio resource management and blocklength optimization when the maximum end-to-end tolerable delay is relatively high. However, reducing the end-to-end delay tolerance, e.g., from L^max=400 to L^max=150, significantly decreases the performance gap between and , indicating that the trajectory design becomes critical for a stricter delay requirement. Additionally, we observe from the figure that as the mission time increases from T=100 s to T=150 s, the EAST performance degrades for all schemes with L^max=400. This trend can be intuitively explained by the fact that EAST is inversely proportional to the mission time T, according to (<ref>). Nevertheless, with a higher T, the UR has more flexibility in finding the best locations for secure information relaying, and hence the total amount of information securely exchanged between Alice and Bob is expected to increase, which also contributes to the EAST enhancement. Thus, owing to such a trade-off, it appears that the EAST should asymptotically
converge to a small but non-zero value as T gets sufficiently large.
Fig. <ref> illustrates the 3D trajectory and velocity profiles of the UR for the different approaches. For , we observe from Figs. <ref>(sim:fig2traj) and <ref>(sim:fig2vel) that the UR attempts to fly at the minimum altitude with maximum velocity while heading towards a location between Alice and Bob, and hovering at that point as long as possible. This solution greatly improves the EAST performance compared with the initial direct-path trajectory with fixed velocity, as already shown in Fig. <ref>. Nevertheless, when both the trajectory design and resource optimization are taken into account as in the proposed design, the UR demonstrates effective 3D navigation, capitalizing on its altitude adjustment ability to considerably enhance the EAST performance in comparison with the other benchmarks.
Fig. <ref> depicts the optimized transmit power profiles of Alice and the UR, as well as both the uplink and downlink coding blocklengths for different schemes. It is evident that the adoption of fixed power allocation and equal blocklengths is suboptimal. We can see from Fig. <ref>(sim:fig4pow) that when the channel quality of the main link is worse than the eavesdropping link, proper resource management results in reserving the transmit power and blocklength resources for the moment when better communication channels can be obtained, for example between timeslots T=20 s and T=74 s. Furthermore, we observe from Fig. <ref>(sim:fig4blocklength) that when the UR is farther from Bob, larger downlink coding blocklengths are adopted, and they reduce in length as the UR approaches Bob. As the UR flies away from Alice, the proposed algorithm efficiently increases Alice's transmit power and uplink blocklength to improve the uplink SR, which ultimately enhances the EAST performance. We note that Alice does not transmit with full power, particularly between timeslots T=20 s and T=45 s, since the secrecy is jeopardized by the enhancement of the wiretap link. As far as is concerned, it is generally expected that the UR will maximize the relaying power for the sake of EAST improvement.
In Fig. <ref>, the SR of both uplink and downlink transmissions are presented versus time for for both the finite and infinite blocklength cases. We can see from the curves that nonzero SR can be achieved when the UAV relay is properly located relative to Alice and Bob for the task of relaying. This goal is achieved between timeslots T=20 s and T=74 s.
Further, the results reinforce the fact that the adoption of SPC leads to notable a decrease in the achievable SR compared with the conventional infinite blocklength assumption.
This underscores the need for a different approach to system design for SPC scenarios as compared to conventional systems, and emphasizes the importance of carefully considering system parameters in order to achieve optimal performance, avoiding dramatically suboptimal designs.
Fig. <ref> depicts the effects of the end-to-end delay tolerance represented by the maximum coding blocklength L^max, as well as the reliability requirement and information leakage constraint on the EAST performance in the proposed scheme. It can be observed from the curves that the larger the maximum blocklength, the higher the EAST up to a certain level for all the settings. Indeed, increasing L^max can potentially increase both the uplink and downlink blocklengths, which in turn improves the overall EAST performance by reducing the value of the subtractive terms in the objective function introduced by the finite blocklength. Also, from Fig. <ref> it can be seen that the EAST experiences a ceiling phenomenon for all the security and reliability levels when L^max is sufficiently increased, implying that no further improvement in EAST can be achieved by the joint power and trajectory design due to the power budget limitations. Furthermore, we can observe that when the reliability and/or security requirements are relaxed, as indicated by larger values for ε_b, ε_r, and η_e, a higher EAST can be achieved by utilizing proper trajectory design and efficient communication resource management.
Fig. <ref> illustrates how the uncertainty in Eve's location impacts the EAST performance for different mission times T={100, 150, 200} s. Our proposed design once again provides the best performance compared with the other methods regardless of Eve's location uncertainty; nonetheless, the performance gap between and tends to reduce as Δ_e gets larger, which indicates the significance of the uncertainty parameter in the system design. Fig. <ref> illustrates that the proposed approach is not highly sensitive to the uncertainty in Eve's location. For example, experiences only about a 10 bps loss in EAST for T=100 s when the location uncertainty increases from the ideal case Δ_e=0 to a relatively high uncertainty, e.g., Δ_e=300 m. shows the highest robustness to the variation in Δ_e, but is unable to exploit scenarios where Δ_e is small.
§ CONCLUSIONS
We have presented the design of a secure and reliable UAV-IoT relaying system with SPC under the assumption of imperfect knowledge of Eve's location. To optimize the EAST performance of the system, an effective joint design of the UAV's 3D trajectory, power control, and uplink and downlink blocklengths was proposed, which ensures the convergence of the optimization problem to a locally optimal solution with low-complexity. Numerical evaluations were conducted to demonstrate the effectiveness of the proposed scheme in terms of the EAST performance compared to benchmarks that only consider either trajectory optimization or resource management. Our results show that both the uplink and downlink blocklengths should adaptively be adjusted to improve the EAST. In conclusion, our joint design for the proposed four-node system is a promising solution for secure and reliable communications in UAV-IoT networks with SPC, particularly in environments with limited power and computing resources. Future work can explore the generalization of our system model with multiple legitimate transceivers and eavesdroppers under strict secrecy and reliability requirements.
IEEEtran
|
http://arxiv.org/abs/2307.04472v1 | 20230710104248 | Partial Vessels Annotation-based Coronary Artery Segmentation with Self-training and Prototype Learning | [
"Zheng Zhang",
"Xiaolei Zhang",
"Yaolei Qi",
"Guanyu Yang"
] | cs.CV | [
"cs.CV"
] |
Partial Vessels Annotation-based Coronary Artery Segmentation
Z. Zhang and X. Zhang—Contributed equally to this work.
Z. Zhang et al.
LIST, Key Laboratory of Computer Network and Information Integration, Southeast University, Ministry of Education, Nanjing 210096, China
[email protected]. of Diagnostic Radiology, Jinling Hospital, Medical School of Nanjing University, Nanjing, China Jiangsu Provincial Joint International Research Laboratory of Medical
Information Processing, Southeast University, Nanjing 210096, China Centre de Recherche en Information Biom´edicale Sino-Fran¸cais (CRIBs), Strasbourg, France
Partial Vessels Annotation-based Coronary Artery Segmentation with Self-training and Prototype Learning
Zheng Zhang1 Xiaolei Zhang2 Yaolei Qi1 Guanyu Yang1,3,4()
August 12, 2023
=======================================================================================================
Coronary artery segmentation on coronary-computed tomography angiography (CCTA) images is crucial for clinical use. Due to the expertise-required and labor-intensive annotation process, there is a growing demand for the relevant label-efficient learning algorithms. To this end, we propose partial vessels annotation (PVA) based on the challenges of coronary artery segmentation and clinical diagnostic characteristics. Further, we propose a progressive weakly supervised learning framework to achieve accurate segmentation under PVA. First, our proposed framework learns the local features of vessels to propagate the knowledge to unlabeled regions. Subsequently, it learns the global structure by utilizing the propagated knowledge, and corrects the errors introduced in the propagation process. Finally, it leverages the similarity between feature embeddings and the feature prototype to enhance testing outputs. Experiments on clinical data reveals that our proposed framework outperforms the competing methods under PVA (24.29% vessels), and achieves comparable performance in trunk continuity with the baseline model using full annotation (100% vessels).
§ INTRODUCTION
Coronary artery segmentation is crucial for clinical coronary artery disease diagnosis and treatment <cit.>. Coronary-computed tomography angiography (CCTA), as a non-invasive technique, has been certified and recommended as established technology in the cardiological clinical arena <cit.>. Thus, automatic coronary artery segmentation on CCTA images has become increasingly sought after as a means to enhance diagnostic efficiency for clinicians. In recent years, the performance of deep learning-based methods have surpassed that of conventional machine learning approaches (e.g. region growing) in coronary artery segmentation <cit.>. Nevertheless, most of these deep learning-based methods highly depend on accurately labeled datasets, which need labor-intensive annotations. Therefore, there is a growing demand for relevant label-efficient learning algorithms for automatic coronary artery segmentation on CCTA images.
Label-efficient learning algorithms have garnered considerable interest and research efforts in natural and medical image processing <cit.>, while research on label-efficient coronary artery segmentation for CCTA images is slightly lagging behind. Although numerous label-efficient algorithms for coronary artery segmentation in X-ray angiograms have been proposed <cit.>, only a few researches focus on CCTA images. Qi et al. <cit.> proposed an elabrately designed EE-Net to achieve commendable performance with limited labels. Zheng et al <cit.> transformed nnU-Net into semi-supervised segmentation field as the generator of Gan, having achieved satisfactory performance on CCTA images. Most of these researches use incomplete supervision, which labels a subset of data. However, other types of weak supervision (e.g. inexact supervision), which are widely used in natural image segmentation <cit.>, are seldom applied to coronary artery segmentation on CCTA images.
Different types of supervision are utilized according to the specific tasks. The application of various types of weak supervision are inhibited in coronary artery segmentation on CCTA images by the following challenges. 1) Difficult labeling (Fig. <ref>(a)). The target regions are scattered, while manual annotation is drawn slice by slice on the planes along the vessels. Also, boundaries of branches and peripheral vessels are blurred. These make the annotating process time-consuming and expertise-required. 2) Complex topology (Fig. <ref>(b)). Coronary artery shows complex and slender structures, diameter of which ranges from 2 mm to 5 mm. The tree-like structure varies individually. Based on these challenges and the insight that vessels share local feature (Fig. <ref>(b)), we propose partial vessels annotation and our framework as following.
Given the above, we propose partial vessels annotation (PVA) (Fig. <ref>(c)) for CCTA images. While PVA is a form of partial annotation (PA) which has been adopted by a number of researches <cit.>, our proposed PVA differs from the commonly used PA methods. More specifically, PVA labels vessels continuously from the proximal end to the distal end, while the labeled regions of PA are typically randomly selected. Thus, our proposed PVA has two merits. 1) PVA balances efficiency and informativity. Compared with full annotation, PVA only requires clinicians to label vessels within restricted regions in adjacent slices, rather than all scattered target regions in each individual slice. Compared with PA, PVA keep labeled vessels continuous to preserve local topology information. 2) PVA provides flexibility for clinicians. Given that clinical diagnosis places greater emphasis on the trunks rather than the branches, PVA allows clinicians to focus their labeling efforts on vessels of particular interest. Therefore, our proposed PVA is well-suited for clinical use.
In this paper, we further propose a progressive weakly supervised learning framework for PVA. Our proposed framework, using PVA (only 24.29% vessels labeled), achieved better performance than the competing weakly supervised methods, and comparable performance in trunk continuity with the full annotation (100% vessels labeled) supervised baseline model. The framework works in two stages, which are local feature extraction (LFE) stage and global structure reconstruction (GSR) stage. 1) LFE stage extracts the local features of coronary artery from the limited labeled vessels in PVA, and then propagates the knowledge to unlabeled regions. 2) GSR stage leverages prediction consistency during the iterative self-training process to correct the errors, which are introduced inevitably by the label propagation process. The code of our method is available at <https://github.com/ZhangZ7112/PVA-CAS>.
To summarize, the contributions of our work are three-fold:
* To the best of our knowledge, we proposed partial vessels annotation for coronary artery segmentation for the first time, which is in accord with clinical use. First, it balances efficiency and informativity. Second, it provides flexibility for clinicians to annotate where they pay more attention.
* We proposed a progressive weakly supervised learning framework for partial vessels annotation-based coronary artery segmentation. It only required 24.29% labeled vessels, but achieved comparable performance in trunk continuity with the baseline model using full annotation. Thus, it shows great potential to lower the label cost for relevant clinical and research use.
* We proposed an adaptive label propagation unit (LPU) and a learnable plug-and-play feature prototype analysis (FPA) block in our framework. LPU integrates the functions of pseudo label initialization and updating, which dynamically adjusts the updating weights according to the calculated confidence level. FPA enhances vessel continuity by leveraging the similarity between feature embeddings and the feature prototype.
§ METHOD
As shown in Fig. <ref>, our proposed framework for partial vessels annotation (PVA) works in two stages. 1) The LFE stage(Sec. <ref>) extracts and learns vessel features from PVA locally. After the learning process, it infers on the training set to propagate the learned knowledge to unlabeled regions, outputs of which are integrated with PVA labels to initialize pseudo labels. 2) The GSR stage (Sec. <ref>) utilizes pseudo labels to conduct self-training, and leverages prediction consistency to improve the pseudo labels. In our proposed framework, we also designed an adaptive label propagation unit (LPU) and a learnable plug-and-play feature prototype analysis (FPA) block. LPU initialize and update the pseudo labels; FPA block learns before testing and improves the final output during testing.
§.§ Local Feature Extraction Stage
In LFE stage, our hypothesis is that the small areas surrounding the labeled regions hold valid information. Based on this, a light segmentation model 𝒮_l is trained to learn vessel features locally, with small patches centering around the labeled regions as input and output. In this manner, the negative impact of inaccurate supervision information in unlabeled regions is also reduced.
§.§.§ Pseudo Label Initialization in LPU.
After training, 𝒮_l propagates the learned knowledge of local feature to unlabeled regions. For each image of shape H× W× D, the corresponding output logit ŷ_1∈ [0,1]^H× W× D of 𝒮_l provides a complete estimate of the distribution of vessels, albeit with some approximation. Meanwhile, the PVA label y_PVA∈{0,1}^H× W× D provides accurate information on the distribution of vessels, but only to a limited extent. Therefore, LPU integrate ŷ_1 and y_PVA to initialize the pseudo label y_PL (Equ. <ref>), which will be utilized in GSR stage and updated during iterative self-training.
y_PL^(t=0)(h,w,d)_∀ (h,w,d) ∈ℝ^H× W× D=
1, y_PVA(h,w,d)=1,
ŷ_1(h,w,d), otherwise.
§.§ Global Structure Reconstruction Stage
The GSR stage mainly consists of three parts: 1) The segmentation model 𝒮_g to learn the global tree-like structure; 2) LPU to improve pseudo labels; 3) FPA block to improve segmentation results at testing.
Through initialization (Equ. <ref>), the initial pseudo label y_PL^(t=0) contains the information of both PVA labels and the knowledge of local features in 𝒮_l. Therefore, at the beginning of this stage, 𝒮_g learns from y_PL^(t=0) to warm up. After this, logits of 𝒮_g are utilized to update the pseudo labels during iterative self-training.
§.§.§ Pseudo Label Updating in LPU.
The principle of this process is that more reliable logit influences more the distribution of the corresponding pseudo label. Based on this principle, first we calculate the confidence degree η^(t)∈ [0,1] for ŷ_2^(t). Defined by Equ. <ref>, η^(t) numerically equals to the average of the logits in labeled regions. This definition makes sense since the expected logit equals to ones in vessel regions and zeros in background regions. The closer ŷ_2^(t) gets to the expected logit, the higher η^(t) (confidence degree) will be.
η^(t) = ∑_h∑_w∑_dy_PVA(h,w,d) ·ŷ_2^(t)(h,w,d)/∑_h∑_w∑_dy_PVA(h,w,d)
Then, a quality control test is performed to avoid negative optimization as far as possible. If the confidence degree η^(t) is higher than all elements in the set {η^(i)}_i=1^t-1, the current logit is trustworthy to pass the test to improve the pseudo label. Then, y_PL^(t) is updated by the exponentially weighted moving average (EWMA) of the logits and the pseudo labels (Equ. <ref>). This process is similar to prediction ensemble <cit.>, which hase been adopted to filter pseudo labels<cit.>. However, different from their methods, where the factor η^(t) is a fixed hyperparameter coefficient and the pseudo labels are updated each or every several epoches, η^(t) in our method is adaptive and a quality control test is performed.
y_PL^(t)=
η^(t)ŷ_2^(t)+(1-η^(t))y_PL^(t-1), η^(t)=max{{η^(i)}_i=1^t}
y_PL^(t-1), otherwise.
§.§.§ Feature Prototype Analysis Block.
Inspired by <cit.>, which generates class feature prototype ρ _c (Equ. <ref>) from the embeddings z^l_i of labeled points in class c, we inherit the idea but further transform the mechanism into the proposed learnable plug-and-play block, FPA block. Experimental experience finds that the output of FPA block has good continuity, for which the FPA output are utilized to enhance the continuity of convolution output at testing.
ρ _c = 1/|ℐ_c |∑_z^l_i∈ℐ_cz^l_i
In the penultimate layer of the network, which is followed by a 1×1×1 convolutional layer to output logits, we parallelly put the feature map Z∈ℛ^C× H× W× D into FPA. The output similarity map O∈ℛ^1× H× W× D is calculated by Equ. <ref>, where Z(h,w,d)∈ℛ^C denotes the feature embeddings of voxel (h,w,d), and ρ_θ∈ℛ^C the kernel parameters of FPA.
O(h,w,d)=exp(-‖ Z(h,w,d)-ρ_θ‖^2)
The learning process of FPA block is before testing, during which the whole model except FPA gets frozen. To reduce the additional overhead, ρ_θ is initialized by one-time calculated ρ _c and fine-tuned with loss ℒ_fpa (Equ. <ref>), where only labeled voxels will take effect in updating the kernel.
ℒ_fpa=∑_h∑_w∑_dy_PVA(h,w,d)· log(O(h,w,d))/∑_h∑_w∑_dy_PVA(h,w,d)
§ EXPERIMENTS AND RESULTS
§.§ Dataset and Evaluation Metrics
Experiments are implemented on a clinical dataset, which includes 108 subjects of CCTA volumes (2:1 for training and testing). The volumes share the size of 512 × 512 × D, with D ranging from 261 to 608. PVA labels of the training set are annotated by clinicians, where only 24.29% vessels are labeled.
The metrics used to quantify the results include both integrity and continuity assessment indicators. Integrity assessment indicators are Mean Dice Coefficient (Dice), Relevant Dice Coefficient (RDice) <cit.>, Overlap (OV) <cit.>; continuity assessment indicators are Overlap util First Error (OF) <cit.> on the three main trunks (LAD, LCX and RCA).
§.§ Implementation Details
3D U-Net<cit.> is set as our baseline model. Experiments were implemented using Pytorch on GeForce RTX 2080Ti. Adam optimizer was used to train the models with an initial learning rate of 10^-4. The patch sizes were set as 128 × 128 × 128 and 512 × 512 × 256 respectively for 𝒮_l and 𝒮_g. When testing, sliding windows were used with a half-window width step to cover the entire volume.
§.§ Comparative Test
To verify the effectiveness of our proposed method, it is compared with both classic segmentation models (3D U-Net <cit.>, HRNet <cit.>, Transunet <cit.>) and partial annotation-related weakly supervised frameworks (EWPA <cit.>, DMPLS <cit.>).
The quantative results of different methods are summarized in Tab. <ref>, which shows that our proposed method outperforms the competing methods under PVA. The competing frameworks (EWPA and DMPLS) had achieved the best results in their respective tasks under partial annotation, but our proposed method achieved better results for PVA-based coronary artery segmentation. It is worth mentioning that the performance in trunk continuity (measured by the indicator OF) of our proposed method using PVA (24.29% vessels labeled) is comparable to that of the baseline model using full annotation (100% vessels labeled).
The qualitative visual results verify that our proposed method outperforms the competing methods under PVA. Three cases are shown in Fig. <ref>. All the cases show that the segmentation results of our method have good overall topology integrity, especially on trunk continuity.
§.§ Ablation Study
Ablation experiments were conducted to verify the importance of the components in our proposed framework (summarized in Tab. <ref>). The performance improvement verifies the effectiveness of pseudo label initialization (PLI) and updating (PLU) mechanisms in the label propagation unit (LPU). PLI integrates the information of PVA labels with the propagated knowledge, and PLU improves the pseudo labels during self-training. With the help of FPA block, the segmentation results gain further improvement, especially on the continuity of trunks.
§ CONCLUSION
In this paper, we proposed partial vessels annotation (PVA) for coronary artery segmentation on CCTA images. The proposed PVA is convenient for clinical use for the two merits, providing flexibility as well as balancing efficiency and informativity. Under PVA, we proposed a progressive weakly supervised learning framework, which outperforms the competing methods and shows comparable performance in trunk continuity with the full annotation supervised baseline model. In our framework, we also designed an adaptive label propagation unit (LPU) and a learnable plug-and-play feature prototype analysis(FPA) block. LPU integrates the functions of pseudo label initialization and updating, and FPA improves vessel continuity by leveraging the similarity between feature embeddings and the feature prototype. To conclude, our proposed framework under PVA shows great potential for accurate coronary artery segmentation while requiring significantly less annotation effort.
splncs04
|
http://arxiv.org/abs/2307.04638v2 | 20230710153526 | DeePTB: A deep learning-based tight-binding approach with $ab$ $initio$ accuracy | [
"Qiangqiang Gu",
"Zhanghao Zhouyin",
"Shishir Kumar Pandey",
"Peng Zhang",
"Linfeng Zhang",
"Weinan E"
] | cond-mat.mtrl-sci | [
"cond-mat.mtrl-sci",
"physics.comp-ph"
] |
fnnumber
|
http://arxiv.org/abs/2307.03974v2 | 20230708132712 | Comparing EventB, $\{log\}$ and Why3 Models of Sparse Sets | [
"Maximiliano Cristiá",
"Catherine Dubois"
] | cs.SE | [
"cs.SE"
] |
Short-time large deviations of the spatially averaged height
of a KPZ interface on a ring
Baruch Meerson
August 12, 2023
=========================================================================================
Many representations for sets are available in programming languages libraries. The paper focuses on sparse sets used, e.g., in some constraint solvers for representing integer variable domains which are finite sets of values, as an alternative to range sequence. We propose in this paper verified implementations of sparse sets, in three deductive formal verification tools, namely , and 3. Furthermore, we draw some comparisons regarding specifications and proofs.
§ INTRODUCTION
Sets are widely used in programs. They are sometimes first-class objects of programming languages, e.g. SETL <cit.> or <cit.>,
but more frequently they are data structures provided in libraries. Many different representations are available, depending on the targeted set operations. In this paper, we focus on sparse sets, introduced by Briggs and Torczon in <cit.>, used in different contexts and freely available for different programming languages (Rust, C++ and many others). In particular,
sparse sets are used in constraint solvers as an alternative to range sequences or bit vectors for implementing domains of integer variables <cit.> which are nothing else than mathematical finite sets of integers. Their use in solvers implementations is motivated by -at least- the two following properties: searching and removing an element are constant-time operations—removing requires only two swapping
operations on arrays; sparse sets are cheap to trail and restore, which is a key point when backtracking.
Confidence on constraint solvers using sparse sets can be improved if the algorithms implementing the main operations are formally verified, as it has been done by Ledein and Dubois in <cit.> for the traditional implementation of domains as range sequences. Hence, the main contribution of this paper is
a verified implementation of sparse sets for representing finite sets of integers in , and 3.
We prove that the implemented operations preserve the invariants and we also prove properties that can be seen as formal foundations of trailing and restoring. As far as we know, this is the first formally verified implementation of sparse sets, whereas it has been done for other representations e.g. <cit.>. All the specifications and proofs can be found here: <https://gitlab.com/cdubois/sets2023.git>.
It has been known for decades that there is no silver bullet for software engineering or software development. The best we can do as software engineers is to increase our toolbox as much as possible and use the best available tool in it for the problem at hand. This software engineer practical principle still applies when it comes to formal development, formal methods and formal verification. In our opinion the Formal Methods (FM for short) community should have as much information as possible about the relative advantages and disadvantages of different FM methods and tools. With the intention to shed some light on the ups and downs of different FM, we specified and verified sparse sets with three different FM techniques. Then, a second contribution of this paper is a comparison of these FM w.r.t. aspects such as expressiveness, specification analysis and automated proof.
§ SPARSE SETS
We deal here with sets as subsets of natural numbers up to N-1, where N is any non null natural number. A sparse set S is represented by two arrays of length N called mapD and domD (as in <cit.>), and a natural number sizeD. The array mapD maps any value v ∈ [0,N-1] to its index ind_v in domD, the value indexed by ind_v in domD is v. The main idea that brings efficiency when removing an element or testing membership is to split domD into two sub-arrays, domD[0,sizeD-1] and domD[sizeD, N-1], containing resp. the elements of S and the elements of [0,N-1] not in S. Then, if S is empty, sizeD
is equal to 0, if S is the full set, then sizeD is N.
Checking if an element i belongs to the sparse set S simply consists in the evaluation of the expression mapD[i]<sizeD. Removing an element from the set consists in moving this element
to domD[sizeD, N-1] (with 2 swaps in mapD and domD and decreasing sizeD). Binding S to the singleton set {v} follows the same idea: moving this element at the first place in domD and assigning the value 1 to sizeD.
In our formalizations, we only deal with two operations consisting in removing an element in a sparse set and bind a sparse set to a singleton set since these two operations are fundamental when solving constraints. In this context, we may also need to walk through all the elements of a variable domain, it means exploring domD[0..sizeD-1]. If minimal and maximal values are required, then they have to be maintained in parallel. This is outside the scope of this work.
§ FORMAL DEVELOPMENT
In this section we succinctly introduce the formal specification language and with more detail the models for sparse sets.
§.§
<cit.> is a deductive formal method based on set theory and first order logic allowing users to design correct-by-construction systems. It relies on a state-based modeling language in which a model, called a machine,
is made of a state and a collection of events allowing for state changes. The state consists of variables constrained by invariants.
Proof obligations are generated to verify the preservation of invariants by events. A machine may use a -mathematical- context which introduces abstract sets, constants, axioms or theorems. A formal design in starts with an abstract machine which is usually refined several times. Proof obligations are generated to verify the correctness of a refinement step.
An event may have parameters. When its guards are satisfied, its actions, if any, are executed, updating state variables. Actions may be -multiple- deterministic assignments, x,y:=e, f, or -multiple- nondeterministic ones, x,y :| BAP(x,x',y,y') where BAP is called a Before-After Predicate relating current (x, y) and next (x', y') values of state variables x and y.
In the latter case, x and y are assigned arbitrary values satisfying the BAP predicate. When using such a non-deterministic form of assignment, a feasibility proof obligation is generated in order to check that there exist values for x' and y' such that BAP(x,x',y,y') holds when the invariants and guards hold. Furthermore when this kind of action is used and refined, the concrete action updating x and y is required to assign them values which satisfy the BAP predicate.
In the following, we use Rodin, an Eclipse based IDE for project management, model edition, refinement and proof, automatic proof obligations generation, model animation and code generation. Rodin supports automatic and interactive provers <cit.>. In this work we used the standard provers (AtelierB provers) and also the SMT solvers VeriT, CVC3 and CVC4. More details about and Rodin can be found in <cit.> and <cit.>.
§.§ formalization
The formalization is made of six components, i.e. two contexts, a machine and three refinements. Context Ctx introduces the bound N as a non-zero natural number and context Ctx1 extends the latter with helper theorems. The high level machine gives the abstract specification. This model contains a state composed of a finite set D, constrained to be a subset of the (integer) range 0..N-1, and two events, to remove an element from D or set D as a singleton set (see Fig. <ref> in which bind is removed for lack of space).
The first refinement (see Fig.<ref>)
introduces the representation of the domain as a sparse set, i.e. two arrays mapD and domD modeled as total functions and also the variable sizeD which is a natural number in the range 0..N. Invariants inv4 and inv5 constrain mapD and domD to be inverse functions of each other.
The gluing invariant inv6 relates the states between the concrete and former abstract machines. So the set domD[0..sizeD-1] containing the elements of the subarray from 0 to sizeD-1 is exactly the set D.
Theorem inv7 is introduced to ease some interactive proofs, it is proved as a consequence of the previous formulas (inv1 to inv6).
It follows directly from a theorem of Ctx1 whose statement is inv7 where domD and mapD are universally quantified. Theorem inv8, also used in an interactive proof, and automatically proved by CVC3, states that domD is an injective function.
Variables mapD and domD are both set initially to the identity function on 0..N-1 and sizeD to N. So invariants are satisfied at the initial state. Machine SparseSets_ref1 refines the events of the initial machine by non deterministic events. So here the remove event assigns the three state variables with values that satisfy invariants and also such that sizeD strictly decreases and removed elements in domD are kept at the same place (properties in bold font). Event bind follows the same pattern (again not shown here).
The second refinement has the same state than the previous refinement (see Fig. <ref>). Its events implement the operations using the new state variables. It is a straightforward translation of the algorithms described in <cit.>.
The only reason to have introduced the intermediate model
SparseSets_ref1 is to express the properties written in bold font and thus generate, in the next refinement, proof obligations which, when discharged, will not only ensure that the events refined in Fig. <ref> preserve the invariants inv1, inv2 …inv6 but also the local properties regarding sizeD and domD[sizeD..N-1] (SIM proof obligations).
The feasibility (FIS) proof obligations generated by the non-deterministic events of SparseSets_ref1 require to prove that there exist values such that the BAP predicate holds. We can prove it using the new values of domD, mapD and sizeD specified in the last refinement as witnesses. The simulation (SIM) proof obligations generated by events of SparseSets_ref2 require to prove that the latter values again satisfy the BAP predicate used in SparseSets_ref1. In order not to do these -interactive- proofs twice, we generalize them and prove them as theorems of the context. Thus to discharge the FIS and SIM proof obligations, we only have to instanciate these theorems to provide a proof.
A last algorithmic refinement, omitted here, refines the remove event in two events, removeLastButOne and removeLast. The former differs from remove only by its more restrictive guard; the latter is dedicated to the case where the element with index sizeD-1 in domD is removed thus avoiding the unnecessary swapping.
§ FORMAL DEVELOPMENT
In this section we briefly present the tool and how we used it to encode the model of sparse sets.
§.§
is a constraint logic programming (CLP) language and satisfiability solver where sets and binary relations are first-class citizens <cit.>. The tool implements several decision procedures for expressive fragments of set theory and set relation algebra including cardinality constraints <cit.>, restricted universal quantifiers <cit.>, set-builder notation <cit.> and integer intervals <cit.>. In previous works has been satisfactory tested against some known case studies <cit.>.
code enjoys the formula-program duality. This means that code can behave as both a formula and a program. When seen as a formula, it can be used as a specification on which verification conditions can be (sometimes automatically) proved. When seen as a program, it can be used as a (less efficient) regular program. Due to the formula-program duality, a piece of code is sometimes called forgram—a portmanteau word resulting from combining formula with proggram.
§.§ formalization
The formalization presented in this paper is the result of translating the abstract specification (i.e., Fig. <ref>) and the second refinement (i.e. Fig. <ref>). Both models can be easily translated into by using the (still under development) state machine specification language (SMSL) defined on top of
(see Fig. <ref> and <ref>) <cit.>. The notions of context and refinement are not available in SMSL. For this reason, refinements introduced in the model have to be manually encoded in . The context is encoded simply as an axiom. In order to ensure that the code verifies the properties highlighted in bold in Fig. <ref> as well as the gluing invariant (i.e., inv6), a few user-defined verification conditions are introduced as theorems. Since the first refinement is introduced to express the properties written in bold, its events have not been encoded in .
Figures <ref> and <ref> list only representative parts of the forgram.
We tried to use the same identifiers as for the models as much as possible. In this way, for example, the invariant labeled as inv6 in the SparseSets_ref1 machine (Fig. <ref>), is named in the forgram. The name of variables in cannot fully complain with those used in the models because requires all variables to begin with a capital letter. So, for example, domD in the SparseSets_ref1 machine becomes in .
As can be seen in Fig. <ref>, the state machine specification language defined on top of allows for the declaration of parameters (similar to context constants), state variables, axioms (similar to context axioms) and invariants. Parameter is used to compute the identity relation on the integer interval [0,N-1] as shown in axiom , which in turn is used in invariant . As is a CLP language implemented on top of Prolog, it inherits many of Prolog's features. In particular, integer expressions are evaluated by means of the predicate. Along the same lines, all set operators are implemented in as constraints. For example, is true when is the identity relation on the set . The term corresponds to the integer interval [0,M].
Invariants named , and correspond to invariant inv1 of the SparseSets_ref1 machine. Splitting invariants in smaller pieces, is a good practice when using as a prover because it increases the chances of automated proofs. implements the negation of invariant . does not automatically compute the negation of user-defined predicates. As a user-defined predicate can contain existential variables, its negation could involve introducing universal quantifiers which fall outside 's decision procedures. Then, users are responsible for ensuring that all predicates are safe.
In invariant we can see the constraint. This constraint implements the notion of restricted universal quantifier (RUQ). That is, for some formula ϕ and set , corresponds to ∀ X.(X ∈ A ϕ(X)). In a constraint it is possible to quantify over binary relations, as is the case of . Hence, we have a quantified ordered pair (), rather than just a variable. Likewise, offers the constraint implementing the notion of restricted existential quantifier (REQ). The important point about REQ and RUQ is not only their expressiveness but the fact that there is a decision procedure involving them <cit.>. In these constraints are used to state a double set inclusion equivalent to the formula domD[0 .. sizeD - 1] = D. If the user is not convinced or unsure about the validity of this equivalence (s)he can use itself to prove it.
Note that is not declared as an invariant because in Fig. <ref> it is a theorem that can be deduced from previous invariants.
Therefore, we introduce it as a simple predicate but then we declare a theorem whose conclusion is . Later, will include as a proof obligation and will attempt to discharge it. Given that is a satisfiability solver, if Φ is intended to be a theorem then we ask it to prove the unsatisfiability of ¬Φ.
Moving into in Fig. <ref> we can see the encoding of the remove operation specified in the SparseSets_ref2 machine of Fig. <ref>, along with two user-defined proof obligations. In , there is no global state so state variables have to be included as explicit arguments of clauses representing operations. Next-state variables are denoted by decorating the base name with an underscore character (e.g., corresponds to the value of in the next state). Another important difference between the and the specifications is that in the latter we can use set unification to implement function application. For instance, is equivalent to the predicate: ∃ y_2, y_5, domD_1. (domD = {sizeD - 1 ↦ y_2, y_1 ↦ y_5}∪ domD_1), where y_1 = mapD(v) (due to the previous set unification). The not-membership constraints following the equality constraint prevent to generate repeated solutions. Hence, when is called with some set term in its fourth argument, this term is unified with . If the unification succeeds, then the images of and are available.
As said before, some user-defined proof obligations are introduced as theorems to ensure that the forgram verifies the gluing invariant (i.e., inv6) and the properties written in bold in machine SparseSets_ref1. Precisely, theorem states that if holds and and its abstract version (not shown in the paper) are executed, then holds in the next state.[ and its abstract version can be distinguished by their arities.]
Likewise, theorem ensures that the second property written in bold in machine SparseSets_ref1 is indeed a property of the forgram. As can be seen, the theorem states that if is executed and the functional image[ is a user-defined predicate computing the relational image through a function— stands for functional image.] of the interval from up to through is , then it must coincide with the functional image of the same interval but through .
Once the specification is ready, we can call the verification condition generator (VCG) and run the verification conditions (VC) so generated:
VCs include the satisfiability of the conjunction of all axioms, the satisfiability of each operation and preservation lemmas for each and every operation and invariant. The last command above will attempt to automatically discharge every VC. Part of the output is as follows:
An answer means that, for some reason, is unable to discharge the VC. Most of the times this is due to some missing hypothesis which, in turn, is due to the way the VCG generates the VCs. Briefly, when it comes to invariance lemmas, the VCG generates them with the minimum number of hypothesis. So, for instance, the invariance lemma named is as follows:
By including minimum hypothesis, will have to solve a simpler goal which reduces the possibilities to have a complexity explosion. If the hypothesis is not enough, the command can be used to find potential missing hypothesis.
In this way, users can edit the VC file, add the missing hypothesis and run the VC again. If more hypotheses are still missing, the process can be executed until the proof is done—or the complexity explosion cannot be avoided.
discharges all the VC generated by the VCG for the present forgram.
§ WHY3 FORMAL DEVELOPMENT
In this section we briefly introduce the 3 platform and describe with some details our specification of sparse sets.
§.§ 3
Why3 <cit.> is a platform for deductive program verification providing
a language for specification and programming, called WhyML, and relies on external automated and interactive theorem provers, to discharge verification conditions. In the context of this paper, we used Why3 with the SMT provers CVC4 and Z3.
Proof tactics are also provided, making 3 a proof environment close to the one of Rodin for interactive proofs. 3 supports modular verification.
WhyML allows the user to write functional or imperative programs featuring polymorphism, algebraic data types, pattern-matching, exceptions, references, arrays, etc. These programs can be annotated by contracts and assertions and thus verified. User-defined types with invariants can be introduced, the invariants are verified at the function call boundaries. Furthermore to prevent logical inconsistencies, 3 generates a verification condition to show the existence of at least one value satisfying the invariant. To help the verification, a witness is explicitly given by the user (see the clause in Fig. <ref>).
The and operators can be used inside post-conditions and assertions to refer to the value of a mutable program variable at some past moment of execution. In particular in a function post-condition refers to the value of term when the function is called. Programs may also contain ghost variables and ghost code to facilitate specification and verification.
From verified WhyML programs, correct-by-construction OCaml programs (and recently C programs) can be automatically extracted.
§.§ 3 formalization
From the 3 library, we use pre-defined theories for integer arithmetic, polymorphic finite sets and arrays. In the latter, we use in particular the operation that exchanges two elements in an array and its specification using the predicate.
We first define a record type, , whose mutable fields are a record of type containing the computational elements of a sparse set representation and a ghost finite set of integer numbers which is the abstract model of the data structure. The type invariant of relates the abstract model with the concrete representation. It is used
to enforce consistency between them. Invariants enforcing consistency between the two arrays and and the bound are attached to the type: lengths of the arrays is , contents are belonging to 0..-1 and the two arrays are inverse of each other, is in the interval 0... These type definitions and related predicates are shown in Fig. <ref>.
Our formalization (see Fig. <ref>, where, again, bind is removed for lack of place) contains three functions,
, and , which update their arguments. They are the straightforward translation of the algorithms in <cit.> in WhyML, except for the supplementary ghost code (the last statement in both and ) which updates the abstract model contained in . Function is a helper function
which is called in the other ones. The contract of makes explicit the modifications of both arrays and , using the predicate defined in the library. Verification conditions for this function concern the conformance of the code to the two post-conditions (trivial as it is ensured by ) and also the preservation of the invariant attached to the type—i.e. mainly that and after swapping elements remain inverse from each other.
Both and act not only on the two arrays and the bound but also on the ghost part, i.e. the corresponding mathematical set . Thus the verification conditions here not only concern the structural invariants related to , and but also the ones deriving from the use of the type, proving the link between the abstract logical view (using finite sets) and the computational one implemented through arrays.
Observe that types and correspond to the state and invariants of the refinements. The abstract specification presented in the first machine becomes a ghost field in WhyML. The invariant of the type corresponds to the gluing invariant (inv6). A similar transposition happens for the operations. Actions in the abstract events, i.e. updating the abstract set, appear as ghost code in WhyML.
All proofs are discovered by the automatic provers except for some proof obligations related to the function. Nevertheless these proofs are simplified thanks to some 3 tactics that inject some hints that can be used by the external provers to finish the proofs.
§ COMPARISON AND DISCUSSION
Set theory is primitive in and whereas Why3 which permits to express other theories, provides a theory for it. Rodin uses provers where set theory is primitive but can also call external provers such as VeriT, Z3 and CVC4—where set theory is not primitive. However a big effort has been done to process set theory in VeriT, which is often recognized as allowing significant improvements in proofs <cit.>.
Why3 relies entirely on external provers where set theory is not primitive. Conversely, is a satisfiability solver that can only work with set theory—and linear integer algebra. It is the only
of the three tools implementing advanced decision procedures for set theory. Likely, this proved to be crucial for being able to be the only tool that automatically discharged all the VC, although it required a simple hypothesis discovery procedure. It should be a concern the time needs to discharge all the VC because with more complex models the resolution time might be prohibitive. It worth to be studied ways of avoiding the algorithmic complexity of the decision procedures implemented in . Results on Computable Set Theory should be revisited (eg. <cit.>). Why3 and Rodin interactive proofs are not numerous and remain quite simple.
In , 51 proof obligations were generated for the whole development, around half of them coming from the first refinement.
37 were proven automatically by the standard provers (AtelierB provers), 18 automatically by SMT provers, mainly VeriT, either directly or after applying the Rodin lasso allowing for adding additional,
backup hypotheses having identifiers in common with
the goal. Only two proof obligations required real human intervention, mainly instantiations of the general theorems introduced in Ctx1 or explicit witnesses introduction in the case of feasibility proof obligations.
After working in the way described in Sect. <ref>, discharges all the 38 VC generated by the VCG in around 7 minutes.
Why3 makes it possible to apply transformations (e.g. split conjunctions) on a proof goal instead of calling an automatic prover on it. Some of these transformations are very simple, e.g. splitting conjunctions, and can then been applied systematically and automatically. Most of the generated VC in our formalization were proven automatically thanks to the split transformation. Only two of them about pieces of type invariants, required human interaction to insert some more complex transformations, e.g a case analysis on indexes in mapD (). At the end, 55 VC were proved by CVC4, except two of them discharged by Z3, in a total amount of time of 30 seconds.
Clearly, all three tools are expressive enough for the problem at hand. However, the specification is probably the most readable. The tools permit to express axioms, invariants and automatically generate similar VC. still needs work to express how two models are linked in terms of abstraction/refinement relations. Writing some key properties proved to be complex in . Indeed, it was necessary to add a somewhat artificial refinement level for Rodin being able to generate the desired VC linking. These properties can be easily defined by the user in . However, in Why3 and , proof obligations are automatically generated from the specifications, in particular the abstract and concrete models can be naturally linked and the tool automatically generates the corresponding VC. In that regard, Why3 and are safer than .
The possibility to count with executable code without much effort enables many lightweight analysis that can be put into practice before attempting complex proofs. In tool where specification and implementation are described by only one piece of code (cf. forgrams). This tool is not the integration of an interpreter and a prover; the same set of rewrite rules are used to compute and prove. In /Rodin there is only a specification—later it can be converted into an executable representation if tools such as ProB are used.
Why3 can execute WhyML programs natively thanks to its interpreter and the command.
Furthermore, once the the program is proved to verify the specification, correct-by-construction OCaml and C programs can be automatically extracted. These programs will be orders of magnitude more efficient than the equivalent forgrams.
§ CONCLUSION
We formally verified the implementation of sparse sets using three formal languages and associated tools, focusing on the operations and correctness properties required by a constraint solver when domains of integer variables are implemented with sparse sets. We compared in particular the several statements of invariants and pre-post properties and their proofs.
As future work, two directions can be investigated. The first one is to complete the formal developments with other set operations. A second one is to implement and verify, in Why3 or , a labeling procedure such as the ones used in constraint solvers, it would need to backtrack on the values of some domains, and thus make use of the theorems proven in this paper. Labeling is native in when the CLP(FD) solver is active.
abbrv
|
http://arxiv.org/abs/2307.04287v1 | 20230710002925 | Generalizing Graph ODE for Learning Complex System Dynamics across Environments | [
"Zijie Huang",
"Yizhou Sun",
"Wei Wang"
] | cs.LG | [
"cs.LG",
"cs.AI",
"cs.CE",
"cs.MA",
"cs.NE"
] |
University of California, Los Angeles
Los Angeles
CA
[email protected]
University of California, Los Angeles
Los Angeles
CA
[email protected]
University of California, Los Angeles
Los Angeles
CA
[email protected]
Learning multi-agent system dynamics has been extensively studied for various real-world applications, such as molecular dynamics in biology,
multi-body system in physics, and particle dynamics in material science.
Most of the existing models are built to learn single system dynamics, which learn the dynamics from observed historical data and predict the future trajectory.
In practice, however, we might observe multiple systems that are generated across different environments, which differ in latent exogenous factors such as temperature and gravity.
One simple solution is to learn multiple environment-specific models, but it fails to exploit the potential commonalities among the dynamics across environments and offers poor prediction results where per-environment data is sparse or limited.
Here, we present (Generalized Graph Ordinary Differential Equations), a machine learning framework for learning continuous multi-agent system dynamics across environments. Our model learns system dynamics using neural ordinary differential equations (ODE) parameterized by Graph Neural Networks (GNNs) to capture the continuous interaction among agents. We achieve the model generalization by assuming
the dynamics across different environments are governed by common physics laws that can be captured via learning a shared ODE function. The distinct latent exogenous factors learned for each environment are incorporated into the ODE function to account for their differences. To improve model performance, we additionally design two regularization losses to (1) enforce the orthogonality between the learned initial states and exogenous factors via mutual information minimization; and (2) reduce the temporal variance of learned exogenous factors
within the same system via contrastive learning. Experiments over various physical
simulations show that our model can accurately predict system dynamics, especially in the long range, and can generalize well to new systems with few observations.
Generalizing Graph ODE for Learning
Complex System Dynamics across Environments
Wei Wang
August 12, 2023
=================================================================================
§ INTRODUCTION
Building a simulator that can understand and predict multi-agent system dynamics is a crucial research topic spanning over a variety of domains such as planning and control in robotics <cit.>, where the goal is to generate future trajectories of agents based on what has been seen in the past. Traditional simulators can be very expensive to create and use <cit.> as it requires sufficient domain knowledge and tremendous computational resources to generate high-quality results[To date, out of the 10 most powerful supercomputers in the world, 9 of them are used for simulations, spanning the fields of cosmology, geophysics and fluid dynamics <cit.>]. Therefore, learning a neural-based simulator directly from data that can approximate the behavior of traditional simulators becomes an attractive alternative.
As the trajectories of agents are usually coupled with each other and co-evolve along with the time, existing studies on learning system dynamics from data usually view the system as a graph and employ Graph Neural Networks (GNNs) to approximate pair-wise node (agent) interaction to impose strong inductive bias <cit.>. As a pioneering work, Interaction Networks (IN) <cit.> decompose the system into distinct objects and relations, and learn to reason about the consequences of their interactions and dynamics. Later work incorporates domain knowledge <cit.>, graph structure variances <cit.>,
and equivariant representation learning <cit.> into learning from discrete GNNs, achieving state-of-the-art performance in various domains including mesh-based physical simulation <cit.> and molecular prediction <cit.>.
However, these discrete models usually suffer from low accuracy in long-range predictions as (1) they approximate the system by discretizing observations into some fixed timestamps and are trained to make a single forward-step prediction and (2) their discrete nature fails to adequately capture systems that are continuous in nature such as the spread of COVID-19 <cit.> and the movements of an n-body system <cit.>.
Recently, researchers propose to combine ordinary differential equations (ODEs) - the principled way for modeling dynamical systems in a continuous manner in the past, with GNNs to learn continuous-time dynamics on complex networks in a data-driven way <cit.>. These Graph-ODE methods have demonstrated the power of capturing long-range dynamics, and are capable of learning from irregular-sampled partial observations <cit.>. They usually assume all the data are generated from one single system, and the goal is to learn the system dynamics from historical trajectories to predict the future.
In practice, however, we might observe data that are generated from multiple systems, which can differ in their environments. For example, we may observe particle trajectories from systems that are with different temperatures, which we call exogenous factors.
These exogenous factors can span over a wide range of settings such as particle mass, gravity, and temperature <cit.> across environments. One simple solution is to learn multiple environment-specific models, but it can fail to exploit the potential commonalities across environments and make accurate predictions for environments with sparse or zero observations. In many useful contexts, the dynamics in multiple environments share some similarities, yet being distinct reflected by the (substantial) differences in the observed trajectories. For example, considering the movements of water particles within multiple containers of varying shapes, the trajectories are driven by both the shared pair-wise physical interaction among particles (i.e. fluid dynamics) and the different shapes of the containers where collisions can happen when particles hit the boundaries. Also, the computational cost for training multiple environment-specific models would be huge. More challengingly, the exogenous factors within each environment can be latent, such as we only know the water trajectories are from different containers, without knowing the exact shape for each of them. Therefore, how to learn a single efficient model that can generalize across environments by considering both their commonalities and the distinct effect of per-environment latent exogenous factors remains unsolved. This model, if developed, may help us predict dynamics for systems under new environments with very few observed trajectories.
Inspired by these observations, in this paper, we propose Generalized Graph ODE (), a general-purpose continuous neural simulator that learns multi-agent system dynamics across environments. Our key idea is to assume the dynamics across environments are governed by common physics laws that can be captured via learning a shared ODE function. We introduce in the ODE function a learnable vector representing the distinct latent exogenous factors for each environment to account for their differences. We learn the representations for the latent exogenous factors from systems' historical trajectories through an encoder by optimizing the prediction goal. In this way, different environments share the same ODE function framework while incorporating environment-specific factors in the ODE function to distinguish them.
However, there are two main challenges in learning such latent exogenous factor representations. Firstly, since both the latent initial states for agents and the latent exogenous factors are learned through the historical trajectory data, how can we differentiate them to guarantee they have different semantic meanings? Secondly, when inferring from different time windows from the same trajectory, how can we guarantee the learned exogenous factors are for the same environment?
Towards the first challenge, we enforce the orthogonality between the initial state encoder and the exogenous factor encoder via mutual information minimization. For the second challenge, we reduce the variance of learned exogenous factors within the same environment via a contrastive learning loss. We train our model in a multi-task learning paradigm where we mix the training data from multiple systems with different environments. In this way, the model is expected to fast adapt to other unseen systems with a few data points. We conduct extensive experiments over a wide range of physical systems, which show that our is able to accurately predict system dynamics, especially in the long range.
The main contributions of this paper are summarized as follows:
* We investigate the problem of learning continuous multi-agent system dynamics across environments. We propose a novel framework, known as , which describes the dynamics for each system with a shared ODE function and an environment-specific vector for the latent exogenous factors to capture the commonalities and discrepancies across environments respectively.
* We design two regularization losses to guide the learning process of the latent exogenous factors, which is crucial for making precise predictions in the future.
* Extensive experiments verify the effectiveness of GG-ODE to accurately predict system dynamics, especially in the long range prediction tasks. also generalizes well to unseen or low-resource systems that have very few training samples.
§ PROBLEM DEFINITION
We aim to build a neural simulator to learn continuous multi-agent system dynamics automatically from data that can be generalized across environments. Throughout this paper, we use boldface uppercase letters to denote matrices or vectors, and regular lowercase letters to represent the values of variables.
We consider a multi-agent dynamical system of N interacting agents as an evolving interaction graph 𝒢^t = {𝒱,ℰ^t}, where nodes are agents and edges are interactions between agents that can change over time. For each dynamical system, we denote e∈ E as the environment from which the data is acquired. We denote X^t,e∈𝒳 as the feature matrix for all N agents and x_i^t,e as the feature vector of agent i at time t under environment e. The edges between agents are assigned if two agents are within a connectivity radius R based on their current locations p_i^ which is part of the node feature vector, i.e. p_i^∈x_i^. They reflect the local interactions of agents and the radius is kept constant over time <cit.>.
Our model input consists of the trajectories of N agents over K timestamps X^t_1:K,e={X^t_1,e, X^t_2,e, …, X^t_K,e}, where the timestamps t_1,t_2⋯ t_K can have non-uniform intervals and be of any continuous values. Our goal is to learn a generalized simulator s_θ:X^t_1:K,e→ Y^t_K+1:T,e that predicts node dynamics in the future for any environment e.
Here Y^∈𝒴 represents the targeted node dynamic information at time t, and can be a subset of the input features. We use y_i^ to denote the targeted node dynamic vector of agent i at time t under environment e.
§ PRELIMINARIES AND RELATED WORK
§.§ Dynamical System Simulations with Graph Neural Networks (GNNs)
Graph Neural Networks (GNNs) are a class of neural networks that operate on graph-structured data by passing local messages<cit.>.
They have been extensively employed in various applications such as node classification <cit.>, link prediction <cit.>, and recommendation systems <cit.>.
By viewing each agent as a node and interaction among agents as edges, GNNs have shown to be efficient for approximating pair-wise node interactions and achieved accurate predictions for multi-agent dynamical systems <cit.>.
The majority of existing studies propose discrete GNN-based simulators where they take the node features at time t as input to predict the node features at time t+1. To further capture the long-term temporal dependency for predicting future trajectories, some work utilizes recurrent neural networks such as RNN, LSTM or self-attention mechanism to make prediction at time t +1 based on the historical trajectory sequence within a time window <cit.>. However, they all restrict themselves to learn a one-step state transition function. Therefore, when successively apply these one-step simulators to previous predictions in order to generate the rollout trajectories, error accumulates and impairs the prediction accuracy, especially for long-range prediction.
Also, when applying most discrete GNNs to learn over multiple systems under different dynamical laws (environments), they usually retrain the GNNs individually for dealing with each specific system environment <cit.>, which yields a large computational cost.
§.§ Ordinary Differential Equations (ODEs) for
Multi-agent Dynamical Systems
The dynamic nature of a multi-agent system can be captured by a series of nonlinear first-order ordinary differential equations (ODEs), which describe the co-evolution of states for a set of N dependent variables (agents) over continuous time t∈ℝ as <cit.>: ż_i^t:=d z_i^t/d t=g(z_1^t, z_2^t⋯z_N^t). Here z_i^t∈ℝ^d denotes the state variable for agent i at timestamp t and g denotes the ODE function that drives the system move forward. Given the initial states z_1^0,
⋯z_N^0 for all agents and the ODE function g, any black box numerical ODE solver such as Runge-Kuttais <cit.> can solve the ODE initial-value problem (IVP), of which the solution z_i^T can be evaluated at any desired time as shown in Eqn <ref>.
z_i^T=z_i^0+∫_t=0^T g(z_1^t, z_2^t⋯z_N^t) d t
Traditionally, the ODE function g is usually hand-crafted based on some domain knowledge such as in robot motion control <cit.> and fluid dynamics <cit.>, which is hard to specify without knowing too much about the underlying principles. Even if the exact ODE functions are given, they are usually hard to scale as they require complicated numerical integration <cit.>. Some recent studies <cit.> propose to parameterize it with a neural network and learn it in a data-driven way. They combine the expressive power of neural networks along with the principled modeling of ODEs for dynamical systems, which have achieved promising results in various applications <cit.>.
§.§ GraphODE for Dynamical Systems
To model the complex interplay among agents in a dynamical system, researchers have recently proposed to combine ODE with GNNs, which has been shown to achieve superior performance in long-range predictions <cit.>. In <cit.>, an encoder-processor-decoder architecture is proposed, where an encoder first computes the latent initial states for all agents individually based on their first observations. Then an ODE function parameterized by a GNN predicts the latent trajectories starting from the learned initial states. Finally, a decoder extracts the predicted dynamic features based on a decoding function that takes the predicted latent states as input. Later on, a Graph-ODE framework has been proposed <cit.> which follows the structure of variational autoencoder <cit.>. They assume an approximated posterior distribution over the latent initial state for each agent, which is learned based on the whole historical trajectories instead of a single point as in <cit.>. The encoder computes the approximated posterior distributions for all agents simultaneously considering their mutual influence and then sample the initial states from them. Compared with <cit.>, they are able to achieve better prediction performance, especially in the long range, and are also capable of handling the dynamic evolution of graph structures <cit.> which is assumed to be static in <cit.>.
We follow a similar framework to this line but aim at generalizing GraphODE to model multiple systems across environments.
§ METHOD
In this section, we present Generalized Graph ODE ( ) for learning complex system dynamics across environments. As depicted in Figure <ref>, consists of four main components that are trained jointly: (1) an initial state encoder for inferring the latent initial states for all agents simultaneously; (2) an environment encoder which learns the latent representations for exogenous factors; (3) a generative model defined by a GNN-based ODE function that is shared across environments for modeling the continuous interaction among agents in the latent space. The distinct latent exogenous factors learned
for each environment are incorporated into the ODE function to
account for their discrepancies, and (4) a decoder that extracts the predicted dynamic features based on a decoding function. We now introduce each component in detail.
§.§ Initial State Encoder
Given the observed trajectories X^t_1:K,e, the initial state encoder computes a posterior distribution of latent initial state q_ϕ(z_i^0,e| X^t_1:K,e) for each agent, from which z_i^0,e is sampled. The latent initial state z_i^0,e for each agent determines the starting point for the predicted trajectory. We assume the prior distribution p(z_i^0,e) is a standard normal distribution, and use Kullback–Leibler divergence term in the loss function to add significant regularization towards how the learned distributions look like, which differs VAE from other autoencoder frameworks <cit.>.
In multi-agent dynamical systems, agents are highly-coupled and influence each other. Instead of learning such distribution separately for each agent, such as using an RNN <cit.> to encode the temporal pattern for each individual trajectory, we compute the posterior distributions for all agents simultaneously (similar to <cit.>).
Specifically, we fuse all trajectories as a whole into a temporal graph to consider both the temporal patterns of individual agents and the mutual interaction among them, where each node is an observation of an agent at a specific timestamp. Two types of edges are constructed, which are (1) spatial edges 𝒱^t that are among observations of interacting agents at each timestamp if the Euclidean distance between the agents' positions r_ij^t,e = ||p_i^ - p_j^||_2 is within a (small) connectivity radius R; and (2) temporal edges that preserve the autoregressive nature of each trajectory, defined between two consecutive observations of the same agent. Note that spatial edges are bidirectional while temporal edges are directional to preserve the autoregressive nature of each trajectory, as shown in Figure <ref>. Based on the constructed temporal graph, we learn the latent initial states for all agents through a two-step procedure: (1) dynamic node representation learning that learns the representation h_i^ for each observation node whose feature vector is x_i^. (2) sequence representation learning that summarizes each observation sequence (trajectory) into
a fixed-dimensional vector through a self-attention mechanism.
§.§.§ Dynamic Node Representation Learning.
We first conduct dynamic node representation learning on the temporal graph through an attention-based spatial-temporal GNN defined as follows:
h_j^l+1(t,e)=h_j^l(t,e)+σ(∑_i^(t^',e)∈𝒩_j^()α_i^l(t',e) → j(t,e)×W_vĥ_i^l(t',e))
α_i^l(t',e) → j(t,e) = (W_kĥ_i^l(t',e))^T(W_qh_j^l(t,e)) ·1/√(d)
ĥ_i^l(t',e)= h_i^l(t',e) + TE(t'-t)
TE(Δ t)_2i=sin(Δ t/10000^2 i / d), TE(Δ t)_2i+1=cos(Δ t/10000^2 i / d)
where σ(·) is a non-linear activation function; d is the dimension of node embeddings. The node representation is computed as a weighted summation over its neighbors plus residual connection where the attention score is a transformer-based <cit.> dot-product of node representations by the use of value, key, query projection matrices W_v,W_k,W_q. The learned attention scores are normalized via softmax across all neighbors. Here h_j^l(t,e) is the representation of agent j at time t in the l-th layer. h_i^l(t^',e) is the general representation for a neighbor which is connected either by a temporal edge (where t'<t and i=j) or a spatial edge (where t=t' and i≠ j) to the observation h_j^l(t,e). We add temporal encoding <cit.> to each neighborhood node representation in order to distinguish the message delivered via spatial and temporal edges respectively. Finally, we stack L layers to get the final representation for each observation node as : h_i^ = h_i^L(t,e).
§.§.§ Sequence Representation Learning
We then employ a self-attention mechanism to generate the sequence representation m_i^e for each agent, which is used to compute the mean μ_i^0,e and variance σ_i^0,e of the approximated posterior distribution of the agent's initial state. Compared with recurrent models such as RNN, LSTM <cit.>, it offers better parallelization for accelerating training speed and in the meanwhile alleviates the vanishing/exploding gradient problem brought by long sequences <cit.>.
We follow <cit.> and compute the sequence representation m_i^e as a weighted sum of observations for agent i:
m_i^e=1/K∑_tσ((a_i^e)^T ĥ_i^ĥ_i^), a_i^e=tanh((1/K∑_tĥ_i^) W_a),
where a_i^e is the average of observation representations with a nonlinear transformation W_a and ĥ_i^ = h_i^ + TE(t). K is the number of observations for each trajectory. Then the initial state is drawn from the
approximated posterior distribution as:
q_ϕ(z_i^0,e| X^t_1:K,e)=𝒩(μ_i^0,e, σ_i^0,e) , μ_i^0,e, σ_i^0,e=f_trans(m_i^e)
z_i^0,e∼ p(z_i^0,e) ≈ q_ϕ(z_i^0,e| X^t_1:K,e)
where f_trans is a simple Multilayer Perceptron (MLP) whose output vector is equally split into two halves to represent the mean and variance respectively.
§.§ Environment Encoder
The dynamic nature of a multi-agent system can be largely affected by some exogenous factors from its environment such as gravity, temperature, etc. These exogenous factors can span over a wide range of settings and are sometimes latent and not observable. To make our model generalize across environments, we design an environment encoder to learn the effect of the exogenous factors automatically from data to account for the discrepancies across environments. Specifically, we use the environment encoder to learn the representations of exogenous factors from observed trajectories and then incorporate the learned vector into the ODE function which is shared across environments and defines how the system evolves over time. In this way, we use a shared ODE function framework to capture the commonalities across environments while preserving the differences among them with the environment-specific latent representation, to improve model generalization performance. It also allows us to learn the exogenous factors of an unseen environment based on only its leading observations.
We now introduce the environment encoder in detail.
The exogenous factors would pose influence on all agents within a system. On the one hand, they will influence the self-evolution of each individual agent. For example, temperatures would affect the velocities of agents. On the other hand, they will influence the pair-wise interaction among agents. For example, temperatures would also change the energy when two particles collide with each other. The environment encoder f_enc^env therefore learns the latent representation of exogenous factors u^e by jointly consider the trajectories from all agents, i.e. f_enc^env: X^t_1:K,e→u^e. Specifically, we learn an environment-specific latent vector from the aforementioned temporal graph in Sec <ref> that is constructed from observed trajectories. The temporal graph contains both the information for each individual trajectory and the mutual interaction among agents through temporal and spatial edges.
To summarize the whole temporal graph into a vector u^e, we attend over the sequence representation m_i^e for each trajectory introduced in Sec <ref> as:
u^e=1/N∑_iσ((b^e)^T m_i^em_i^e), b^e=tanh((1/N∑_im_i^e) W_b),
where W_b is a transformation matrix and the attention weight is computed based on the average sequence representation with nonlinear transformation similar as in Eqn (<ref>). Note that we use different parameters to compute the sequence representation m_i^e as opposed to the initial state encoder. The reason is that the semantic meanings of the two sequence representations are different: one is for the latent initial states and another is for the exogenous factors.
§.§.§ Time Invariance.
A desired property of the learned representation for exogenous factors u^e is that it should be time-invariant towards the input trajectory time window. In other words, for the same environment, if we chunk the whole trajectories into several pieces, the inferred representations should be similar to each other as they are describing the same environment.
To achieve this, we design a contrastive learning loss to guide the learning process of the exogenous factors. As shown in Figure <ref>, we force the learned exogenous factor representations to be similar if they are generated based on the trajectories from the same environment (positive pairs), and to be apart from each other if they are from different environments (negative pairs). Specifically, we define the contrastive leanring loss as follows:
ℒ_contra =-logexp(sim(f_enc ^env (X^t_1: t_2, e), f_enc ^env (X^t_3: t_4, e)) / τ)/∑_e^'≠ eexp( s i m (f_enc ^env (X^t_1: t_2, e, f_enc ^env (X^t_5: t_6, e^') / τ)..
where τ is a temperature scalar and sim(·, ·) is cosine similarity between two vectors. Note that the lengths of the observation sequences can vary. The detailed generation process for positive and negative pairs can be found in Appendix <ref>.
§.§.§ Orthogonality.
features two encoders that take the input of observed trajectories X^t_1:K,e for learning the latent initial states and the latent exogenous factors respectively. As they are designed for different purposes but are both learned from the same input, we disentangle the learned representations from them via a regularization loss defined via mutual information minimization.
Mutual information measures the dependency between two random variables X,Z <cit.>. Since we are not interested in the exact value of the mutual information, a lower bound derived from Jensen Shannon Divergence <cit.> could be formulated as
I_JSD(X, Z)= E_P_X Z[-sp(-M(x, z))] -E_P_X P_Z[sp(M(x, z))],
where P_X P_Z is the product of the marginal distributions and P_X Z is the joint distribution. sp(w)=log(1+e^w) and M is a discriminator modeled by a neural network to compute the score for measuring their mutual information.
According to recent literature <cit.>, the sample pair (positive pairs) (x,z) drawn from the joint distribution P_X Z are different representations of the same data sample, and the sample pair (negative pairs) drawn from P_X P_Z are different representations from different data samples. We therefore attempt to minimize the mutual information from the two encoders as follows
ℒ_MI
=𝔼_e∈ E,i[-sp(-Ψ(z_i^0,e, u^e))]-𝔼_e∈ E× e'∈ E,i[sp(Ψ(z_i^0,e, u^e'))]
where Ψ is a MLP-based discriminator. Specifically, we force the latent initial states z_i^0,e for all agents from environment e to be dissimilar to the learned exogenous factors u^e. And construct negative pairs by replacing the learned exogenous factors from another environment as u^e'. The generation process for positive and negative pairs can be found in Appendix <ref>.
§.§ ODE Generative Model and Decoder
§.§.§ ODE Generative Model
After describing the initial state encoder and the environment encoder, we now define the ODE function that drives the system to move forward. The future trajectory of each agent can be determined by two important factors: the potential influence received from its neighbors in the interaction graph and the self-evolution of each agent. For example, in the n-body system, the position of each agent can be affected both by the force from its connected neighbors and its current velocity which can be inferred from its historical trajectories.
Therefore, our ODE function consists of two parts: a GNN that captures the continuous interaction among agents and the self-evolution of the node itself. One issue here is how can we decide the neighbors for each agent in the ODE function as the interaction graph is evolving, the neighbors for each agent are dynamically changing based on their current positions, which are implicitly encoded in their latent state representations z_i^t,e, z_j^t,e. We propose to first decode the latent node representations z_i^t,e, z_j^t,e with a decoding function f_dec to obtain their predicted positions p_i^t,e, p_j^t,e at current timestamp. Then we determine their connectivity based on whether their Euclidean distance r_ij^t,e = ||p_i^ - p_j^t,e||_2 is within the predefined radius R. This can be computed efficiently by using a multi-dimensional index structure such as the k-d tree. The decoding function f_dec is the same one that we will use in the decoder.
To incorporate the influence of exogenous factors, we further incorporate u^e into the general ODE function to improve model generalization ability as:
d z_i^/dt = g(z_1^, z_2^⋯z_N^) = ∑_j∈𝒩_i f_GNN(z_i^, z_j^) + f_self(z_i^)
z_i^ = f_env(z_i^||u^e)
where || denotes concatenation and f_GNN can be any GNN that conducts message passing among agents. f_self, f_env are implemented as two MLPs respectively. In this way, we learn the effect of latent exogenous factors from data without supervision where the latent representation u^e is trained end-to-end by optimizing the prediction loss.
§.§.§ Decoder
Given the ODE function g and agents' initial states z_i^0,e for i=1,2⋯ N, the latent trajectories for all agents are determined, which can be solved via any black-box ODE solver. Finally, a decoder generates the predicted dynamic features based on the decoding probability p(y_i^t,e | z_i^t,e) computed from the decoding function f_dec as shown in Eqn <ref>. We implement f_dec as a simple two-layer MLP with nonlinear activation. It outputs the mean of the normal distribution p(y_i^t,e | z_i^t,e), which we treat as the predicted value for each agent.
z_i^t_1,e⋯z_i^t_T,e = ODESolve(g, [z_1^0,e,z_2^0,e⋯z_N^0,e],(t_1⋯ t_T))
y_i^t,e ∼ p(y_i^t,e | z_i^t,e) = f_dec(z_i^t,e)
§.§ Training
We now introduce the overall training procedure of . For each training sample, we split it into two halves along the time, where we condition on the first half [t_1,t_K] in order to predict dynamics in the second half [t_K+1,t_T]. Given the observed trajectories X^t_1:K,e, we first run the initial state encoder to compute the latent initial state z_i^0,e for each agent, which is sampled from the approximated posterior distribution q_ϕ(z_i^0,e| X^t_1:K,e). We then generate the latent representations of exogenous factors u^e from the environment e via the environment encoder. Next, we run the ODE generative model that incorporates the latent exogenous factors to compute the latent states for all agents in the future. Finally, the decoder outputs the predicted dynamics for each agent.
We jointly train the encoders, ODE generative model, and decoder in an end-to-end manner. The loss function consists of three parts: (1) the evidence lower bound (ELBO) which is the addition of the reconstruction loss for node trajectories and the KL divergence term for adding regularization to the inferred latent initial states for all agents. We use Z^0,e to denote the latent initial state matrix of all N agents. The standard VAE framework is trained to maximize ELBO so we take the negative as the ELBO loss; (2) the contrastive learning loss for preserving the time invariance properties of the learned exogenous factors; (3) the mutual information loss that disentangles the learned representations from the two encoders. λ_1, λ_2 are two hyperparameters for balancing the three terms. We summarize the whole procedure in Appendix <ref>.
ℒ = ℒ_ELBO + λ_1ℒ_contra + λ_2 ℒ_MI
ℒ_ELBO(θ,ϕ) = -𝔼_Z^0,e∼∏_i=1^Nq_ϕ(z_i^0,e| X^t_1:K,e)[log p_θ(Y^t_K+1:T,e)]
+KL[∏_i=1^Nq_ϕ(z_i^0,e |X^t_1:K,e) p(Z^0,e)]
§ EXPERIMENTS
§.§ Experiment Setup
§.§.§ Datasets
We illustrate the performance of our model across two physical simulations that exhibit different system dynamics over time: (1) The Water dataset <cit.>, which describes the fluid dynamics of water within a container. Containers can have different shapes and numbers of ramps with random positions inside them, which we view as different environments. The dataset is simulated using the material point method (MPM), which is suitable for simulating the behavior of interacting, deformable materials such as solids, liquids, gases [<https://en.wikipedia.org/wiki/Material_point_method>]. For each data sample, the number of particles can vary but the trajectory lengths are kept the same as 600. The input node features are 2-D positions of particles, and we calculate the velocities and accelerations as additional node features using finite differences of these positions. The total number of data samples (trajectories) is 1200 and the number of environments is 68, where each environment can have multiple data samples with different particle initializations such as positions, velocities, and accelerations.
(2) The Lennard-Jones potential dataset <cit.>, which describes the soft repulsive and attractive interactions between simple atoms and molecules [<https://en.wikipedia.org/wiki/Lennard-Jones_potential>]. We generate data samples with different temperatures, which could affect the potential energy preserved within the whole system thus affecting the dynamics. We view temperatures as different environments. The total number of data samples (trajectories) is 6500 and the number of environments is 65. Under each environment, we generate 100 trajectories with different initializations. The trajectory lengths are kept the same as 100. The number of particles is 1000 for all data samples. More details about datasets can be found in Appendix <ref>.
§.§.§ Task Evaluation and Data Split
We predict trajectory rollouts across varying lengths and use Mean Square Error (MSE) as the evaluation metric.
Task Evaluation.
The trajectory prediction task is conducted under two settings: (1) Transductive setting, where we evaluate the test sequences whose environments are seen during training; (2) Inductive setting, where we evaluate the test sequences whose environments are not observed during training. It helps to test the model's generalization ability to brand-new systems.
Data Split.
We train our model in a sequence-to-sequence setting where we split the trajectory of each training sample into two parts [t_1,t_K] and [t_K+1,t_T]. We condition on the first part of observations to predict the second part. To conduct data split, we first randomly select 20% environments whose trajectories are all used to construct the testing set X_test^Induct in the inductive setting. For the remaining trajectories that cover the 80% environments, we randomly split them into three partitions: 80% for the training set X_train, 10% for the validation set X_val and 10% for the testing set in the transductive setting X_test^trans. In other words, we have two test sets for the inductive and transductive settings respectively, one training set and one validation set.
To fully utilize the data points within each trajectory, we generate training and validation samples by splitting each trajectory into several chunks that can overlap with each other, using a sliding window. The sliding window has three hyperparameters: the observation length and prediction length for each sample, and the interval between two consecutive chunks (samples). Specifically, for the Water dataset, we set the observation length as 50 and the prediction length as 150. We obtain samples from each trajectory by using a sliding window of size 200 and setting the sliding interval as 50. For the Lennard-Jones potential dataset, we set the observation length as 20, the prediction length as 50, and the interval as 10. The procedure is summarized in Appendix <ref>. During evaluations for both settings, we ask the model to roll out over the whole trajectories without further splitting, whose prediction lengths are larger than the ones during training. The observation lengths during testing are set as 20 for the Lennard-Jones potential dataset and 50 for the Water dataset across the two settings.
§.§ Baselines
We compare both discrete neural models as well as continuous neural models where they do not have special treatment for modeling the influence from different environments. For discrete ones we choose: NRI <cit.> which is a discrete GNN model that uses VAE to infer the interaction type among pairs of agents and is trained via one-step predictions; GNS <cit.>, a discrete GNN model that uses multiple rounds of message passing to predict every single step; LSTM <cit.>, a classic recurrent neural network (RNN) that learns the dynamics of each agent independently. For the continuous models, we compare with NDCN <cit.> and Social ODE <cit.>, two ODE-based methods that follow the encoder-processor-decoder structure with GNN as the ODE function. The initial state for each agent is drawn from a single data point instead of a leading sequence. CG-ODE <cit.> which has the same architecture as our model, but with two coupled ODE functions to guide the evolution of systems.
§.§ Performance Evaluation
We evaluate the performance of our model based on Mean Square Error (MSE) as shown in Table <ref>. As data samples have varying trajectory lengths, we report the MSEs over three rollout percentages regarding different prediction horizons: 30%, 60%, 100% where 100% means the model conditions on the observation sequence and predicts all the remaining timestamps.
Firstly, we can observe that consistently outperforms all baselines across different settings when making long-range predictions, while achieving competitive results when making short-range predictions. This demonstrates the effectiveness of in learning continuous multi-agent system dynamics across environments. By comparing the performance of LSTM with other methods, we can see that modeling the latent interaction among agents can indeed improve the prediction performance compared with predicting trajectories for each agent independently. Also, we can observe the performance gap between and other baselines increase when we generate longer rollouts, showing its expressive power when making long-term predictions. This may be due to the fact that is a continuous model trained in a sequence-to-sequence paradigm whereas discrete GNN methods are only trained to make a fixed-step prediction. Another continuous model NDCN only conditions a single data point to make predictions for the whole trajectory in the future, resulting in suboptimal performance. Finally, we can see that has a larger performance gain over existing methods in the inductive setting than in the transductive setting, which shows its generalization ability to fast adapt to other unseen systems with a few data points. Figure <ref> visualizes the prediction results under the transductive setting for the Water dataset.
§.§.§ Ablation Studies
To further analyze the rationality behind our model design, we conduct an ablation study by considering three model variants: (1) We remove the contrastive learning loss which forces the learned exogenous factors to satisfy the time invariance property, denoted as -w/o ℒ_contra; (2) We remove the mutual information minimization loss which reduces the variance of the learned exogenous factors from the same environment, denoted as -w/o ℒ_MI. (3) We share the parameters of the two encoders for computing the latent representation m_i^e for each observation sequence in the temporal graph, denoted as shared encoders. As shown in Table <ref>, all three variants have inferior performance compared to , verifying the rationality of the three key designs. Notably, when making long-range predictions, removing ℒ_MI would cause more harm to the model than removing ℒ_contra. This can be understood as the latent initial states are more important for making short-term predictions, while the disentangled latent initial states and exogenous factors are both important for making long-range predictions.
§.§.§ Hyperparameter Study
We study the effect of λ_1/λ_2, which are the hyperparameters for balancing the two regularization terms that guide the learning of the two encoders, towards making predictions under different horizons. As illustrated in Figure <ref>, the optimal ratio for making 30%, 60%, 100% rollout predictions are 2, 1,0.5 respectively, under both the transductive and inductive settings. They indicate that the exogenous factors modeling plays a more important role in facilitating long-term predictions, which is consistent with the prediction errors illustrated in Table <ref> when comparing -w/o ℒ_MI with -w/o
ℒ_contra. However, overly elevating ℒ_MI would also harm the model performance, as the time invariance property achieved by ℒ_contra is also important to guarantee the correctness of the learned latent initial states, which determines the starting point of the predicted trajectories in the future.
§.§.§ Sensitivity Analysis.
can take arbitrary observation lengths to make trajectory predictions, as opposed to existing baselines that only condition on observations with fixed lengths. It allows the model to fully utilize all the information in the past. We then study the effect of observation lengths on making predictions in different horizons. As shown in Figure <ref>, the optimal observation lengths for predicting the rollouts with 20, 40, and 50 steps are 20, 25, 35 in the inductive setting, and 15, 25, 30 in the transductive setting. When predicting long-range trajectories, our model typically requires a longer observation sequence to get more accurate results. Also, for making predictions at the same lengths, the inductive setting requires a longer observation length compared with the transductive setting.
§.§ Case Study
We conduct a case study to examine the learned representations of the latent exogenous factors on the Lennard-Jones potential dataset. We first randomly choose one data sample for each of the 65 temperatures and visualize the learned representations of exogenous factors. As shown in Figure <ref> (a), the representations of higher temperatures are closer to each other on the right half of the figure, whereas the lower temperatures are mostly distributed on the left half. Among the 65 temperatures, 20% of them are not seen during training which we circled in black. We can see those unseen temperatures are also properly distributed, indicating the great generalization ability of our model. We next plot the representations for all data samples under temperatures 2.5 and 3.5 respectively as shown in Figure <ref> (b). We can see that the learned representations are clustered within the two temperatures, indicating our contrastive learning loss is indeed beneficial to guide the learning process of exogenous factors.
§ CONCLUSION
In this paper, we investigate the problem of learning the dynamics of continuous interacting systems across environments. We model system dynamics in a continuous fashion through graph neural ordinary differential equations. To achieve model generalization, we learn a shared ODE function that captures the commonalities of the dynamics among environments while design an environment encoder that learns environment-specific representations for exogenous factors automatically from observed trajectories. To disentangle the representations from the initial state encoder and the environment encoder, we propose a regularization loss via mutual information minimization to guide the learning process. We additionally design a contrastive learning loss to reduce the variance of learned exogenous factors across time windows under the same environment. The proposed model is able to achieve accurate predictions for varying physical systems under different environments, especially for long-term predictions. There are some limitations though. Our current model
only learns one static environment-specific variable to achieve model generalization. However, the environment can change over time such as temperatures. How to capture the dynamic influence of those evolving environments remain challenging.
This work was partially supported by NSF 1829071, 2031187, 2106859, 2119643, 2200274, 2211557, 1937599, 2303037, NASA, research awards from Amazon, Cisco, NEC, and DARPA #HR00112290103, DARPA #HR0011260656. We would like to thank Mathieu Bauchy, Han Liu and Abhijeet Gangan for their help to the dataset generation procedure and valuable discussion throughout the project.
ACM-Reference-Format
§ APPENDIX
§.§ Datasets
We conduct experiments over two datasets: The Water dataset and the Lennard-Jones potential dataset. As introduced in Sec <ref>, the edges between agents are assigned if the Euclidean distance between the agents' positions r_ij^t,e = ||p_i^ - p_j^||_2 is within a (small) connectivity radius R.
The connectivity radius for the two datasets is set as 0.015 and 2.5 respectively. The number of particles is kept the same as 1000 for all trajectories in the Lennard-Jones potential dataset, while in the Water dataset, each data sample can have a varying number of particles, and the maximum number of particles is 1000.
§.§.§ Data Split.
Our model is trained in a sequence-to-sequence mode, where we split the trajectory of each training sample into two parts [t_1,t_K] and [t_K+1,t_T]. We condition on the first part of observations to predict the second part. To fully utilize the data points within each training sample, we split each trajectory into several chunks with three hyperparameters: the observation length and prediction length for each sample, and the interval between two consecutive chunks (samples). We summarize the procedure in Algorithm <ref>, where K is the number of trajectories and d is the input feature dimension.
§.§.§ Input Features and Prediction Target.
For the Water dataset, the input node features are 2-D positions p_i^, and we additionally calculate the 2-D velocities and accelerations using finite differences of these positions as v_i^ = p_i^ - p_i^t-1,e, a_i^t = v_i^t,e - v_i^t-1,e = p_i^ - 2p_i^t-1,e + p_i^t-2,e. For positions, velocities, and accelerations, we precompute their mean and variance across all samples and normalize them with z-score. For the Lennard-Jones potential dataset, the input node features are 3-D positions, velocities, and accelerations. We train the model to predict the future positions for each agent along the time for both datasets.
§.§ Software and Experiment Environment
We implement our model in PyTorch. All experiments are conducted on a GPU powered by an NVIDIA A100. For all datasets, we train over 100 epochs and select the one with the lowest validation loss as the reported model. We report the average results over 10 runs. Encoders, the generative model, and the decoder are jointly optimized using Adam optimizer <cit.> with a learning rate 0.005. The batch size for the Water dataset is set as 128, and for the Lennard-Jones potential dataset. is set as 256. Note that the batch size denotes the number of data samples generated as in Alg <ref>.
§.§ Implementation Details
We now introduce the implementation details of our model.
§.§.§ Initial State Encoder.
The initial state encoder aims to infer latent initial states for all agents simultaneously via a two-step procedure: Firstly, the encoder computes the structural representation for each observation node by the use of a spatial-temporal GNN. We set the number of GNN layers l as 2 and the hidden dimension as 64 across all datasets. LayerNorm <cit.> is employed to provide training stability in our experiment. Next, a self-attention-based sequence representation learning procedure computes the sequence representation for each agent and samples the initial state from it. We use a 2-layer MLP as f_trans in Eqn <ref> with latent dimensions as 128 and activation function as Tanh.
§.§.§ Environment Encoder.
The environment encoder learns the latent representations of exogenous factors based on the observed trajectories. The architecture is the same as the initial state encoder but are using two sets are parameters with the same hyperparameter settings introduced in Sec <ref>.
Contrastive Learning Loss Sampling.
The contrastive learning loss ℒ_contra shown in Eqn <ref> is designed to achieve the time invariance properties of the learned exogenous factors. Specifically, we sample the positive pairs X^t_1:t_2,e, X^t_3:t_4,e using two strategies: (1) The intra-sample generation, where ^t_1:t_2,e, X^t_3:t_4,e are from the same training sample but representing two different time windows. We achieve this by randomly selecting two timestamps within each training sample to serve as t_1, t_3 respectively, and then set the window size as the observation length L to get t_2 = t_1 + L , t_4 = t_3+L. (2) The cross-sample generation, where ^t_1:t_2,e, X^t_3:t_4,e are from two different samples within the same environment e. Specifically, for each training sample, we first randomly choose another sample under the same environment. Then we generate t_1, t_3 by randomly selecting one timestamp for each of them. Finally, we calculate t_2,t_4 by adding the observation length. To generate negative pair X^t_5:t_6,e^' for each X^t_1:t_2,e, we first randomly select one another environment e', from which we randomly pick one data sample. Similarly, we then randomly select one timestamp within that data sample to serve as t_5 and then obtain t_6 as t_6 = t_5 + L. The temperature scalar τ in Eqn <ref> is set as 0.05.
Mutual Information Minimization Loss Sampling. To disentangle the representations of the latent initial states and the exogenous factors, we design the mutual information minimization loss in Eqn <ref> as a regularization term during training. We conduct the sampling procedure for positive and negative pairs as follows: For each training sample, we pair the latent initial states z_i^0,e of all the N agents with the learned exogenous factors u^e, thus constructing N positive pairs. To generate negative pairs, we randomly select another environment e' and pair it with the latent initial states of all agents within one training sample. Thus we obtain the same number of positive and negative pairs during training. The discriminator Ψ is implemented as a two-layer MLP with hidden dimension and out dimension as 128 and 64 respectively.
§.§.§ ODE Function and Solver.
The ODE function introduced in Eqn <ref> consists of two parts: the GNN f_GNN that captures the mutual interaction among agents and f_self that captures the self-evolution of agents. We use the following two-layer message passing GNN function as f_GNN:
v → e: 𝐞_(i, j)^l_1(t,e) =f_e^1([𝐳_i^t,e||𝐳_j^])
e → v: 𝐳_j^l_1(t,e) =f_v^1(∑_i ≠ j𝐞_(i, j)^l_1(t,e))
v → e: 𝐳_j^l_2(t,e) =f_e^2([𝐳_i^l_1(t,e)||𝐳_j^l_1(t,e)])
where || denotes concatenation, f_e^1, f_v^1, f_e^2 are two-layer MLPs with hidden dimension size of 64. We use 𝐳_j^l_2(t,e) as output representation for agent j at timestamp t from f_GNN. The self-evolution function f_self and the transformation function f_env are also implemented as two-layer MLPs with hidden dimension of 64. We use the fourth-order Runge-Kutta method from torchdiffeq python package <cit.> as the ODE solver, which solves the ODE systems on a time grid that is five times denser than the observed time points. We also utilize the Adjoint method described in <cit.> to reduce memory usage.
§.§ Pseudo-Code of Training
|
http://arxiv.org/abs/2307.04865v1 | 20230710192453 | Social inequalities that matter for contact patterns, vaccination, and the spread of epidemics | [
"Adriana Manna",
"Júlia Koltai",
"Márton Karsai"
] | physics.soc-ph | [
"physics.soc-ph",
"cs.SI",
"stat.AP"
] |
Social inequalities that matter for contact patterns, vaccination, and the spread of epidemics
Adriana Manna^1 Júlia Koltai^2,3,4Márton Karsai^1,4,5*
^1Central European University, Quellenstraße 51, 1100 Vienna, Austria
^2Computational Social Science - Research Center for Educational and Network Studies, Centre for Social Sciences, Tóth Kálmánutca 4,Budapest, 1097, Hungary
^3Department of Social Research Methodology, Faculty of Social Sciences, Eötvös Loránd University, Pázmány Péter s étány 1/A, Budapest, 1117, Hungary.
^4 National Laboratory for Health Security, Hungary.
^5 Rényi Institute of Mathematics, Reáltanodautca 13-15, Budapest, 1053, Hungary.
^*Corresponding author: [email protected]
===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
Individuals socio-demographic and economic characteristics crucially shape the spread of an epidemic by largely determining the exposure level to the virus and the severity of the disease for those who got infected. While the complex interplay between individual characteristics and epidemic dynamics is widely recognized, traditional mathematical models often overlook these factors. In this study, we examine two important aspects of human behavior relevant to epidemics: contact patterns and vaccination uptake. Using data collected during the Covid-19 pandemic in Hungary, we first identify the dimensions along which individuals exhibit the greatest variation in their contact patterns and vaccination attitudes. We find that generally privileged groups of the population have higher number of contact and a higher vaccination uptake with respect to disadvantaged groups. Subsequently, we propose a data-driven epidemiological model that incorporates these behavioral differences. Finally, we apply our model to analyze the fourth wave of Covid-19 in Hungary, providing valuable insights into real-world scenarios. By bridging the gap between individual characteristics and epidemic spread, our research contributes to a more comprehensive understanding of disease dynamics and informs effective public health strategies.
keywords: social inequalities, epidemics, human behaviour, mathematical models, contact matrices
§ INTRODUCTION
Individuals' socio-demographic and economic characteristics are among the most significant factors that shape the dynamics of epidemic spreading processes. They not only influence the epidemic outcome in the hosting population but largely determine the course and severity of the disease for those who got infected <cit.>. There is a widespread agreement that pandemics disproportionately affect certain population groups rather than others <cit.>. Health-related inequalities in the burden of an epidemic partly arise from differences in the level of exposure to viruses and bacteria. These are related to differences in social interactions, mobility patterns and work-related conditions, which are aggravated by disparities in the ability to be compliant with non-pharmaceutical interventions (NPIs), such as self-isolation, home-office and avoiding crowded places <cit.>. At the same time, inequalities in the severity and fatality of a disease can be accounted for the heterogeneity in preexisting individual health conditions, protection attitudes and access to medical care, which are themselves related to socio-demographic and economic factors <cit.>.
Although it is widely recognized that socioeconomic inequalities play a crucial role in the transmission dynamics of diseases, traditional mathematical approaches have often overlooked these factors. Indeed, the state-of-the-art framework of modelling infectious diseases incorporates stratification of the population according to age groups <cit.>, while discarding other potential relevant heterogeneities between groups of individuals belonging to different socioeconomic strata. They commonly ignore the mechanisms through which these heterogeneities come into play - both directly and indirectly, in the different phases of an epidemic process. In traditional epidemiological models, contact patterns are usually represented in the aggregate form of an age contact matrix (C_ij), which encodes information on the average number of contacts that individuals of different age groups have with each other <cit.>. Moreover, not only the description of contact patterns is limited to an age structure but also other epidemiological-relevant factors, such as vaccination uptake, infection fatality rates <cit.> or susceptibility <cit.> are usually described only by considering differences between different age groups. While age is unarguably one of the most important determinant of these characters, the current literature falls short to understand the role of other social, demographics, and economic factors in shaping human behaviour that are relevant to the epidemic spreading. In recent years, researchers have advocated including social aspects in infectious disease modelling, arguing that the epidemic modelling community is lacking a deep understanding of the mechanisms through which the socioeconomic divide translates into heterogeneities in the spread of infectious diseases <cit.>.
With this in mind, we aim to shed light on these mechanisms to address the following interrogatives: which are the most important individual characters and corresponding sub groups of the population that differentiate the most their epidemic-relevant behaviours; and, how these differences translate into epidemiological outcomes? We address these questions by analyzing a large survey dataset coming from the MASZK study carried out in Hungary during the Covid-19 pandemic <cit.>. This data collects information on individuals' face-to-face interaction patterns in different contexts and other epidemiological-related behavioural patterns and opinions such as travel habits, vaccination attitude, or mask-wearing. The MASZK study consists of 26 cross-sectional representative surveys carried out in each month between 2020/04 and 2022/06 (for more details on the data see MM). By considering the course of the pandemic in the country, we aggregate the data in six periods covering four epidemic waves (Ws) and two interim periods (IPs) as demonstrated in Fig. <ref>a.
Throughout this study we are mainly interested in the dynamics and most influential determinants of social contacts, that were recorded in the data as reported proxy interactions between pairs of individuals who spent at least 15 minutes within 2 meters from each other on a given day. Outside of home, we distinguish between two context where social interactions may evolve. We differentiate between work contacts that emerge at the workplace (or at school) of respondents (or their minors) and community contacts that they evolved elsewhere than home or work. Meanwhile, we do not take into account household contacts in our study as we assume they do not change significantly during the different phases of the pandemic.
Through the analysis of contact patterns, our aim is to show existing significant differences among sub-groups of different socio-demographic characters,
when accounting for the effect of age. Particularly, we demonstrate that dimensions such as employment situation and education level play a crucial role in determining contact numbers and vaccination uptake during a pandemic. Additionally, by proposing a new data-driven mathematical framework, which explicitly considers further social dimensions, other than age, we analyze the impact of such differences in terms of epidemic outcomes. Finally, by focusing on the Hungarian Covid-19 pandemic scenario, we reveal the unequal impact of the pandemic in terms of individuals belonging to different socio-economic statuses, where we differentiate individuals by their employment situation and income level. Note that although all the models have been completed on each pandemic periods, for the demonstration of our findings we exclusively show results about the 4th wave in the main text, while reporting our findings concerning other periods in the Supplementary Information.
§ RESULTS
§.§ The main determinants of human contact patterns
Human contact patterns represent the routes of infectious disease spreading by shaping the underlining transmission chain among susceptible individuals. During the Covid-19 pandemic, many aspects of human behaviour has experienced drastic interruption in most countries worldwide. This was largely due to the implementation of non-pharmaceutical interventions (NPIs) that were installed to mitigate the spreading and other effects of the pandemic. They aimed at controlling the number of contacts, as well as influencing individual attitudes, to change the ways humans meet and interact with each other <cit.>. Their effects are evident from Fig. <ref>a, where the average number of daily contacts in Hungary are shown. These numbers increase during interim periods (IPs) when the numbers of daily infection cases are low, and decrease during the epidemic waves (Ws) when infection risk is high, this way sensitively reflecting the adaptive behaviour of people throughout the pandemic.
Although at the aggregate level these patterns are clear, there are non-trivial disparities at the level of individuals that may result in diverse contact patterns for given sub-groups of the population. To explore these effects, in our statistical analysis, we focus on several socioeconomic dimensions, that, interacting with age, may significantly affect the number of contacts that individuals have. We consider various socio-demographic variables such as individual's education, employment, income, gender, settlement type, actual chronic or acute disease or smoking habits (for more details and definitions see MM and SI). As a first observation, in Fig. <ref>b we show the distribution of the maximum confidence level of the effects of these variables that they had on the number of contacts in interaction with age, during each period (for definition see MM). In these distributions a higher value indicates that a given variable explains better differences in the number of contacts, given the age of individuals. Based on these results employment, education and income level were found to be the three most important dimensions in determining the number of contact. This observation stands if we consider the overall number of contacts including both work and community relationships, and it is true as well if we only consider community contacts (with results shown in the SI).
To further investigate the ways individuals of different characters adapt their number of contacts to the actual epidemiological situation, in Fig <ref> we show the average number of contacts over time decoupled by education level (Fig <ref> a,c) and employment situation (Fig <ref> b,d) for adult individuals older than 15 years old (for the corresponding plot decoupled by age groups see SI). Results in panel (a) suggest that high and mid-educated individuals have consistently higher number of contacts in the community layer throughout the observed period. In addition, these groups are able to better adapt to the epidemiological situation and NPIs by decreasing their contact numbers during epidemic waves and increasing again during interim periods. At the same time, low-educated individuals maintain a lower number of contacts over time with smaller variation, reflecting their limited social environment and adaption capacities. By looking at the contact dynamics at workplaces it is evident that only highly educated individuals were able to adapt to the epidemiological situation, while the low- and mid-educated people presented similar dynamics and had less flexibility to adapt during the different pandemic periods. Interestingly, mid-educated individuals reported a higher number of contacts at work particularly during the 2nd and the 3rd epidemic waves.
When we group people by their employment situation, it makes sense to compare groups in the community layer. From Fig <ref>b it is evident that employed people maintain more contacts even outside of their workplace as compared to not-employed individuals, which is a clear sign of behavioural differences between these two groups. Meanwhile, employed individuals follow somewhat similar contact dynamics as high-educated people (see Fig. <ref>d, signalling some correlation between these two groups.
From an epidemic modelling perspective, the most convenient way to code interaction patterns between different groups is via contact matrices that represent a social network at an aggregate level. Contact matrices allow models to depart from the homogeneous mixing assumption, i.e. taking all individuals to meet with the same probability. Instead, they allow the introduction of non-homogeneous mixing patterns between different groups, while keeping the model computationally more feasible as compared to contact network based simulations. Conventionally, epidemic models incorporate C_i,j age contact matrices that code the average number of contacts between people from different age-groups (for formal definition see MM). Nevertheless, age contact matrices could be further stratified by other socio-demographic characters that influence the contact numbers of individuals. In Fig <ref>e-i we show the age contact matrices decoupled by education level and employment situation (C_di,j) for the 4th epidemic wave for the adult population (See MM for more details and SI for the corresponding matrices including children). These matrices have been computed by considering community, work and household contacts together. The emerging large differences between these matrices demonstrate clearly that beyond age, the identified variables, i.e. education and employment status, induce significant differences in the contact patterns of people. Although these variables may not be independent of the age of people, the observed distinct patterns suggest more complex mechanisms controlling contact patterns among sub-groups that can be explained by age alone.
§.§.§ Beyond age stratification
We demonstrated that social inequalities significantly influence human contact patterns, thereby shaping the network of proxy social interactions. This is critically important for the propagation of diseases as they determine the transmission chain of an infection spreading among a susceptible population. Consequently, incorporating the contact pattern differences among individuals of different socioeconomic background into epidemiological models is crucial. This could help to understand the unequal spread and uneven burden what an epidemic could impose for the different socio-demographic groups of a society. To this end, we propose a simple mathematical framework based on the extension of a conventional age-structured SEIR compartmental model <cit.>, which we call the extended SEIR model. The conventional SEIR model assumes that each individual in a population is in one of the mutual exclusive states of Susceptible (S), Exposed (E), Infected (I) or Recovered (I). Transitions of an individual between these states are controlled by rates (SE, EI, IR) with the λ rate influenced by the frequencies of interactions between age groups coded in a C_i,j age-contact matrix. The proposed extended model incorporates C_d̅,i,j age contact matrices instead, that are decoupled along important socio-demographic dimensions d̅ to model epidemic spreading in different sub-groups of the population (See MM for further details).
Particularly, we analyze the impact of decoupled age-contact matrices along four dimensions: employment situation, education level, settlement, and income level (for exact definitions and possible variable values see MM).
Taking the decoupled contact matrices as input we simulate the spread of infectious disease among an entirely susceptible population using both the conventional SEIR and the extended SEIR models. Having fixed the epidemiological parameters such as the transmission rates and seeding strategy, other input parameters like the population distributions and contact matrices have been estimated from data as we explain in the SI in more details.
The proposed model allows us to investigate how differences in contact patterns along diverse social groups translate into an unequal burden of the epidemic. To quantify these differences in the epidemic outcome, we measure the attack rate defined as the population wise normalised fraction of individuals who contracted the infection from a given group. To follow the distribution of the people along the investigated dimensions, we show their population fractions in the different age groups in Fig. <ref>a-d. Meanwhile, in Fig <ref>e-h we depict the attack rates calculated using the extended SEIR models for different age and socio-demographic groups (and as reference only for age - see grey solid lines). Results are shown for the cases when we decouple each age-group along the four dimensions analyzed. As anticipated by the statistical analysis, employment, and education produce the largest differences between groups in terms of attack rate by age.
Interestingly, the group of employed people happened to be the most infected group in all age groups, while mid and high educated individuals are more infected among those, who are 45-60 years old. When decoupling age contact matrices by settlement and income, although differences appear smaller between groups, high-income individuals and the ones living in the capital are more infected, particularly elderly ones with age 60+. These modelling results suggest the interesting overall conclusion that privileged people of the population report a higher attack rate, thus they are typically the most infected group relative to their population size.
These results also demonstrate that the extended SEIR model is able to capture differences introduced by the considered socio-demographic variable and, in this way, to model the epidemic impact on the different groups of the population. These differences are also visible at the population level. In Fig <ref>i we show the differences between the attack rates predicted by the conventional and theextended SEIR models, for each age group and overall too. It is evident from these results that models using contacts only stratified by age may underestimate (negative difference) the size of the epidemic in different age groups or in the whole population. For example, our simulations based on data from the 4th wave demonstrate that the conventional SEIR model could predict higher attack rates for each age group with respect to the extended SEIR, which considers differences in contact numbers along the employment situation or the education level too.
It is important to highlight that the uneven age distribution within the different subgroups sometimes reduces or annul the effect of difference in the contact patterns when we are computing aggregate quantities at the population level. This explains why, even if there is a significant difference in contact patterns, the difference in the attack rates only spans a small range between the two models.
§.§ Vaccination uptake and contact number differences
Beyond the crucial role played by the network of face-to-face interactions, individual vaccination attitudes may also substantially affect epidemiological outcomes by decreasing the morbidity and mortality. By applying the same pipeline of statistical analysis as we explained above, we identified the dimensions along which individuals made different decisions in terms of vaccination, given their age. In this case, the interaction with age is particularly important given that the Covid-19 immunization strategy implemented in Hungary followed an age-stratified outreach by prioritizing elderly individuals <cit.>. Interestingly, the statistical analysis in this case indicated income as the most important dimension along which individuals made different vaccination decisions (See SI for the results of the statistical analysis). Fig <ref>a-d show the percentage of vaccinated individuals by age and the investigated dimensions, during the 4th wave of the pandemic. Although by the 4th epidemic wave the vaccination saturated in Hungary, the effects of the age-dependent vaccination policy is clearly visible. More strikingly, we find that privileged groups of the population were more likely to get vaccinated against the Covid-19 virus. Convincingly, this observation is valid for all age groups and periods considered in the analysis (See SI for the corresponding figures for the other periods).
To consider these observations, we model the vaccination uptake in the extendend SEIR framework. More precisely, we define the probability of getting vaccinated (i.e. immune or recovered from the point of the infection) to be dependent in this case on both the age and the subgroup of the population considered. Using this extended SEIR model, we are able to compare the effects of vaccination uptake, while keeping fixed the structure of contacts. Fig <ref>e-h shows the averted attack rate due to vaccination with respect to the non-vaccination scenario. We consider the probability of getting vaccinated along the four different investigated dimensions separately. In all of these scenarios the gain in averted infection is strongly dependent on the subgroup membership. As expected, the groups with higher vaccination uptakes are the ones, which reduce their attack rate the most in the vaccination scenario. However, this pattern is not linear. For example, among individuals aged 60+, although the not-employed people report a higher vaccination uptake, they are the ones that gain the less in terms of averted infections. This is because these individuals, having a low number of contact, are already protected from exposure to the virus thus they gain less from vaccination. It should be noted that our focus is solely on the number of infected individuals. It is important to acknowledge that alternative conclusions may arise if the number of averted deaths is taken into consideration. In any case, these results clearly show that although several other factors affect the outcome of an epidemic, by neglecting differences in vaccination uptake and the effects of vaccination campaigns among subgroups in modelling, we miss an important determinant, which significantly influence the final outcome of an epidemic.
§.§ Stratified modelling of the Hungarian scenario
To provide an example of how the proposed mathematical framework can be applied to a real case scenario, we model the 4th Covid-19 wave in Hungary between 09/2021 and 01/2022. As the statistical analysis showed that employment and income are the most important dimensions along which, respectively, contact patterns and vaccination uptake change the most, here we divide the population into subgroups by considering simultaneously these two additional dimensions other than age. In addition, we introduce a new compartment D to our SEIR model, that represents a dead state that infected individuals may enter with transmission rate ID.
To simulate the SEIRD for the 4th Covid-19 wave in Hungary we calibrate our model using the Approximate Bayesian Computation (ABC) method <cit.> on the total number of daily deaths from 09/2021 to 01/2022 <cit.>. Details about the fitting method and calibrated results are summarised in the SI.
The results of the simulated model are presented in Figure <ref>, which shows the daily fraction of newly infected (panels (a)-(c)) and new dead (panels (d)-(f)) cases for different employment, income, and age groups. As expected, these curves suggests that group of employed people extracted the infection with the highest rate as compared to not employed others. At the same time, in terms of socioeconomic status and age, more affluent and younger people got infected more during the simulated epidemic wave. On the other hand, strikingly the contrary trend is suggested in terms of mortality rates. From the simulations we found that although not-employed, low-income and older individuals appeared with the lowest infection rates, they evolved with the highest mortality rate as compared to other groups.
Considering that the fatality rate of infected individuals depends on their age, these observation can be largely accounted to the fact that not-employed and low-income individuals are also the oldest one in the population. To further explore the correlation between these dimensions, we examine the attack rates and mortality rates by age groups separately in each of the subgroups of the population stratified by income level and employment status (Fig. <ref>g-h). These results confirm an overall decreasing infection rate by age and that not-employed individuals experience the lowest attack rate in each age group. Further stratification of this group by income level reveals a clear pattern, with high-income people exhibiting a higher infection rate compared to mid-income and low-income individuals. In contrast, the infection pattern among income levels of employed individuals is age-dependent. For young employed individuals in age group 15-30, high and mid-income individuals register a higher share of infections, while among those older than 30, low-income individuals exhibit a higher infection rate compared to mid and high-income individuals. An exception is represented by the age group 60-70, for which the most infected group is still the high income. In terms of mortality we find an increasing trend by age, otherwise we can conclude a similar pattern. Once we account for the correlations among the dimensions of interest and age, our results show that employed people die with a higher rate, while in terms of income, other than the group of age 15-30 and 60-70, the lower income people suffered more death according to our simulations. The extreme decrease of mortality rate for the employed high-income group is due to data sparsity in the survey data, recording only a few data point in this category.
§ DISCUSSION
There are several particular factors that may determine how an infection would turn-out for a given person. Some of them are coded genetically, or determined by physiological conditions, but many of them are environmental and correlate with one's socio-demographic characters. In any society people show uneven patterns along numerous social, demographic, and economic characters, like age, income or employment status. These characteristics not only induce medical disparities between people (as in immunity, overall health conditions, or chronic diseases) but they naturally translate to differences in adaption capacities and other behavioural patterns, letting certain groups more exposed to infection. The simultaneous actions of all these factors lead to observable inequalities in terms of epidemic burden between different groups at the population level.
This study highlights the significant impact that social determinants have on human behaviours that are relevant to epidemic transmission. Specifically, exploiting the data of the MASZK study <cit.>, we show that contact patterns and vaccination uptake are influenced by socioeconomic factors. Our findings suggest that contact patterns are shaped by social factors not only in their absolute values but also in the extent to which they fluctuate in response to extraordinary events, such as a lockdown or curfew interventions. Specifically, our statistical analysis shows that socioeconomic factors such as employment situation and education level played a significant role in determining contact numbers and vaccination uptake during the COVID-19 pandemic in Hungary. Additionally, we find that privileged groups tend to have a higher number of a contact and are the ones able to better adapt to the epidemiological situation and NPIs by adjusting their number of contacts. Contrarily, less privileged groups maintain a lower number of contacts over time with smaller oscillations. We also find that privileged groups, such as those with higher education, income, and employment status, were more likely to get vaccinated.
We propose a mathematical framework that extends the well-known age-stratified approach to model infectious diseases by explicitly accounting for differences in contact patterns and vaccination uptake for specific sub-groups of the population.
This method allows us to better understand the mechanisms underlying the emergence of inequalities in epidemiological outcomes. Results demonstrate that traditional epidemiological models, that only consider age, could overlook crucial heterogeneities along other social and demographic aspects that may impact the spreading of an epidemic.
By simulating a pandemic period in Hungary, we reveal the unequal health-related impact of the COVID-19 pandemic along individuals belonging to different socioeconomic groups. Although the higher number of contacts translates into higher attack rates for privileged individuals, the age structure and the vaccination decision of such groups translate into lower mortality rates for these individuals, while disadvantaged groups are the one suffering higher mortality. These results are in line with the empirical findings of <cit.> for the 2nd and 3rd Covid-19 waves in Hungary.
Due to the limitation of the survey collection methodology, contact patterns of individuals can be differentiated only by the characteristics of participants.
Indeed, the only information we know about the contacted peers is their age, while their other characteristics remain unknown. Thus, our extended SEIR model can only account for age-contact matrices that are decoupled along other social dimension of the participants (ego). In other words, although our model incorporates additional social dimensions, given the sub-group the ego belongs to, it still only considers the average number of contacts stratified by the age group of the contacted (alter). In order to introduce a generalised matrix <cit.> stratified along multiple socio-demographic dimensions of the contactee, we would need information about such dimensions. Such information can be collected via detailed contact diaries <cit.>, which are based on the reports of the respondents about the peers, and commonly suffer from recall biased and other limitations.
By shedding light on the complex interplay between social, demographic and economic factors and disease transmission dynamics, our findings underline the need for a new mathematical framework for epidemic modelling that accounts for multidimensional inequalities. This would help us to better understand the socially stratified consequences of an epidemic and to highlight non-negligible inequalities between different socio-demographic groups. Additionally, incorporating social factors into epidemiological models will provide a valuable tool to design and evaluate targeted NPIs to cope more efficiently with the spread of an infectious disease.
§ MATERIALS AND METHODS
§.§ Data description
The data used in this study comes from the MASZK survey study <cit.>, a large data collection effort on social mixing patterns made during the COVID-19 pandemic. It was carried out in Hungary from April 2020 to July 2022 on a monthly basis. The data was collected via cross-sectional anonymous representative phone surveys using CATI methodology and involved a 1000 large nationally representative sample each month. During the data collection participants were not asked information that could be used for their re-identification. The data collection was fully complying with the actual European and Hungarian privacy data regulations and was approved by the Hungarian National Authority for Data Protection and Freedom of Information <cit.>, and also by the Health Science Council Scientific and Research Ethics Committee (resolution number IV/3073- 1 /2021/EKU).
The primary goal of the data collection effort was to follow how people changed their social contact patterns during the different intervention periods of the pandemic. Relevant to this study, the questionnaires recorded information about the proxy social contacts, defined as interactions where the respondent and a peer stayed within 2 meters for more than 15 minutes <cit.> at least one of them not wearing mask. Approximate contact numbers were recorded between the respondents and their peers from different age groups of 0–4, 5–14, 15–29, 30–44, 45–59, 60–69, 70–79, and 80+. Contact number data about underage children were collected by asking legal guardians to estimate daily contact patterns.
Beyond information on contacts before and during the pandemic, the MASZK dataset provided us with an extensive set of information on social-demographic characteristics (gender, education level etc.), health condition (chronic and acute illness etc.), financial and working situation (income, employment status, home office etc.), and attitude towards Covid-19 related measures and recommendations (attitude towards vaccination, mask-wearing etc.) of the participants. In order to study different stages of the pandemic, we consider six epidemiological periods including three epidemic waves (Ws) and three interim periods (IPs) (see Fig <ref>a).
On the collected data a multi-step, proportionally stratified, probabilistic sampling procedure was elaborated and implemented by the survey research company using a database that contained both landline and mobile phone numbers. The survey response rate was 49 percent, which is expressly higher than the average response rate (being between 15-20 percent) of telephone surveys in Hungary. The sample is representative for the Hungarian population aged 18 or older by gender, age, education and domicile. Sampling errors were corrected using iterative proportional post-stratification weights. After data collection, only the anonymised and hashed data was shared with people involved in the project after signing non-disclosure agreements.
§.§ Sociodemographic dimensions
The sociodemographic dimensions that we analyze are the following: (i) education level, which can have three possible levels: low, mid and high; (ii) employment situation, which can be either employed or not-working, including students and retirees individuals; (iii) income can have three possible levels: low, mid and high; (iv) gender refers to the biological gender and can be either female or male; (v) settlement, which refers to the area where individual live and can be either capital, rural or urban; (vi) chronic disease is a boolean dimension indicating if an individual is affected by any chronic disease; (vii) acute disease is a boolean dimension indicating if an individual is affected by any acute disease; and (viii) smoking is a boolean dimension indicating if an individual is a smoker or not. A detailed explenation of these variables is provided in the SI.
§.§ Statistical analysis
In order to build an epidemiological model that explicitly takes into account social inequalities we need to identify, which are the main dimensions, that interacting with age affect contact patterns the most. To identify these dimensions, we model the expected number of contacts of respondents i using a negative binomial regression <cit.> as defined in Eq. <ref>:
μ_i = α + β_1 age_group_i + β_2 X_i + β_3 age_group_i* X_i + ϵ_i,
where age_group_i is the age class of i; X_i is the variable of interest (e.g., education, income etc.), age_group_i*X_i is the interaction term of age group and the variable of interest, and ϵ_i is the error term. Given μ_i, we define λ_i=exp(μ_i) to be the expected number of contacts for respondent i. Then we model the reported number of contacts for respondent i, y_i, as
y_i ∼ Neg-Bin(λ_i, ϕ)
where ϕ∈ [1, ∞) is a shape parameter that is inversely related to over-dispersion: the higher ϕ is estimated to be, the closest y_i’s distribution is to a Poisson distribution with rate parameter λ_i.
We build a model <ref> for each variable of interest (X). Particularly, the interaction term between age_group_i and the variable of interest allows us to examine, whether there are differences in the effect of X_i on the number of contacts in the different age groups. To be able to provide a meaningful description of the interactions, we analyse the average marginal effect (AME) <cit.> of X_i on the number of contacts for different age groups, defined as:
[ AME_X_i = 1/n∑_i=1^n∂μ_i/∂ age_group_i; ; ∂μ_i/∂ age_group_i = β_1 + β_3 X_i. ]
Working mostly with categorical variables (e.g., education level or employment situation), we can calculate different AMEs for all categories of the categorical variables in each age_group. For each age group, we consider the maximum confidence level, at which the AMEs of two categories of the categorical variable show a statistically significant difference. Finally, to summarize the results, we consider the median of these maximum confidence levels. Therefore, we have one value for each variable of interest, which characterise the strength of the interaction between age and the variable of interest on the number of contacts. By following this procedure for each of the variables of interest, we are able to rank the variables according to their importance in driving differences in contact patterns additionally to age, in different periods of the Covid-19 pandemic. The pipeline of these analyses are illustrated in Fig. <ref>.
Following the same methodology we investigate the dimensions that - in interaction with age - affect the most the probability of getting vaccinated against Covid-19. In this case, we model this probability using a logistic regression model instead of a negative binomial, as the dependent variable was binary and not a count one. In the Supplementary Information we provide the details of this models and we report their relative results.
§.§ Decoupled contact matrices
Conventionally, to compute age-contact matrices C_ij we divide a population into sub-groups according to their age and calculate the average number of contacts that individuals in age class i have with individuals in age class j <cit.>. Here, instead, we further stratify individuals from each age class i according to various dimensions, like employment status, settlement or education level.
In detail, we decouple the conventional age contact matrix C_ij into D number of matrices, one for each of the sub-groups of the dimension that we want to take into account. More precisely, let d̅ be the sub-group of the dimension considered and let d̅∈ 1,..., D. We can write
C_d̅i,j = CT_d̅i,j/N_d̅i,
where CT_d̅i,j is the total number of contacts that individuals of age class i and belonging to sub-group d̅ have with individuals in age class j, regardless of the sub-group, to which the contacted individuals belong; and N_d̅i is the total number of individuals in age class i and sub-group d̅.
For example, to differentiate between employed and not-employed individuals, we compute two age contact matrices: C_employed,i,j and C_not-employed,i,j. Note that here we are considering each of these dimensions (e.g., employment, education level, settlement) separately and only include one dimension additionally to age. However, this framework can be extended to any number of dimensions considered simultaneously, in this case, the length of the d̅ vector will correspond to the number of combinations of the levels of the dimensions considered.
§.§ The epidemiological model
In order to investigate the effect of the decoupled contact matrices on the dynamic of infectious disease transmission, we propose a simple mathematical framework as an extension of the conventional age-structured SEIRD compartmental model <cit.>.
The conventional SEIRD model is defined on a population where individuals are assigned to five compartments based on their actual state: susceptible (S), exposed (E), infected (I), recovered (R) and dead (D). The model further defines the transition rates of individuals from one compartment to another by incorporating for each age class a given force of infection, which includes the average number of contacts with all the other age classes. The model proposed here extends this definition by taking into account not only the age structure of the contacts in the population but also their differences along a set of other dimensions d̅, such as education level, income level and employment situation.
The model can be described by a set of ordinary coupled differential as presented in Eq. <ref>:
[ Ṡ_d̅,i = -λ_d̅,i S_d̅,i; Ė_d̅,i = λ_d̅,i S_d̅,i -ϵ E_d̅,i; İ_d̅,i = ϵ E_d̅,i - μ I_d̅,i; Ṙ_d̅,i = μ (1-IFR_i) I_d̅,i; Ḋ_d̅,i = μ IFR_i I_d̅,i.; ]
Here i indicates the age group of the ego, j indicates the age group of the peer, d̅ represents a vector of dimensions to which the ego belongs, β is the probability of transmission given a contact,ϵ is the rate at which individuals become infectious, μ is the recovery rate, IFR is the infection fatality rate, and C_d̅ is the age contact matrix corresponding to dimensions d̅.
In this equation system we rely on the concept and of force of infection that is defined as:
[ λ_d̅,i(t)= β∑_jC_id̅,j/N_jI_j ]
Further we rely on the definition of the infection fatality rate (IFR_i), that is defined as fraction of infected individuals that died. See Supplementary Information for the details on the implementation of the numerical simulations.
§ ACKNOWLEDGMENT
The authors gratefully thank to Alessandro Vespignani, Eszter Bokáni, Alessia Melegaro and Filippo Trentini for useful discussions. A.M. and M.K. were supported by the Accelnet-Multinet NSF grant. J.K. and M.K. acknowledges funding from the National Laboratory for Health Security (RRF-2.3.1-21-2022-00006). M.K. acknowledges support from the ANR project DATAREDUX (ANR-19-CE46-0008); the SoBigData++ H2020-871042; the EMOMAP CIVICA projects.
§ SUPPLEMENTARY INFORMATIONS
§ DATA PRE-PROCESSING
All the analysis on the number of contacts have been performed after having delete the outliers at the 99% percentile with respect to the period of interest.
All the results presented in this work have been computed by accounting each participant according to it's representative weight. The weight has been provided by the survey company as described in the MM section of the main text.
§ SOCIO-DEMOGRAPHIC DIMENSIONS
The MASZK dataset provided us with an extensive set of information on social-demographic characteristics of the participants. In this section we provide a detailed explanation of all the variables used in this work.
* Education level which can have three possible levels: low, mid and high. We considered low educated individuals those with any primary school degrees, mid educated those holding any diploma or certificate from professional schools, and high educate those with an university education or above (eg. BSc, MSc, PhD).
* Employment situation which can be either employed or not-employed. In the not-employed category we include students and retirees individuals.
* Income level can have three possible levels: low, mid and high. In particular, individuals were asked to report their perceived income with respect to the average using a scale from 1 to 10. We consider as low income individual those that answered from 1 to 4, mid income those who answered 5 or 6 and high income those that answered 7 or above.
* Gender refers to the biological gender and can be either female or male.
* Settlement which refer to the area where individual live and can be either capital, rural or urban.
* Chronic disease is a boolean dimension indicating if an individual is affected by any chronic disease or not.
* Acute disease is a boolean dimension indicating if an individual is affected by any acute disease or not.
* Smoking is a boolean dimension indicating if an individual is a smoker or not. We consider as non smoker individuals that declared them self as not-smoker or that stopped, while we consider smokers individuals that smoke frequently or occasionally.
§ STATISTICAL MODEL
In this section we provide all the results of the additional statistical analysis that we performed in order to support the findings presented in the Result section of the main text.
§.§ Model on contacts
§.§.§ Accounting for the high number of zeros
Due to the implementation of NPIs (lockdowns and curfews) throughout the pandemic, people were forced, when possible, to reduce or completely reset their number of contacts. Thus, when analysing the distribution of the number of contacts we found a high presence of zeros.
To test the robustness of the results, coming from the negative binomial regression model (nb), to this zero-inflated mechanism we implemented two additional models:
* After having excluded the observations where the number of contacts were zero we re-run the negative binomial regression model on the non-zero number of contacts (nb_n contacts>0).
μ_i = α + β_1 age_group_i + β_2 X_i + β_3 age_group_i* X_i + ϵ_i
* We modelled the probability to have at least a contact using a logistic regression model (logit).
log P_i(any contact=1) = α + β_1 age_group_i + β_2 X_i + β_3 age_group_i* X_i + ϵ_i
where age_group_i is the age class of i; X_i is the variable of interest (e.g., education, income etc.), age_group_i* X_i is the interaction term of age group and the variable of interest, and ϵ_i is the error term.
In model <ref> given μ_i, we define λ_i=exp(μ_i) to be the expected number of contacts for respondent i. While in model <ref> P_i(any contact=1) indicate the probability for respondent i to have at least one contact.
By applying the same methodology as explained in the Method section we computed the max significance level by age for each of the variables and period considered in the analysis for model <ref> and <ref>. Fig <ref> shows the results of the three models that we implemented. Interestingly, we can see that the same qualitative patterns result from both the negative binomial models, that is if we include the observations where the number of contacts is 0 (nb), o we discard it (nb_n contacts>0) (Fig. <ref>a,b). In particular, although the maximum confidence level computed with these models differ in terms of variability, education, employment and income seem to remain the most significant dimensions in terms of explaining differences in contact numbers among subgroups of the population. The logistic regression model (logit) (Fig. <ref>c), shows similar results regarding the education and employment situation while it indicates that the other variable analyzed are not actually determining if individuals actually have contacts or not.
§.§.§ Community contacts
Furthermore, we also investigated the contacts happening exclusively in the community layer, excluding the ones happening at work.
We show here the results of the statistical analysis on the number of contacts in the community layer. Particularly in Fig. <ref> we show the results of the three models we are implementing: 1. negative binomial regression (nb), 2. negative binomial regression only on positive observations of the number of contacts (nb_n contacts>0), and 3. logistic regression to model the probability of having at least one contact (logit). In all these models the dependent variable (y_i) refers to the community-level contacts of the individuals. Results are similar to the one run considering all the types of contacts together. Indeed, also in this case results indicate that education, employment and income are the most significant dimensions in terms of explaining differences in contact numbers in the community layer among subgroups of the population. Also the logistic regression model (logit) (Fig. <ref>c), shows similar results indicating that only education and employment are significant in determining if individuals actually have contacts or not in the community layer.
§.§ Model on vaccination
We model the probability of getting vaccinated against Covid-19 using a logistic regression model instead of a negative binomial, as the dependent variable was binary.
Namely, we model the probability of getting vaccinated for respondent i using logistic regression as defined in Eq. <ref>:
log P_i(vax=1) = α + β_1 age_group_i + β_2 X_i + β_3 age_group_i* X_i + ϵ_i
where age_group_i is the age class of i; X_i is the variable of interest (e.g., education, income etc.), age_group_i*X_i is the interaction term of age group and the variable of interest, and ϵ_i is the error term.
By applying the same methodology as explained in the Method section we computed the max significance level by age for each of the variables and period considered in the analysis (Fig. <ref>)
§ CONTACTS
§.§ Average number of contacts in the community and in the work layer by sub-groups
In the main text, we show the evolution of contacts over time decoupled by education level and employment. For completeness, here we report the same figures for income level Fig <ref>-a,c and settlement Fig <ref>-b,d.
Looking at the community contact we can observe that there is a clear rank among the income levels in their number of contacts, with high-income individuals having the highest number of contact and low-income individuals having the lowest. The same is arguable for the individuals living in the capital, which appear to be the most active in the community layer, while, individuals living in rural areas are the less active.
To what concern the contacts at work, we can clearly observe that high-income individuals were the ones who would better adapt to the epidemiological situation, while mid and low income maintained a fairly stable number of contact over time at the workplace. Particularly, they reported a higher number of contacts at the workplace during the Covid-19 waves. A similar conclusion can be drawn when we decoupled individuals according to their settlement. In this case, individuals living in rural areas appear to be the most active in the workplace while the ones living in the capital tend to have a lower number of interactions in the workplace.
§.§ Average number of contacts in the community layer by sub-groups and age groups
To show the robustness of our finding over different age groups in Fig <ref> we report the evolution of community contacts over time decoupled by education level, employment, income level and settlement. While the correlation with age influences the magnitude of differences among the examined sub-groups, the conclusion discussed in the main text appears to be still valid.
§.§ Contact Matrices
For each of the periods considered in this study, we computed the age contact matrix C_ij considering the whole population as shown in Fig <ref>. These are the matrices that have been fed to to the conventional SEIR model. Instead for the extended SEIR model, we computed the decoupled contact matrices (C_di,j) considering different dimensions as well: (i) employment situation (Fig <ref>), (ii) education level (Fig <ref>),(iii) settlement (Fig <ref>), and (iv) income level (Fig <ref>).All the matrices have been computed considering the contact at work, in the community and in the household. Detailed explanation of the computation of such matrices is provided in the Method section.
§ VACCINATION UPTAKE
Here we show the probability of getting vaccinated against COVID-19 given age and another dimension of interest. Namely, we consider (i) employment situation, (ii) education level,(iii) settlement, and (iv) income level. From Fig. <ref> we can observe that privilege groups of the population tend to have higher vaccination uptake across all age groups. This fining is consistent over the four different periods considered in the analysis.
§ EPIDEMIC MODELS
In this section, we report the ODE's equation of the (i) conventional age-stratified SEIRD model and, (i) extended SEIRD where beyond the age stratification we can differentiate the population along others dimension of interest (d̅). Specifically, the reported equations for the extended SEIRD model account also for vaccination.
§.§ Conventional age-stratified SEIRD
Let's consider an infectious disease that can be described with a Susceptible-Exposed-Infected-Recovered-Death model <cit.>. The epidemic dynamic is encoded in the set of differential equations in Eq. <ref>. Where, i indicates the age group of the ego, j indicates the age group of the alter, β is the probability of transmission given a contact, ϵ is the rate at which individuals become infectious, μ is the recovery rate, C_ij is the age contact matrix, and IFR_i is the infection fatality rate by age group.
[ Ṡ_i = -λ_i S_i; Ė_i = λ S_i -ϵ E_i; İ_i = ϵ E_i-μ I_i; Ṙ_i = μ I_i; Ḋ_i = μ IFR_i I_i ]
The force of infection is defined as in eq. <ref>
λ_i= β∑_jC_ij/N_j I_j
§.§ Extended SEIRD with vaccination
We extend the conventional SEIRD model to account for different vaccination uptake of different groups of the population. Namely, each of the compartments is now considered separately for vaccinated and unvaccinated individuals. We define as g_1 and g_2 the efficiency of the vaccination respectively against infection and against death. In addition, we consider the delay in the official registrations of deaths by adding a new compartment Da and a delay of Δ^-1 days. The equations of the model are presented in equation <ref>.
[ Ṡ_d̅,i = -λ_d̅,i S_d̅,i; Ṡv̇_d̅,i = -(1-g_1)λ_d̅,i Sv_d̅,i; Ė_d̅,i = λ_d̅,i S_d̅,i -ϵ E_d̅,i; Ėv̇_d̅,i = (1-g_1)λ_d̅,i Sv_d̅,i-ϵ Ev_d̅,i; İ_d̅,i = ϵ E_d̅,i - μ I_d̅,i; İv̇_d̅,i = ϵ Ev_d̅,i- μ Iv_d̅,i; Ṙ_d̅,i = μ (1-IFR_i) I_d̅,i; Ṙv̇_d̅,i = μ (1-(1-g_2)IFR_i) Iv_d̅,i; Ḋ_d̅,i = μ IFR_iI_d̅,i; Ḋv̇_d̅,i = μ (1-g_2) IFR_i Iv_d̅,i; Ḋ_̇ȧ_d̅,i = Δ^-1Ḋ_d̅,i; Ḋv̇_̇ȧ_d̅,i = Δ^-1Ḋv̇_d̅,i; ]
The force of infection is defined as in eq. <ref>
[ λ_d̅,i(t)= β∑_jC_d̅,ij/N_j [I_j+Iv_j] ]
§ EPIDEMIC SIMULATIONS
We developed stochastic, discrete-time, compartmental models using chain binomial processes to simulate the transitions among compartments. Specifically, at each time step t, the model samples the number of individuals in group (d̅,i) and compartment X transitioning to compartment Y from PrBin(X_d̅,i(t),p_X_d̅,iY_d̅,i(t)). Here, p_X_d̅,iY_d̅,i(t) represents the transition probability.
To illustrate this, let's consider the number of individuals in the group (d̅, i) and compartment S that at time t become exposed transiting to compartment E. Thus, the number of individuals in S_d̅, i(t) getting exposed are extracted from a PrBin(S_d̅, i(t), λ_d̅, i(t)) where λ_d̅, i(t) is the force of infection.
The model has been initialize by computing the population distributions from the MASZK data, while we set the Hungarian population size to 9.750.000.
The decoupled contact matrices have been computed considering the contacts happening at work, in the community and in with family members.
While, the epidemiological parameters are set to realistic values to closely simulate the characteristics of Covid-19. These values are retrieved from the literature. In particular, ϵ is set to 2.4; γ is set to 1/6.6 <cit.>.
The transmission rate β is computed in each of the periods using the Next Generation Matrix approach <cit.> on the aggregate age-contact matrices corresponding to the periods analysed. We fixed R0 = 2.5 and we derived β using Eq. <ref>.
R_0 = β/μρ(C_ij)
Where ρ(C_ij) is the spectral radius of the mage contact matrix.
In the simulations in which we introduce vaccination, we respectively set the g_1 =0.6 and g_2=0.8 <cit.>.
The initial size of the epidemic is set to 5. All the results in the main text refer to the median over 500 simulations of the model.
§.§ Impact of different contact patterns
In this section, we show the results of the extended SEIR model when differences in age contact matrices are considered for different sub-groups of the population. Particularly, we model age contact interactions differentiating individuals along their employment situation, education level, settlement and income level. For each of the dimensions considered, we run the extended SEIR model. We look at (i) and how the prediction of this model differs from the conventional SEIR and (ii) how the attack rate differs for different subgroups as a result of their differences in contact patterns.
Specifically, in Fig. <ref> we show the difference between attack rate by age group as predicted by the extended SEIR model and the conventional SEIR model. The results are shown for each of the periods considered in the analysis. Again, as demonstrated in the main text, the conventional SEIR model tend to overestimate the attack rate by age group with respect to extended SEIR model, particularly when employment situation and education are taken into account.
In Fig. <ref> we show the output of each of the extended SEIR models in terms of attack rate by age by differentiating along the subgroups taken into account. Results are shown for each of the periods considered in the analysis.
As shown in the main text the analysis over the other periods confirms that, employed and highly educated individuals happened to be the most infected groups in all age groups. When decoupling age contact matrices by settlement and income, although differences appear smaller between groups, high-income individuals and the ones living in the capital are more infected, particularly elderly ones with age 60+.
§.§ Impact of different vaccination uptake
In order to show the impact of different vaccination uptake here we show the averted attack rate by age due to vaccination (Fig. <ref>). Specifically, we show the difference among the attack rate by age, for the different subgroups as predicted by the extended SEIR in the non-vaccination scenario with respect to the one in which individuals get vaccinated according to their age and subgroup- as shown in Fig. <ref>.
The findings from the additional periods support the observations discussed in the main text. Indeed, Fig. <ref> clearly shows that vaccination benefits are disproportionately advantageous for more privileged population groups.
§ MODEL CALIBRATION
We calibrate a SEIRD model with vaccination by modelling differences in contacts patterns and vaccination uptake among employed and not employed individuals belonging to different education levels.
In particular, we calibrate the free parameters of the model using an Approximate Bayesian Computation (ABC) technique <cit.> . First, we define the prior distributions of the free parameters P(θ), a number of accepted sets N, an error metric m(E,E'), and a tolerance δ. We start sampling a set of parameters θ from P(θ), and generate an instance of the model using these parameters. Then, using the chosen error metric we compare an output quantity E' of the model with the corresponding real quantity E: if m(E,E') < δ then we accept the set θ, otherwise we reject it. We repeat this accept/reject step until N parameter sets are accepted. The empirical distribution of the accepted sets is an approximation of their real posterior distribution. Finally, we generate an ensemble of possible epidemic trajectories sampling parameter sets from the posteriors distributions. In this work, we consider the following free parameters and prior distributions:
* the transmission rate parameter β: the prior distribution is set to U(0.02,0.15)
* the delay in reporting deaths Δ: the prior distribution is set to U(5,20)
* the initial recovered population R: the prior distribution is set to U(700K,2800K)
* the initial exposed population E: the prior distribution is set to U(100,3K)
* the initial infected population I: the prior distribution is set to U(200,9K)
We calibrate our model on the aggregate number of daily deaths from 09/2021 to 01/2022.
For simplicity, as the percentage of those who were vaccinated was quite stable <cit.> in the period considered we assume that the population got vaccinated at time 0.
As an error metric, we use the Median Absolute Percentage Error (MdAPE). We also set the number of accepted sets N = 3000 and the tolerance δ = 0.3 In Fig <ref> are shown the posterior distributions of the parameters calibrated through the ABC rejection algorithm.
The fixed parameters of the model have been informed from the literature. In particular:
* the efficacy of the vaccine against infection and against death, is modelled as a normal distribution with mean respectively g_1 = 0.7, g_2 = 0.8, and standard deviation 0.05 <cit.>. This choice has been made to account for the variability of the efficacy of the different vaccination types and against the different variants.
* the recovery rate μ = 1/2.5 <cit.>
* the incubation period ϵ = 1/4 <cit.>
* the infection fatality rate by age IFR_i is set as in Table <ref> <cit.>
In Fig. <ref> we report the number of daily real and simulated deaths (median and 95% CI).
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|
http://arxiv.org/abs/2307.05725v1 | 20230711184931 | Estimates of the Height and Date of the 25th Cycle of Solar Activity | [
"V. N. Obridko",
"D. D. Sokoloff",
"M. M. Katsova"
] | astro-ph.SR | [
"astro-ph.SR"
] |
ISSN 0236-2457 DOI:10.24412/0236-2457-2023-1658-1-4
ASTRONOMICHESKII TSIRKULYAR
Published by the Eurasian Astronomical Society
and Sternberg Astronomical Institute
№ 1658, July 2023
Estimates of the Height and Date of the 25th Cycle of Solar Activity
V.N. Obridko^1, D.D. Sokoloff^2, and
M.M. Katsova^3
^1IZMIRAN, Troitsk, Moscow, Russia
E-mail: [email protected]
^2IZMIRAN, Troitsk, Moscow, Russia
^3Sternberg Astronomical Institute of the Lomonosov Moscow State University, Moscow, Russia
Received July 8, 2023
Abstract.
Further development of the work of Obridko et al. [1] based on recent data
confirms the assumption that the 25th cycle of solar activity is a
medium-low cycle. Its height is expected to be 125.2±5.6, and the expected
date of the maximum phase is the end of 2023 or the first quarter of 2024.
§ INTRODUCTION
Obridko et al. [1] analyzed the evolution of the large-scale magnetic field
on the Sun during the last four cycles from 1976 to early 2022. WSO data
(<http://wso.stanford.edu/>) have been used. Special attention
was paid to the effect of prolonged cycles of solar activity. The term appeared
in literature in 1988 [2, 3], although during that decade, observational
evidence appeared indicating that the magnetic activity of one cycle
overlapped for some period of time (often up to several years) with that of
the previous cycle [4]. As observed on the surface, the extended solar
cycle starts during the sunspot maximum at high latitudes and consists of a
relatively short polarward branch (described as "rush to the poles") and a
long equatorward branch, which continues through the solar minimum and the
following sunspot cycle [5], see also [6] for review.
The polaward and equatorward waves appear almost simultaneously and have
opposite predominant polarity of the magnetic field. There are periods when
two “rush to the poles” waves with the field of opposite signs coexist in
the Sun, one of which has nearly reached the pole and the other has just
appeared at mid latitudes. In such periods we can see three zones of
alternating polarity in each hemisphere.
The moment, when all three types of waves are simultaneously present on the
disk and, as a whole, six intermittent zones are observed, exactly
corresponds to the zonal harmonic with l=5. We propose to call this
time interval "the overlapping phase". As seen below, the overlapping phase
can be quantitatively described in terms of the 5th zonal magnetic field
harmonics, which, in this connection, can be referred to as the height of
the overlapping phase. During the overlapping phase, three activity waves
coexist on the solar surface, which lead to the appearance and enhancement
of the odd zonal harmonic with l = 5.
The maximum amplitude of this harmonic dramatically decayed over the past
four cycles similar to the cycle amplitude recorded in sunspot numbers. A
particularly strong decline in the value of the g_5,0 harmonic is
observed after 2000, and this led to the low Cycle 24. Of course, four
cycles are not enough to provide a convincing statistics; however, it seems
plausible that Cycle 25 will not be much higher than Cycle 24.
§ NEW ESTIMATES
At present, new data have appeared that support this conclusion.
* In the middle of 2022, an increase of the fifth zonal harmonic was recorded. Although this peak is not high and is much lower than in Cycles 21, 22, and 23, it certainly indicates the near onset of a low sunspot maximum. In Cycles 21, 22 and 23, the maximum of this harmonic was ahead of the sunspot maximum by no more than 1-1.5 years.
* In March 2023, a reversal of the polar magnetic field was recorded, and in June 2023, half the sum of the field values at both poles turned to zero. This usually indicates the proximity of the sunspot maximum. At the same time, in Cycles 21 and 22, the field reversal virtually coincided with the date of the sunspot maximum; in the low Cycle 24, the polarity reversal was ahead of the sunspot maximum by about a year.
* The relation between the magnitude of the polar field and the height of the upcoming sunspot maximum number has been often used for forecasting. At present, reliable measurements of the polar field strength in Cycles 22, 23, and 24 are available. The relationship between the polar field strength and the height of the upcoming cycle is described by the formula:
SSN_max=36.405+1.3666 B_pol
The last maximum value of B_pol equal to 65 μ T was recorded in
summer 2019 (see the site <http://wso.stanford.edu/Polar.html>). Hence,
the predicted value of SSN in Cycle 25 is 125.2±5.6. This is only a few
units higher than in Cycle 24 (116.4).
The authors are grateful to Dr. T.Hoeksema for access to the the site
<http://wso.stanford.edu>
We acknowledge the support of the Ministry of Science and Higher Education
of the Russian Federation under the grant 075-15-2020-780 (VNO and MMK) and
075-15-2022-284 (DDS).
§ REFERENCES
1.Obridko V. N., Shibalova A. S., and Sokoloff, D. D., MNRAS 523, 1, 982 (2023).
2. Altrock R. C., in: Solar and Stellar Coronal Structure and Dynamics, ed. by R.C.Altrock. pp 414–420 (1988).
3. Wilson P. R., Altrock R. C., Harvey K. L., Martin S. F., and Snodgrass H. B., Nature 333, 748 (1988).
4. Leroy J. L. and Noens J. C., A&A 120, L1 (1983).
5. Kosovichev A., Pipin V., and Getling A., in: American Astronomical Society Meeting Abstracts. p. 304.05 (2021).
6. McIntosh S. W. et al., Sol. Phys. 296, 189 (2021).
|
http://arxiv.org/abs/2307.04367v1 | 20230710064801 | Explanation Needs in App Reviews: Taxonomy and Automated Detection | [
"Max Unterbusch",
"Mersedeh Sadeghi",
"Jannik Fischbach",
"Martin Obaidi",
"Andreas Vogelsang"
] | cs.SE | [
"cs.SE"
] |
Explanation Needs in App Reviews: Taxonomy and Automated Detection
Max Unterbusch
University of Cologne
[email protected]
Mersedeh Sadeghi
University of Cologne
[email protected]
Jannik Fischbach
Netlight Consulting GmbH | fortiss GmbH
[email protected]
Martin Obaidi
Leibniz University Hannover, Software Engineering Group
[email protected]
Andreas Vogelsang
University of Cologne
[email protected]
August 12, 2023
============================================================================================================================================================================================================================================================================================================================================================================================================================
Explainability, i.e. the ability of a system to explain its behavior to users, has become an important quality of software-intensive systems. Recent work has focused on methods for generating explanations for various algorithmic paradigms (e.g., machine learning, self-adaptive systems).
There is relatively little work on what situations and types of behavior should be explained. There is also a lack of support for eliciting explainability requirements.
In this work, we explore the need for explanation expressed by users in app reviews. We manually coded a set of 1,730 app reviews from 8 apps and derived a taxonomy of Explanation Needs.
We also explore several approaches to automatically identify Explanation Needs in app reviews. Our best classifier identifies Explanation Needs in 486 unseen reviews of 4 different apps with a weighted F-score of 86%.
Our work contributes to a better understanding of users' Explanation Needs. Automated tools can help engineers focus on these needs and ultimately elicit valid Explanation Needs.
Explainability, Requirements, NLP
§ INTRODUCTION
Software systems are becoming more intelligent and ubiquitous than ever before, increasing the criticality of their impact on humans. Driven by modern artificial intelligence, it is becoming increasingly difficult for an external user, but also for the developers of these systems, to understand their inner workings and thus their decisions and actions. The ability to provide explanations—a natural ability of humans—is therefore considered an important capability of software systems. As such, explainability is now accepted as a critical quality attribute <cit.> and represents an emerging topic in the field of RE <cit.>.
Researchers have explored the foundations of explainability from different angles. There are several approaches to generating explanations for different algorithmic paradigms.
However, there has been relatively little focus in the literature on what users actually need explanations for <cit.>.
This lack of knowledge limits our ability to effectively elicit explainability requirements and apply existing explanation generation methods.
Thus, the first problem we address in this paper is as follows: We lack knowledge about what users need explanations for.
App reviews have been overlooked as a potential source of Explanation Needs. Pagano and Maalej <cit.> found that app reviews contain valuable RE-related information because they represent rich and readily available textual data that provides insights into thousands of user experiences.
Unlike interview or survey data, app reviews are collected “in the field” under natural circumstances. Users are motivated enough to publish their opinions about an app; they are not forced or paid to do so. This underlines the importance that users place on their concerns. In addition, users are not asked about any specific aspect. The review messages are open to any feedback the users want to give to the app vendors or developers.
We set out to understand users' need for explanation, which we refer to as Explanation Need. Our focus is to characterize the occurrence of Explanation Needs in app reviews and to investigate the types of Explanation Needs that users express. We conducted a qualitative analysis of 1,730 English app reviews of 8 different apps. As a result, we propose a taxonomy of Explanation Needs in app reviews to help developers and researchers distinguish between different types. One of the key benefits of the taxonomy is that it enables researchers and engineers to extract explainability requirements in a systematic and rigorous manner. By categorizing users' Explanation Needs from their perspective into distinct categories, the taxonomy highlights areas where a system may lack transparency or fail to meet users' expectations. This, in turn, provides valuable insight into the types of explanations that are most needed.
Our qualitative analysis shows that Explanation Needs in app reviews are valuable and contain rich information, but are relatively sparse. Explanation Needs have only appeared in about 5% of the app reviews studied. However, manually analyzing app reviews can be challenging due to the sheer volume of reviews and the varying levels of detail and insight they provide.
Tool support to filter the reviews for relevant content would be valuable to allow development and stakeholders to efficiently exploit this source of information <cit.>. We identify this as the second problem addressed in this paper: We lack tool support to automatically identify Explanation Needs in app reviews. To support the use of app reviews, we investigated several classifiers (rule-based, traditional ML, and transformer approaches) to automatically detect Explanation Needs in app reviews. We evaluated and compared the classifiers in a 10-fold cross-validation on an extended set of 5,078 manually labeled app reviews. In addition, we evaluated our baseline rule-based approach and our best-performing classifier on an additional set of 486 unseen and unmodified reviews of 4 new apps to test how well the approaches generalize and perform in a realistic setting.
Our best-performing classifier—a fine-tuned BERT model—achieved a weighted F-score of 93% in a 10-fold cross-validation and a weighted F-score of 86% when evaluated on unseen data. We make the following contributions:
* We provide a taxonomy of Explanation Needs derived from a large set of app reviews.
* We provide a performance analysis of several classifier approaches to detect Explanation Needs automatically in app reviews.
* We publish a set of 5,564 app reviews that we manually labeled according to our proposed taxonomy.
* To strengthen transparency and facilitate replication, we make our code, dataset, and trained models publicly available.[10.5281/zenodo.7740411 .]
§ TERMINOLOGY AND RELATED WORK
§.§ Explainability and User Needs in Explanations
Explainability has gained significant attention from various research fields, including Human-Computer Interaction, Cyber-Physical Systems, and Psychology <cit.>. Since 2019, when it was proposed as a non-functional requirement <cit.>, it has become a trending topic within the SE and RE communities <cit.>. Research has shown that explainability can enhance trustworthiness, transparency, accountability, fairness, ethics, and other quality aspects by overcoming the black box nature of software systems <cit.>. Chazette et al. developed a concise definition of explainability that meets the requirements of SE and RE communities <cit.>: A system S is explainable with respect to an aspect X of S relative to an addressee A in context C if and only if there is an entity E (the explainer) who, by giving a corpus of information I (the explanation of X), enables A to understand X of S in C. The explainer entity does not have to be the system itself. Achieving explainability depends on specific variables: the system's aspect, the addressee, and the context. Accordingly, Kohl <cit.> and Chazette <cit.> emphasize the significance of identifying users' specific needs for explanations and providing customized explanations correspondingly. Indeed, in cases where users do not require explanations, ensuring explainability may not be necessary <cit.>.
Studying app reviews for explanation need identification is a relatively under-researched area. Consequently, a taxonomy of Explanation Needs can aid in advancing knowledge and eliciting requirements for developing explainable systems. Constructing taxonomies provides numerous benefits, including supporting the communication of complex concepts, revealing relationships between entities, and uncovering knowledge gaps. In a similar approach for a different domain, Sadeghi et al. <cit.> developed a taxonomy of reasons for Explanation Needs. They primarily distinguish between four categories of situations requiring explanations: Training, Interaction, Debugging, and Validation, yet the authors focused on Interaction. For Interaction, the taxonomy further breaks hierarchically down into disobedience, failure, and context-aware behavior. That work considered the system, the user, and the environment in their taxonomy; in contrast, our focus will be on the user only.
§.§ App Store Mining and Classifying App Reviews
Pagano et al. <cit.> conducted a
comprehensive analysis of app stores to determine their usefulness for
requirements engineering. They collected over a million app reviews and found that feedback messages can facilitate communication between users and developers. However, they discovered that a significant amount of the feedback collected was of poor quality and lacked informative value. They argue that although app stores can facilitate user-centered RE through the use of user feedback, it is essential to employ appropriate tools and techniques to filter and pre-process relevant contributions. In response to the need for tool support in app store mining, the RE community developed various solutions to extract valuable insights from app store reviews. Guzman and Maalej <cit.> proposed a method to filter features mentioned by users and extract corresponding sentiments, allowing for a detailed analysis of user experience with individual app features. Chen et al. presented a tool that filters app reviews, groups and ranks them, and provides visualizations of the insights <cit.>.
Particularly relevant to this paper are contributions that classify app reviews according to predefined labels, such as problem reports, inquiries, and user experience, or non-functional requirements such as reliability, usability, and portability. To achieve this classification, researchers typically use traditional ML and DL methods for classifying app reviews into various categories <cit.>. Active Learning strategies have also been experimented with, which can help reduce human labor and improve classification accuracy in certain scenarios <cit.>.
Recently, BERT achieved state-of-the-art performance classifying English app reviews into feature requests, problem reports, and irrelevant <cit.>. In this paper, we compare a simple rule-based approach as a baseline, different ML-based approaches, and a DL-based approach using the BERT-Base model <cit.> for detecting Explanation Needs in reviews automatically.
§ CHARACTERIZATION OF EXPLANATION NEEDS
We define an Explanation Need as a knowledge gap that a user intends to close and present our findings on such needs in app reviews in this section. To consider a review as an Explanation Needs, the user must explicitly raise a question or express a need for an explanation. Rhetorical questions ([sic] “What the hell?”) do not qualify as Explanation Needs as they are not intended to elicit an answer. Direct requests ([sic] “Please could you please check it?”) are also excluded since they do not indicate a specific gap in knowledge.
It is important to note that we distinguish between Explanation Needs and Explainability Need, a non-functional requirement identified for software systems. On the other hand, Explanation Needs are needs perceived by users. Following the formatting of Chazette et al.'s definition of explainability <cit.>, we formally define Explanation Needs as:
An addressee A has incomplete knowledge about an aspect X of system S in context C and requests a corpus of information I provided by an entity E that allows A to understand X of S in C.
§.§ Study Design
In the endeavor to identify users' Explanation Needs, this research aims to explore the potential of app reviews as a source of information. By analyzing the rich textual data of reviews, we seek to uncover the types of explanations that users are looking for. To guide our investigation, we formulated the following research questions (RQ):
RQ1: What types of Explanation Needs have been expressed in app reviews?
RQ2: How prevalent are Explanation Needs and their types in app reviews?
Answering RQ1 is crucial for identifying common issues faced by users and prioritizing areas for improvement in app development. It aims to identify and understand users' Explanation Needs in app reviews, guiding the development of more transparent and user-friendly software systems. To answer our research question, we undertake a qualitative analysis to develop a taxonomy for Explanation Needs in app reviews. The provision of conception classification and taxonomy is generally valuable since it provides a standardized framework and facilitates a common ground to communicate and research in emerging fields of knowledge <cit.>. As depicted in Figure <ref>, the qualitative analysis toward addressing RQ1 involved three phases: (1) Dataset Selection, (2) Analysis and Preliminary Taxonomy Extraction (3) Verification and Taxonomy Finalization.
Phase 1.
In the first phase, we selected the datasets for our analysis.
The original dataset used in our study was assembled by Brunotte <cit.>. Although a more recent version of the dataset exists with a larger number of reviews, we focused our analysis on a subset of 1,730 reviews provided to us directly by the authors. It allowed us to conduct our analysis more targeted and manageable. In the remainder of this paper, we refer to this dataset as . comprise app reviews from eight distinct apps available on the Apple App Store and Google Play. The domains represented in span several categories, including health and wellness, finance, technology, and lifestyle, making it well-suited for exploring the nature of user feedback and Explanation Needs in mobile app reviews. Table <ref> provides an overview of this dataset.
Phase 2.
Using the dataset as our basis, we extracted the preliminary taxonomy of Explanation Needs. A single coder initially analyzed all 1,730 app reviews based on the definition of Explanation Needs outlined in <ref>. The coder then filtered out 1,600 reviews that did not express any Explanation Need, and the remaining 130 cases were labeled as Explanation Need on a tentative basis. While there was a possibility that some of these cases could be excluded by the other coders in subsequent phases, these 130 cases still provided a foundation for further analysis in terms of categorization and taxonomy extraction. Following the template by Saldaña <cit.>, the coder also developed a codebook to maintain, organize, and share the codes with the other authors.
The initial coding resulted in an early version of the taxonomy, which was subject to further refinement through extensive discussions and revisions by the authors involved in the study. Hence, as this phase's output, a preliminary taxonomy was generated, which classified different types of Explanation Needs and established boundaries between them. Nevertheless, at this point, the codebook yet had rather generic and fuzzy definitions of the categories or loose criteria for differentiating them. Therefore, we proceed to the next phase to further verify the applicability of the taxonomy and codebook.=-1
Phase 3.
In the final phase, we aimed to verify and refine the preliminary taxonomy by involving two other coders. We sampled 130 app reviews tentatively identified as Explanation Needs by the first coder, plus a random selection of 70 reviews that were not labeled as such. The resulting dataset was shuffled and divided equally between the coders, with each responsible for categorizing their respective half as Explanation Need or not. For the reviews categorized as Explanation Need, the coders then had to check if they could be classified under one of the leaf nodes of the preliminary taxonomy. The goal was to ensure the preliminary taxonomy and codebook's completeness and accuracy and identify any deficiencies.
The coders then engaged in several rounds of discussions and classification. During the first iteration, the coders compared the labels assigned by the initial coder to the new labels the additional coders gave. From the 130 cases identified by the initial coder as Explanation Need, 48 cases were excluded by either of the new coders. So we were left with 82 app reviews that the new coders also tentatively labeled as Explanation Need, with each case being assigned a specific type of explanation. During the second iteration, all the coders went through these 82 reviews to further discuss and evaluate each case.
Moreover, at this point, coders attempted to prune and/or extend the taxonomy categorization to produce the final taxonomy and to consolidate their descriptions and boundaries recorded in the codebook. Throughout the last iteration, 5 additional app reviews that did not meet the requirements and specifications of the final taxonomy were excluded, resulting in a total of 77 cases labeled as Explanation Need.
§.§ Results: A Taxonomy of Explanation Needs
As shown in Figure <ref>, the taxonomy has a hierarchical structure and consists of two levels. We refer to the lowest level elements, namely , , , , and , as categories of Explanation Needs. To make the categories more tangible, we included a non-exhaustive list of aspects for each category. These aspects are more concrete groupings of related and typical Explanation Needs that we could observe in the data. However, they are not part of the taxonomy in a narrow sense.
Given the Explanation Need <ref>, a key distinction we make in the first level of our taxonomy is whether such a need for some explanation is an issue's primary or secondary concern. More precisely, if the user perceives their lack of knowledge as the only issue, then the Explanation Need becomes a Primary Concern, whereas if they see other substantial problems aside from their knowledge gap, it becomes a Secondary Concern. In the latter case, an underlying problem exists, typically a deficiency, which substitutes the Explanation Need as the primary concern. Therefore, offering an explanation may increase the overall understanding of the situation, but an explanation alone cannot solve the underlying problem.
As depicted in Figure <ref>, the Explanation Needs belonging to the , , and categories represent a primary concern. In general, is when users are unfamiliar with the system or particular features, either because they are new to it or the system's features have been changed. We found the following aspects to characterize best:
* Instruction. Users seek instructions for achieving specific goals, such as how to use a system, feature, or settings option. This aspect requires that the users clearly intend what they aim to do. Instruction aspect excludes reviews if there is an identifiable deficiency, such as an error or failure (see aspect Fix). Example [sic]: “How do you edit from this app???”.
* Features Offered. Users seek information about specific or general systems' features or functionality. Therefore, users are unaware of what the system can exactly do. Example [sic]: “... is there anyway to sort this out ...?”.
* Effect-Of. Users want to obtain information on the potential outcomes of specific actions. The users know how to perform such an action but are not sure what the impact will be. Example [sic]: “If I invest in dividend paying stocks, will the dividends be added to my portfolio?”.
The next category is , including aspects that arise in the ordinary operation of a user familiar with the system. These aspects assume expected behavior, not accounting for deficiencies such as errors or failures. The category was found to encompass the following aspects:
* Algorithm. Users struggle to comprehend why a system generates a particular output, wanting to know the factors that influenced the computation. The output is unique to each user, therefore the programmed logic that is the same for all users is not included in this aspect (see aspect Design Decisions). Example [sic]: “In the last 3 months my credit went up a total of 10 points and then dropped down 7 points December 2. This doesn't make sense.”
* Design Decision. Users wonder why things are a particular way (status quo) or not a certain way (counterfactual). It is not an output of the system that might be individual to each user, but the programmed logic, which the developers have agreed on. Hence, in contrast to the Algorithm aspect, the Design Decisions are the same for multiple (if not all) users. Example [sic]: “why does the app force portrait mode?”
* Signification. Users seek clarification on definitions, visual elements (such as symbols, colors, and highlighting), information visualizations, or related issues in order to understand the system's intended meaning. Example [sic]: “I like this app, but when there may be something in red I just don't understand. Does it means something is wrong?”
The last category in the primary concerns is category. It represents general Explanation Needs that are not necessarily provoked during the interaction with a system. Further, aspects to be explained may be shaped by overarching business goals or specific project or process requirements <cit.>. Here we determined the following aspects:
* Mission. Users seek clarification on the system's purpose, utility, and vision, with a particular focus on specific features and the system as a whole. Example [sic]: “Why do we need to access this app to get the information we used to get by phone from the doctor?”
* Purchase & Subscription. Users inquire about purchase or subscription matters, such as feature exclusivity in premium. This aspect only applies when there are multiple product lines with varying purchase or subscription plans. Example [sic]: “Do I have to pay for it on all devices?”
* Privacy. Users express privacy concerns regarding data collection, processing, and forwarding practices, as well as legal privacy rights and app permissions (e.g., GPS activation). If the inquiry is not focused on privacy but rather on the aspects that affect software decisions, it falls under the Algorithm aspect. Example [sic]: “Not sure why you need date of birth to register a navigation app, very suspicious as far as I'm concerned.”
Moving to the secondary concerns, we have and categories. Accordingly, here the Explanation Need is only the secondary concern of users, and there is a substantial underlying problem (at least in the user's perception) that is their primary concern. Overall, the aspects are somewhat reproachful and the primary concern typically is a subjective deficiency from the user's point of view.
* Change. Users seek explanations for changes to a system, including modifications to the user interface or workflow. This aspect is more critical than genuine. However, it does not necessarily involve the need for re-learning the system, which is covered by the Instruction aspect. Example [sic]: “It just keeps getting worse. Why do you do this?”
* Feature Gap. Users want to know why a feature is incomplete or missing. This aspect doesn't cover cases where a feature is not supported for an individual user's use case (see aspect Compatibility). Example [sic]: “Why would you have a database where you can only add and not edit or delete?”
* Compatibility. Users are confused by a feature(s) not being supported or compatible with their use case. So, they are prevented from using a set of features due to external conditions that are not part of the system. This aspect excludes errors or failures. Example [sic]: “Only big downfall is that USA account holders for some reason ... cannot use the boost feature. No clue why and no one has given answers to why it doesn't work.”
Finally, the category describes a situation with an undeniable objective deficiency such as an error or failure <cit.> in the system. It differs with , where the primary concern is a subjective deficiency in a user's eyes. We found the following aspects to be typical for :
* Fix. Users ask about fixes or workarounds to solve errors/ failures or ask whether errors/failures are known to the developers. Example [sic]: “Anyone experiencing the same or know what to do about it?”.
* Cause. Users ask for the underlying faults that cause errors, failures, or obviously erroneous outputs. They are interested in knowing the cause of the errors/failures to potentially attempt to fix them themselves. On the contrary, they do not ask for any support (see aspect Fix). Example [sic]: “Is it a loading problem or a glitch??”.
* Confusing Message. Users feel misled by rare messages (such as uninformative or incongruous alerts) and assess the messages as incomplete, inaccurate, or erroneous. The messages can potentially be faulty explanations. Example [sic]: “I constantly get warnings that I don't have enough shares to sell and I cannot find any solutions”.
§.§ Discussion of Results
Through a rigorous study of app reviews, we have developed the Explanation Needs taxonomy, which addresses RQ1 and provides a valuable resource for researchers and developers seeking to understand the concerns and requirements of end-users. By categorizing user needs in the taxonomy, we can better recognize and address various requirements in a more systematic manner, ultimately improving the quality, transparency, and user-friendliness of the application. The proposed taxonomy serves as an enabler, allowing for a more effective approach to addressing user needs and fostering a deeper understanding of the end-user experience. As such, the Explanation Needs taxonomy has significant implications for app development and can contribute to the development of more explainable systems that better meet the needs of users.
With the Explanation Needs taxonomy, we were able to tackle the RQ2, which aimed to gain a more statistical view of the types of Explanation Needs expressed in app reviews. So we applied the taxonomy to multiple sets of data, composed of 5,564 reviews in total. Table <ref> provides an overview of all the datasets used in this paper. As discussed in Section <ref>, the taxonomy extraction was based on the and the final labeling was achieved through several rounds of cross-checking to ensure the validity and reliability of our findings.
However, to gain deeper insights into the types of information and Explanation Needs in the app reviews and to further assess the coverage and applicability of our taxonomy, we also labeled the reviews of our extended datasets, which we create for classifier implementation and validations (see Section <ref> for more details). The labeling process of the rest of the data (i.e., the app reviews 9 to 22 in Table <ref>) was carried out after consolidating the taxonomy and codebook, the latter of which provides complete information on inclusion and exclusion criteria, as well as typical and atypical examples. Following this, a single coder categorized the app reviews in and that had already been labeled as Explanation Needs (see Section <ref> for more details).
Besides the description of the apps, source and number of reviews, Table <ref> provides a breakdown of the distribution of different types of Explanation Needs per app. It shows the number of occurrences of each type of Explanation Needs for each app, as well as the total number and percentage of Explanation Needs across all apps. By examining this table, we can answer the RQ2 by identifying the areas where users require the most explanations. This analysis can help shed light on the nature and extent of Explanation Needs in app reviews.
For example, it shows that the majority of cases fall under the Primary Concerns category, accounting for 52.3% of all app reviews. This implies that users' primary issue with the app is their lack of understanding and knowledge, without any substantial problems aside from it. This finding highlights the importance of addressing users' primary concerns and providing sufficient explanations to enhance their overall understanding of the app's functionality. Furthermore, the category is the most frequent type within the Primary Concerns and accounts for 20.7% of the total number of Explanation Needs across all apps. This means that a significant proportion of user feedback in app reviews is related to ordinary interaction with the system. As users engage with the app, they may encounter unexpected behaviours, have questions about design decisions, or need clarification on the meaning of certain visual elements or notions. Accordingly, it is not surprising to have a relatively high number of types since these issues could arise regardless of the app's specific functionality, and, therefore, could be relevant to a wide range of users. Additionally, the category may be particularly salient to users, as it directly affects their experience using the app, and they may be more likely to leave reviews on these types of issues. Similarly, the category stands out with the second-highest percentage of Explanation Needs in the primary concern, accounting for 18.6% of all Explanation Needs, indicates that users frequently encounter difficulties in understanding how to use certain features or functionalities of the app. This finding highlights the importance of providing concise instructions or tutorials to help users learn how to use the app effectively.
Overall, the high percentage of and indicates that the app's user interface or design could be improved. Our results hence may suggest that the application design and development should primarily focus on the usability of the apps by making them more intuitive and user-friendly.
Another interesting observation is that the category, which is classified as a secondary concern, has the highest percentage of Explanation Needs at 32.3%. This could be attributed to its subjective nature, as the primary concern of this category is a perceived deficiency from the user's point of view, which may be difficult to address directly. Additionally, this deficiency is not necessarily related to a specific bug or technical issue, but rather a mismatch between the user's expectations and the app's performance or features. This finding suggests that users are more likely to express their discontentment and frustration in reviews. Last but not least, our qualitative analysis also reveals an important insight. We found that although app reviews provide a wealth of information about users' Explanation Needs, the proportion of reviews that contain such information is relatively low, at only 5.1%. This indicates a need for more efficient and automated techniques to extract useful content from reviews. Therefore, our study has motivated us to pursue our second contribution, which is described in more detail in Section <ref>. By developing machine learning-based approaches to extract Explanation Needs from reviews, we hope to improve the efficiency and effectiveness of analyzing large volumes of user feedback.
§.§ Threats to Validity
A potential threat to internal validity is the use of quantitative coding, which can be interpretive and subjective. This means that our analysis may be influenced by our own biases or assumptions, which could affect the accuracy of our findings. Poor English and typos in some reviews can also lead to inaccurate conclusions, but we made a conscious effort to evaluate unintelligible reviews. In addition, a threat to external validity could be survivorship bias, as our results may not be representative of those with low technological literacy, as they may be less likely to write and publish app reviews in the first place. Also, the we used in our taxonomy extraction is relatively small, with only a few cases of Explanation Needs observed (4.6% as shown in Table <ref>). Accordingly, it might limit the generalizability of our taxonomy categories. However, to mitigate the potential threat of a small sample, we conducted a thorough and saturated coding process and verified the validity of our taxonomy categories on an extended dataset.
§ AUTOMATIC DETECTION OF EXPLANATION NEEDS
§.§ Corpora Creation
To determine the best method for detecting Explanation Needs in a structured way, we follow the recommendations by Dell’Anna et al. <cit.>. They stress that the results of a simple cross-validated experiment do not allow to draw definite conclusions about the performance of a classifier in an operational context. In other words, we cannot necessarily infer from such an experiment whether the classifier is able to generalize and is thereby suitable for use on unseen data in practice. Hence, we evaluate our approaches on two datasets:
CrossVal-DS. We use this dataset to train and compare all models applying 10-fold cross-validation. The main purpose of is to compare the performance of different NLP classifiers and to select the best-performing method. It includes all reviews of created in Section <ref>. However, this dataset with 77 Explanation Needs is not sufficient for training an NLP classifier. Accordingly, we extend the dataset with further reviews and manually label them with respect to the tags “explanation need” and “no explanation need”. We make use of a dataset collected by Maalej et al. <cit.> that has already been utilized in the RE community to classify app reviews into problem reports, inquiries, and irrelevant ones <cit.>. Additionally, we collect further app reviews from 9 popular apps, using custom Python web scraping tools for the Apple App Store[<https://pypi.org/project/app-store-scraper/>] and Google Play Store[<https://pypi.org/project/google-play-scraper/>]. For each of the apps, we scraped as many reviews as possible and then drew a random sample of 100 reviews to include an equal-sized subset of the reviews per app. A detailed overview of is provided in Table <ref>. In total, comprises 5,078 reviews of which 261 contain Explanation Needs (5.14%).=-1
General-DS. To investigate the generalizability of the best-performing classifier, we apply it to a set of unseen reviews that are not associated with any of the apps contained in . Specifically, we scrape and annotate reviews about the four randomly selected apps called WeChat, Memrise, Duolingo, and GitHub (see Table <ref>). The main purpose of is to report the performance of our best classifier in a realistic setting. In total, comprises 486 reviews of which 24 contain Explanation Needs (4.94%).
§.§ Annotation Validity
To verify the reliability of our annotations, we calculated the inter-annotator agreement in terms of Cohen's Kappa <cit.>. We involved a total of four annotators in the creation of and and assessed the inter-rater reliability on the basis of 485 reviews that each have been labeled by two out of the four annotators.
In case of a high imbalance of ratings, Cohen's Kappa is low and indicates poor inter-rater reliability even if there is a high agreement between the raters (Kappa paradox <cit.>). Thus, Cohen's Kappa is not meaningful in such scenarios. Consequently, Cohen's Kappa should always be reported together with the percentage of agreement and other paradox-resistant measures (e.g., Gwet's AC1 measure <cit.>). We calculated all measures (see Table <ref>) using the cloud-based version of AgreeStat[<https://www.agreestat.com/>]. Cohen's Kappa and Gwet's AC1 can both be interpreted using the taxonomy developed by Landis and Koch <cit.>: values ≤ 0 as indicating no agreement and 0.01–0.20 as none to slight, 0.21–0.40 as fair, 0.41–0.60 as moderate, 0.61–0.80 as substantial, and 0.81–1.00 as almost perfect agreement. Table <ref> demonstrates that the inter-rater agreement of our annotation process is reliable as we achieve an average percentage of agreement of 95%. Despite a high agreement of over 90%, Cohen's Kappa yields a relatively low value, which paradoxically suggests only moderate agreement. A more meaningful assessment is provided by Gwet's AC1 as it did not fail in the case of prevalence and remains close to the percentage of agreement. The achieved Gwet's AC1 of 0.945 indicates a nearly perfect agreement. Therefore, we assess and as reliable and suitable for the implementation and evaluation of our Explanation Need detection approach.
§.§ Methods
We define the detection of Explanation Needs as a binary classification problem, in which we are given a certain review 𝒳 and we are required to produce a nominal label y ∈𝒴 = {explanation need, no explanation need}. Since app store reviews are written in natural language, we build our classifier based on different methods established for NLP.
Rule-based Approach.
Instead of using a random classifier as the baseline approach, we involve simple regex expressions for the detection of Explanation Needs. We iterate through all reviews in the test set and check if a question mark or the word “why” is contained. We hypothesize that both expressions might be a feasible indicator for the presence of an Explanation Need. Following this assumption, we classify a review as an Explanation Need if it contains at least one of the two expressions and vice versa.
Machine Learning-based Approach.
We investigate the use of supervised ML models that learn to predict Explanation Needs based on a labeled dataset. Specifically, we employ established binary classification algorithms: NB, SVM, RF, DT, LR, AB, and KNN. To determine the best hyperparameters for each binary classifier, we apply Grid Search, which fits the model on every possible combination of hyperparameters and selects the most performant. We use two different methods as word embeddings: BoW and TF-IDF. In Table <ref> we report the classification results of each algorithm as well as the best combination of hyperparameters.
Deep Learning-based Approach.
With the rise of DL, more and more researchers are using DL models for NLP tasks. In this context, the BERT model <cit.> is prominent and has already been used for question answering and named entity recognition. BERT is pre-trained on large corpora and can therefore easily be fine-tuned for any downstream task without the need for much training data (Transfer Learning). In our paper, we make use of the fine-tuning mechanism of BERT and investigate to which extent it can be used for the detection of Explanation Needs. First, we tokenize each app store review. BERT requires input sequences with a fixed length (maximum 512 tokens). Therefore, for reviews that are shorter than this fixed length, PAD are inserted to adjust all reviews to the same length. Other tokens, such as the CLS, are also inserted in order to provide further information on the review to the model. CLS is the first token in the sequence and represents the whole review (i.e., it is the pooled output of all tokens of a review). For our classification task, we mainly use this token because it stores the information of the whole review. We feed the pooled information into a single-layer feedforward neural network that uses a softmax layer, which calculates the probability that a review contains an Explanation Need or not.
§.§ Evaluation Procedure
is strongly imbalanced as only 261 are positive samples. To avoid the class imbalance problem, we apply Random Under Sampling. We randomly select reviews from the majority class and exclude them from the dataset until a balanced distribution is achieved. Our final dataset consists of 522 reviews of which 261 contain an Explanation Need and the other 261 do not. We follow the idea of cross-validation and divide the dataset into a training, validation, and test set. We opt for 10-fold cross-validation as a number of studies have shown that a model that has been trained this way demonstrates low bias and variance <cit.>. Please note that undersampling stands in conflict with our goal to understand how well our classifier generalizes and performs in a realistic setting. Hence, we do not undersample allowing us to report our final results on a realistically distributed test corpus.
We use standard metrics for evaluating our approaches, such as Precision, Recall, and a weighted F-measure. Since a single run of a k-fold cross-validation may result in a noisy estimate of model performance, we repeat the cross-validation procedure five times and average the scores from all repetitions.
Since our classifier is supposed to assist development teams by detecting relevant Explanation Needs in reviews automatically, we favor Recall over Precision. A high Recall corresponds to a greater degree of automation of Explanation Need detection because it is easier for users to discard FP than to manually detect FN. Consequently, we seek high Recall to minimize the risk of missed Explanation Needs and acceptable Precision to ensure that the development teams are not overwhelmed by FP. To attain a accumulated, single metric from Precision and Recall, the simple F-Measure (F1) is frequently used in binary classification tasks. It is defined as the harmonic mean between Precision and Recall, and thus assigns equal importance to both metrics. To account for our preference for Recall over Precision, it is imperative to make adjustments to the way in which the two metrics are weighted. We evaluate our approaches based on a weighted F-Measure:
F_β = (1+β^2) ·Precision ·Recall/(β^2 ·Precision) + Recall
where β is the ratio to which Recall is more important than Precision <cit.>. Berry <cit.> defines β as follows:
β= time_a ·λ/time_v
where time_a is the average time that a human would need to assess an artifact manually (i.e., the time spent by a human determining whether a particular review is an Explanation Need or not), and time_v is the average time that a human would need to verify whether a positive detection by a tool is actually a True Positive (i.e., the time spent by a human neglecting a FP detection of an Explanation Need). Further, λ is the inverse of the share of relevant artifacts within all artifacts. In other words, λ is the average number of artifacts that an analyzer would need to investigate in order to find a single relevant artifact. In our case, λ is calculated as follows:
λ= (285/5564)^-1 ≈19.52
because we identified a total of 285 Explanation Needs in our dataset of 5,564 reviews. Thus on average, one out of 19.52 app reviews contains an Explanation Need. Since the time required to vet a single answer of our classifier is no more than the time required to manually check if an app review contains an Explanation Need, the weight ratio β is equal to λ. Hence, we define β as 19.52.
§.§ Experimental Results
In the following, we describe the results of our experiments. First, we compare the performance of different NLP classifiers on . Second, we investigate the generalizability of the best-performing method on .
Selection of Best-Performing Method
Table <ref> reveals that our shallow rule-based approach shows a strong performance in detecting Explanation Needs. It achieves a high F_19.52 score for both classes and is able to demarcate between reviews that contain Explanation Needs and those that do not. In comparison, all ML-based approaches exhibit a significantly poorer performance. For example, DT trained on TF-IDF embeddings achieves a Macro-F_19.52 score of 58% (deterioration of 35% compared to the baseline approach). The best performance in this category is achieved by RF trained on BoW embeddings with a Macro-F_19.52 score of 76%. Our experiment shows that the choice of sentence embedding has no significant effect on the performance of the ML-based approaches. Most of the approaches achieve a Macro-F_19.52 score of about 70% regardless of the applied sentence embedding. Our fine-tuned BERT model, on the other hand, shows a considerably stronger performance and achieves a Macro-F_19.52 score of 93%. Interestingly, despite its rich language understanding, the BERT model fails to outperform our simple rule-based approach. In fact, both approaches achieve the same Macro-F_19.52 score and posses consequently the same predictive power. Our experiments thus show that both approaches are suitable for identifying Explanation Needs in . To investigate the generalizability of the rule-based approach and the BERT model, we apply both approaches to a larger set of unseen reviews written for other apps contained in .=-1
Generalizibility of Best-Performing Method
When applied to unseen data, both approaches show a clear performance drop in the detection of Explanation Needs (see Table <ref>). While both approaches continue to show very high F_19.52 scores for the “no explanation need” class, the F_19.52 score for the “explanation need” class has decreased significantly. The largest performance drop is evident in the rule-based approach, which only shows an F_19.52 score of 67% in detecting explanation needs across all reviews of all four apps. Similarly, the trained BERT model fails to match the very good F_19.52 score of 94% that it could achieve when applied to the balanced training set. Instead, it achieves a score of 79% on the unseen data, which corresponds to a decrease of 15%. Overall, the BERT model outperformed the rule-based approach and achieved a significantly better Macro-F_19.52 score of 86%.
The higher Macro-F_19.52 score is mainly attributable to the fact that the BERT model shows a significantly better Recall with regard to the Explanation Need class. In other words, the BERT model identified more Explanation Needs in the reviews than the rule-based system. Our experiment demonstrates that this performance deviation does not depend on a specific app about which the respective reviews were written. In fact, when applied to the reviews about WeChat, Duolingo and Github, the BERT model exhibits better performance. In the case of the reviews about Memrise, it achieves the same Recall as the rule-based approach. Both the rule-based approach and the BERT model show the most significant performance loss with regard to Precision and generate a great number of FP. Using both approaches, two of three reviews that are supposed to contain an Explanation Need are FPs, causing high filtering costs for practitioners.
§.§ Discussion of Results
Our experiments show that the rule-based approach achieves the same performance as the BERT model when evaluated on , but performs worse when applied to unseen data. The rule-based approach fails to recognize more than 30% of the Explanation Needs and seems to generalize less effective than the BERT approach. When analyzing the data in , we see that the detection of Explanation Needs cannot be broken down to the presence of questions and question words. Explanation Needs do not necessarily contain question marks or question words. In many cases, questions are formulated but question marks are not included: Would you please keep us updated on what's going on. I have several texts and don't know how to keep them. Don't want to lose it. The BERT model understands the semantics of sentences better and dependents less on the sentence's syntax. The rule-based approach could be extended by adding more interrogatives (e.g., how) and interrogative verbs (e.g., don't understand) to enhance the Recall of the approach, however, this may lead to an unreasonable increase in FPs. The resulting filtering effort would diminish the use of the approach in practice.
From a critical point of view, our best classifier does not perform flawlessly. It does not identify all Explanation Needs in and predicts a number of FPs. We argue that the recall value needs to be improved above 90% to qualify the approach for practical use. Otherwise, the practitioners would have to go through the reviews manually to detect false negatives, which is time-consuming given the high number of reviews and the fact that Explanation Needs rarely occur. The achieved precision value of 37% is not optimal, but in our view still justifiable. It is much easier for the practitioner to neglect two false positives from 3 reviews predicted as Explanation Needs than to go through 20 reviews manually to discover a single Explanation Needs.
Our classifier marks a first step toward automatic Explanation Need detection. Further studies should focus on optimizing the classifier in terms of recall. We hypothesize that the extension of the training set and the use of further language models might be beneficial. So far, we have only focused on the BERT-Base model <cit.>, although other studies <cit.> show that alternative models such as RoBERTa can achieve even better performance. To assist practitioners in filtering FPs, it may also be useful to have the classifier mark the specific clause in each review that has caused the review to be categorised as Explanation Needs <cit.>. This will help practitioners to understand the inner workings of the classifier and also increase its acceptance.
§.§ Threats to Validity
A threat to internal validity are the annotations themselves as an annotation task is subjective to a certain degree. To minimize the bias of the annotators, we performed two mitigation actions: First, we conducted a workshop prior to the annotation process to ensure a common understanding of Explanation Needs. Second, we assessed the inter-rater agreement by using multiple metrics (Gwet's AC1 etc.).
Despite our efforts to make the labeling process as transparent and systematic as possible, there may still be some variability in the resulting gold standard, e.g., misinterpretation of the users' intention, blurred boundaries between the categories, too broad or too narrow judgement, or human mistakes. Using the adjusted F_β-score as an evaluation metric poses a threat to construct validity. We used an adjusted β value of 19.52, which was calculated based on the frequency of Explanation Need occurrences in app reviews. This value is in the order of β values calculated for other “needle in the haystack” tasks <cit.>. However, it is possible that the value may deviate when calculated based on another dataset. Our results have shown that generalization of our tested classifiers is fairly moderate when applied to unseen, dissimilar test data. This may indicate that more data is needed to train a classifier that generalizes better.
Lastly, app reviews are not the only relevant source of user feedback <cit.>.
§ CONCLUSION
This work is a further step towards user-centered explainability engineering.
It contributes to a better understanding of users' Explanation Needs and lays the foundation for future research and development in this area. The proposed taxonomy of Explanation Needs provides a rigorous approach for extracting explainability requirements from app reviews, ensuring that they meet users' expectations. In addition, our approach represents the first step towards automatic explanation need detection and reduces the manual effort required by engineers and researchers to identify Explanation Needs in reviews. To facilitate practical use of the approach, it needs to be optimized for recall so that practitioners can efficiently focus on eliciting valid Explanation Needs. Finally, our published set of manually labeled app reviews will enable researchers in the field to improve their own models and approaches for detecting Explanation Needs.
§ ACKNOWLEDGEMENTS
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No.: 470146331, project softXplain (2022-2025).
IEEEtran
|
http://arxiv.org/abs/2307.04125v1 | 20230709083942 | Bounced Model of Droplet on Moving Substrate | [
"Chengwu Liu"
] | physics.flu-dyn | [
"physics.flu-dyn"
] |
[email protected]
https://orcid.org/0000-0001-9067-1892
School of Physics, Shandong University, Jinan 250100, China.
School of Physics, Shandong University, Jinan 250100, China.
Firstly, we get the completely bouncing criteria Cr for droplet on moving substrate. The bouncing without splashing condition is Cr>1. Then, we mainly research the effect of wind field for droplet, and get the completely bouncing criteria Cr_wind for droplet with wind.
Lastly, we get the contact angle of droplet on the moving substrate and calculate the Time Independent Reynolds Equation with rho and μ are constant.
Bounced Model of Droplet on Moving Substrate
Chengwu Liu
August 12, 2023
============================================
§ INTRODUCTION
The questions of droplet on a surface are related to the interaction of interface.
There is a micrometer-size gas film in the interface between liquid and solid. This gas film was firstly observed by the way of snapshoot <cit.>. The evolutionary process of gas film was firstly observed by X-Ray technology <cit.> at the moment of contacting. They found that the gas film evolve to a bubble with spending on microsecond-size time. E. Sawaguchi <cit.> found that the distribution of thickness of droplet on a moving surface is similar to saddle surface. In addition, the hydrophobicity of droplet on a moving surface is enhanced and is similar to Leidenfrost effect <cit.>. Therefore, the interaction between liquid and solid would be affected by the motion of surface. In this paper, we will talk about how these parameters affect the hydrophobicity in section 2.
Ted Mao <cit.> assumed a critical bounced state to deal with the question of bounced on motionless surface and got a critical bounced criteria E_ERE^*. Actually, the gas film on motionless surface is different from gas film on moving surface. So the interaction of liquid and solid on motionless surface is also a little different from one on moving surface. In this paper, we will talk about this question in section 3.
Droplet also might splash on a solid surface. We have some models to describe splashing <cit.><cit.><cit.><cit.><cit.>. In this paper, we will talk about how extra wind affects the splashing and bounced of droplet in section 4.
§ DROPLET ON THE MOVING SUBSTRATE
The final states of droplet after it impacts on moving substrate are various. We name bouncing without splashing as completely bouncing, bouncing and splashing as partially bouncing. We find out many physical phenomenons about droplet on the moving substrate with lots of experiments(The experiments conditions are in the supplemental material<cit.>. ). For instance, there is the critical speed which is the shift of bouncing and retention. We will deeply research the completely bouncing condition and critical speed below.
§.§ Completely Bouncing Criteria for droplet on Moving Substrate
We will research the phenomenons without splashing in this part.
Droplet will spread, retract, bounce/retain after impacting on the moving substrate. According to the different interaction modes on the solid-liquid interface, the interface can be divided into three areas. As the figure 2(a) shown, the first one is gas film(thickness h∼ 10μm). The cohesive force of gas film acts as normal capillary force to inhibit bouncing. The second one is solid-liquid interface. The interaction of liquid and solid is Van der Waals force. The third one is molecule area. The interaction of this part mainly include the Derjaguin Disjoining Pressure Π( h) =Π_vdW+Π_EL+Π_struc.+Π_steric. In these areas, inhibition and promotion effects for bouncing should be researched in detial.
§.§.§ Inhibition Effects
Firstly, the maximal viscous dissipation of droplet could be approxed is D≤ V_G=mgh∼ 10^-3J with enery conservation(we order the surface of substrate is the zero gravitational potential surface.) Nextly, the dissipation of droplet also due to the microstructure on the surface of substrate. The roughness could be evaluated by the size of roll angle in general. We could ignore the dissipation of this part with considering a small roll angle(smooth surface). Acturally, we use the substrate with small roll angle in our experiments(more detials in supplemental material<cit.>). Then, we will consider the interaction between liquid and solid on the gas film and solid-liquid interface. The attraction is Van der Waals force mainly. And it is the potential force and plays an obvious role on micrometer scale justly. Therefore, the vdW force between solid and liquid has no effect on bouncing. However, some gas are carried in the gas film during the droplet falling process. This gas film will go away from the system constituted by droplet and solid substrate during the bouncing process. And the capillary force of gas film will dissipate the energy of droplet. So, the cohesive force of gas film will inhibit bouncing. Nextly, we will explain it detially.
First of all, this process is different from the previous section. In this section, the pressure of gas film p changes continuously over time, and it's relatively large. The kinetic equation could be given through analysing the droplet as the figure 2(b) shown.
( p-p_⊖) S=md^2H/dt^2+mg
where p_⊖ is atmospheric pressure, H is the vertical positionsl coordinaten of droplet, S is the contact area of liquid and solid. Apparently, we could find out H and S change over time through observing experiments results. So p also should change over time as figure 2 shown. Then the dissipation due to the cohesive force of gas film is be considered below.
The gas film is divided into two parts g_1, g_2 with h thickness as figure 2(d) shown. And two parts at a distance of d∼ 10^-9m(d is the distance of two molecule). Hence, the per unit area vdW potential between g_1 and g_2 is
w_g =-A/12π[ 1/d^2+1/( d+2h) ^2-2/( d+h) ^2]h≫ d-A/12π d^2
where A∼ 10^-19J is Hamker constant. We assume that the thickness of gas film changes from h_0 to h from the state of maximum spreading to bouncing exactly, τ' is the time interval of this process. Then the energy dissipation during retraction process is
Q_cap=| ∫_h_0^h-∂ w_g/∂ dSdh| =| ∫_0^τ'AS/6π d^3dh/dtdt |
Then we expand h to linear term and consider the ideal gas hypothesis.
dh/dt=h-h_0/t, d^3=kT/p
We could give the energy dissipation due to capillary force with equation <ref>.
Q_cap ≈| ∫_0^τ'-ApS( h-h_0) /6π d^3tSdt|
[Sh_0=V_0]Sh=V| ∫_0^τ'Ap( V-V_0) /6π kTtdt|
[p_0V_0=ν RT]pV=ν RTAν R/6π k| ∫_0^τ'1-( p/p_0) /tdt|
This is a improper integral. In order for the improper integral is convergent. We have
lim_t→ 0+1-( p/p_0) /t=lim_t→ 0+-1/p_0dp/dt=a⇒dp/dt=-ap_0
Therefore, we have the pressure p changes linearly with time below the above approximation. Then bring equation <ref> to equation <ref>.
Q_cap =Aν R/6π k| ∫_p_0^p^⊖dp/p_0|=Aν N_A/6πp_0-p^⊖/p_0=Aν N_A/6πaτ'∼ 10^-3J
where ν is the amount of substrate of gas film, N_A is Avogadro constant. We could conclude that the order of energy dissipation due to capillary force is same as the gravitational potential energy. So it couldn't be ignored.
Lastly, let's consider the viscous dissipation. It's so difficult to calculate that we have to estimate it. Let's consider an undemanding toy model.
Firstly, the viscous dissipation should relateto spreading factor with a certain function. The viscous dissipation which droplet impacts on a smooth motionless substrate relate to ( d_m/D)^2<cit.> during spreading process, and relate to ( d_m/D)^2.3<cit.> during retraction process. Hence, we associate the spring oscillator model of two degrees of freedom with this question(Two springs placed perpendicular to each other horizontally on a smooth substrate). We assume that the viscous dissipation is E_diss∼ QE_p max, where E_p max is the maximum spreading "elastic potential energy" for the first time, Q is similar to quality factor.
E_diss∼α k_n ( D_n max-D_0)^2 +β k_t ( D_t max-D_0) ^2
Considering that the effect of tangential mainly due to surface tension and viscous shear stress T_ν, the effect of normal mainly due to surface tension. We could give the k_n and k_t using above model.
k_t ∼aγD_t max+bT_ν/D_t max-D_0
k_n ∼a' γD_n max/D_n max-D_0
where α, β , a, b, a' is constant, T_ν∼η VD_n maxD_t max/δ is the viscous shear stress<cit.>, δ is the thickness of gas film estimated by LLD's law<cit.>, V is the speed of substrate, the maximum tangential spreading diameter on moving substrate<cit.> and the maximum spreading diameter on motionless substrate<cit.> are respectively D_t max/D_0∼We^1/4Ca^1/6 and D_max/D_0∼We^1/4. And the substrate speed has no effect on normal maximum spreading diameter. Therefore, D_n max∼We^1/4. Then, we could estimate the viscous dissipation with scaling law.
§.§.§ promotion effects
Firstly, one of the promotion effects is initial kinetic energy E_total=1/2ρ( 1/6π D_0^3 ) U^2, where U is the speed which droplet impacts the moving substrate. Next, as the figure 2(c) shown. The movment of solid substrate caused the flow of air beacuse air is the viscous fluid. Hence, the pressure around substrate will be decreased by wind field. So, it has a lift force F_L∼ρ_gU_t^2h on droplet, where ρ_g=1.185kg/m^3 is the density of air(25℃), ρ=997kg/m^3 is the density of water(25℃), U_t∼ 1m/s is wind speed around the substrate, h∼ 10^-3m is the maximum spreading thickness of droplet, D∼ 10^-3m is the initial diameter of droplet. So, we have
ρ DU^2/ρ_gU_t^2h∼ 10^3
So lift force could be ignored when the speed of substrate is little(U∼ 1m/s). The situation of big wind speed will be discussed in section 4.
§.§.§ Completely Bouncing Criteria for droplet on Moving Substrate
Hence, we could conclude that the initial kinetic energy E_total promote to bounce, viscous dissipation E_diss and energy dissipation due to capillary force Q_cap inhibit bouncing.
Considering a imaginary state which droplet bounce exactly. Then, we could get the condition of bouncing using energy conservarion. We have
E_total+mgD/2>E_Diss+Q_cap+mgD/2
So, we can get the completely bouncing criteria for droplet on moving substrate.
Cr=πWe^3/2√(γ/D_0ρ)/2Aν N_A/π D_0^2γ ( p_0-p^⊖/p_0 ) +12 [ ( βWe^1/4Ca^1/6+β'D_0We^1/2Ca^1/2/lc ) ( We^1/4Ca^1/6-1 ) +αWe^1/4 ( We^1/4-1 ) ]
where l_c=( γ/ρ g) ^-1/2 is the Capillary length of water, rho is the density of liquid, γ is the liquid-gas surface tension, α, β, β' are the constants to be determined. The numerator is shown that the effect of initial kinetic energy. The left term of denominator is shown that the effects of capillary force in the gas film. The right term of denominator is shown that the effects of viscous dissipation. The bouncing without splashing condition is
Cr>1
And completely bouncing criteria for droplet on moving substrate Cr_V, which is only due to the speed of substrate.
Cr_V=A/B+( CV^1/6+C'V^1/2) ( DV^1/6-1)
where A,B,C,C',D are indepent to substrate speed V. The droplet will retain on the substrate if Cr_V≤ 1, and bounce on the substrate if Cr_V>1.
In a word, the completely bouncing criteria for droplet on moving substrate is relate to capillary number Ca and Weber number We which both are the initial state parameters of the droplet and moving substrate, as figure 2(e) shown. And we also explain the experimental phenomenons which the final state of droplet will shift with changing substrate speed as figure 2(f) shown. We also could find out the critical speed which is the shift of bouncing and retention as figure 2(f) shown. These results show that droplet might undergo the transformation of "Retention to Bounced" or "Retention to Bounced to Retention". But we didn't consider the dissipation due to the microstructure on the surface of substrate. It's also a complex question
§ THE EFFECT OF WIND FIELD FOR DROPLET
In the discussion of previous section, we find that a lift force due to the wind field couldn't be ignored with high substrate speed. The lift force promote to bounce. We even find out the splashingofdroplet in further experiments. So we design the experiment(more detials are in supplemental material<cit.>) to illustrate the importance of wind field for the final state of droplet as figure 3(a)(b)shown.
We conclude that the final state of droplet is affected by both the initial height of droplet(We) and the speed of wind(Ca) from figure 3(b). And the state of droplet will shift to splashing and partially bouncing with creasing the wind speed and the initial height. The state of droplet will shift to splashing with only creasing the initial height. The state of droplet with small initial height will shift to splashing and partially bouncing if only creasing the wind speed. And state also could shift from retention to completely bouncing/from completely bouncing to retention. We conclude that the spreading factor et. al. could change with the wind speed from figure 3(c). Next, we will research the causes of these phenomenons.
§.§ The Transition Between Retention and Completely Bouncing
Firstly, we assume that (1)all flows could be ragard as the isentropic flows. (2)The boundary conditions between liquid and gas obey Navier boundary conditions, i.e., V⃗_droplet=V⃗_air(3)Wind field is 2D incompressible laminar flow, i.e., ∇·V⃗=0, V_z=const..
We take two circuits C_droplet and C_air around the interface between liquid and gas. The velocity circulations respectively are
Γ_droplet=∮_C_dropletV⃗_droplet·dl⃗=Γ_air=∮_C_airV⃗_air·dl⃗=Γ
These velocity circulations Γ are constant because of assumation 2. And the interaction of wind field on the droplet could be divided into vertical and horizontal direction. The bouncing state mainly depends on the vertical interaction.
L⃗=ρ_aV⃗_∞×Γ⃗_air=-ρ_aΓV⃗_∞×k⃗
It can be seen from assumation 3
∇×L⃗=-ρ_aΓ[ ( k⃗·∇)V⃗_∞+( ∇·k⃗) V⃗_∞-( V⃗_∞·∇) k⃗-( ∇·V⃗_∞) k⃗] =0
So the lift force L⃗ is potential force. We assmue the initial height of droplet is h_0 and the lift force potential of initial position is 0. Then the lift force potential is
V_L( y) =-∫ -ρ_aΓV⃗_∞×k⃗·j⃗dy=-ρ_aΓ∫ V_∞xdy=ρ_aΓ∫_h_0^yV_∞xdy
So the completely bouncing criteria for droplet is
Cr_wind=ρ gπ D_0^3( h_0-0.5D_0) /6/E_Diss+Q_cap+ρ_aΓ∫_h_0^D_0/2V_∞xdy
The bouncing without splashing condition is
Cr_wind>1
The final state of droplt could shift between bouncing and retention beacuse Γ∫_h_0^D_0/2V_∞xdy could bigger/smaller than 0.
§.§ The Transition Between Splashing and without Splashing
Then, considering the transition between splashing and without splashing. The front of droplet maybe generate the liquid finger during the spreading process. Liquid finger will be not only affected by the lubrication force of bottom gas and the attraction of top gas<cit.>F_L=K_lν_gV_t+K_uρ_gV_t^2H_t, but also affected by the wind field as figure 3(e) shown. The effect of extra wind field is shown as lift force F_wind,L( V), where V is the speed of wind speed. Hence, we could introduce the F_wind,L( V) to R&G model.
β^2_Wind=F_L+F_wind,L/2γ
So, the state of droplet maybe have the transition between splashing and without splashing with the wind field.
§ THE BALANCE OF DROPLET ON THE MOVING SUBSTRATE
§.§ The Contact Angle of Droplet
The bottom of droplet will generate a very thin gas film when it impact on a substrate<cit.>. The gas film will be saddle shape stably on moving substrate<cit.>. The pressure of gas film is p∼ 10Pa<cit.>, the gas film thickness is δ∼ 10μ m. We also could approx the Knudsen number of gas film is K_n=λ/h∼ 10^-4. So the gas film could be regard as the continuons flow. If we assume that pressure is a constance on the direction perpendicular to moving substrate, the thickness and pressure of lubrication gas obey the Reylond Equation:
∂/∂ x(h^3/μ∂ p/∂ x)+∂/∂ y(h^3/μ∂ p/∂ y)=6U∂ h/∂ x
where U is the speed of moving substrate, μ is the dynamic viscosity of lubrication gas. This equation shows that the relation of distribution between thickness and pressure. We could give the distribution of pressure with measuring the distribution of thickness.
Then we research the effect of contact angle on moving surface. Considered the difference form gas film on motionless and moving substrate originates from the moving substrate. So, we focus on this element, then analyse this question with minimum energy principle.
We order that the area of gas film is D and O is the center of area D, as shown in the figure1, we research the infinitesimal area D_i with infinitesimal angle and radial length R_i. It is full of air and saturated vapour in area D_i, the pressure of vapour obeys the Clapeyron Equation p=p_0exp(-L_v,m/RT). And the molecular number and pressure of two components(saturated vapour and air) obey that
1=N_air/N+N_H_2O/N
, 1=p_air/p+p_H_2O/p
The mean kinetic energy of two components can be given that e̅_air=5/2kT, e̅_H_2O=3kT, if air is regarded as diatomic molecule. So we could give the internal energy of gas film
E_k=∑_i=1^n∬_D_inhe̅dσ=∑_i=1^n∬_D_i5/2ph+hp_0/2exp(-L_v,m/RT)dσ
Hence, assumed that the front of droplet has a infinitesimal virtual displacement δ R_i. Approximately, the interfacial energy between solid and gas remain unchanged because of the gas film in the solid-gas interface. So the variation in energy of system is δ E_i=(Δ L_icosθ_L γ_LG+Δ L_iγ_SL)δ R_i+δ E_ki. And a stabilized system must obey that
δ E_i/δ R_i=0
A combination of equation <ref> and equation <ref> leads to
cosθ_Li=cosθ_0-1/γ_LG[ 5/2ph+hp_0/2exp( -L_v,m/RT) ]-γ_SG/γ_LG
where p=p(R_icosθ_Δ L_i,R_isinθ_Δ L_i),h=h(R_icosθ_Δ L_i,R_isinθ_Δ L_i), i.e., p and h are the pressure and thickness of soild-liquid interface boundary respectively.
Hence, we can get the mean contact angle with intergrating contact angle cosθ_Li along the boundary.
cosθ_L=cosθ_0-1/Lγ_LG∮_L[ 5/2ph+hp_0/2exp( -L_v,m/RT) ]dl-γ_SG/γ_LG
where θ_0 is the contact angle obeying the Young Equation, L is the circumference of solid-liquid interface boundary, L_v, m is the latent heat of phase transition from liquid to gas phase. T is the temperature of gas film. The element on the right side of equation <ref> is the influence of gas film. The element on the middle of equation <ref> is the influence of moving substrate.
We could conclude that the contact angle on moving substrate is 8^∘ bigger than the one obeyed Young Equation in the room tempurature approximately. So the hydrophobicity will be reinforced on the moving substrate. And the contact time<cit.><cit.>, spreading factor<cit.>, bouncing et.al. will change with the change of hydrophobicity between the droplet and substrate. Then we will elucidate them in detial.
§.§ The analytical solution for Reynolds Equation
Firstly, we could get another equation which describes the gas film on the moving surface from Reynolds transport equation:
∂ h/∂ t+∇·( h𝐮) =0, ∂ h/∂ t=0
where 𝐮 could be seen as the surface speed U𝐢+V𝐣 because of the Navier Boundary Conditions.
Then, the problem is solving the differential equations:
{
h∂^2p/∂ x^2+h∂^2p/∂ y^2+3( ∂ h/∂ x∂ p/∂ x+∂ h/∂ y∂ p/∂ y) =0
U∂ h/∂ x+V∂ h/∂ y=0
.
Then, we order that h( x,y) =h_X( x) h_Y( y), p( x,y) =p_X( x) p_Y( y). We could get equation <ref> through bringing these to equation <ref>.
1/p_X^2d^2p_X/dx^2+1/p_Y^2d^2p_Y/dx^2+3/p_X^2p_Yh_Xdh_X/dxdp_X/dx+3/p_Y^2p_Xh_Ydh_Y/dydp_X/dy=0
Then, finding the derivative of equation <ref> with respect to x and y in turn. We can get
d^2p_X/dx^2-Cdp_X/dx+C^' p_X^2=0
dp_Y/dydh_Y/dy-Cp_Y^2h_Y/3=C^'p_Yh_Y/3
C, C^'are constant. And we could esaily find the solution of equarion <ref>.
h_X=C_h_1exp( -λ/Ux) , h_Y=C_h_2exp( λ/Vy)
bring them to equation <ref>, we can get
∫dp_Y/C_2^' p_Y+C_2p_Y=y
So the p_Y is
p_Y=C_2/-C_2^' +exp( -y+C_2^''/C_2)
Then, we solve the equation <ref> with series method. Considering the series solution p_X=∑_n=0^∞a_nx^n. Then, we can get
a_n+2( n+2) ( n+1) -Ca_n+1( n+1) +2C^'( a_0a_n+a_1+a_n-1+⋯ +a_n/2a_n/2) =0, n is an even.
a_n+2( n+2) ( n+1) -Ca_n+1( n+1) +2C^'( a_0a_n+a_1+a_n-1+⋯ +a_( n-1) /2a_( n+1) /2) =0, n is an odd.
And the radius of convergence R obey that
lim_n→∞[ n+2+2C^'/n+1( a_0R^2+a_1R^3+⋯ +a_n/2R^n/2+2) ]-CR=0, n is an even.
lim_n→∞[ n+2+2C^'/n+1( a_0R^2+a_1R^3+⋯ +a_( n-1) /2R^( n+3) /2) ]-CR=0, n is an odd.
In addition, we can get equation <ref> when R=1.
lim_n→∞[ n+2+2C^' q( n/2+1) /n+1-C] ≤ lim_n→∞[ n+2+2C^'/n+1( a_0R^2+a_1R^3+⋯ +a_n/2R^n/2+2) ]-CR
≤lim_n→∞[ n+2+2C^' q^'( n/2+1) /n+1-C], n is an even.
lim_n→∞[ n+2+2C^' w( ( n+1) /2) /n+1-C] ≤ lim_n→∞[ n+2+2C^'/n+1( a_0R^2+a_1R^3+⋯ +a_( n-1) /2R^( n+3) /2) ]-CR
≤lim_n→∞[ n+2+2C^' w^'(( n+1) /2) /n+1-C], n is an odd.
where q=min{ a_0, a_1,⋯, a_n/2}, q^'=max{ a_0, a_1,⋯, a_n/2} and w=min{ a_0, a_1,⋯, a_( n-1) /2}, w^'=max{ a_0, a_1,⋯, a_( n-1) /2}.
If the equation <ref> is right, the equation <ref> and <ref> would be wrong. So the R is either ∞ or 1<R<∞.
So p( x,y) is
p=( a_0+a_1x+Ca_1-C^' a_0^2/2x^2+⋯) ( C_2/-C_2^' +exp( -y+C_2^''/C_2) ) , 1<x≤∞
So h( x, y) is
h( x, y) =h_X· h_Y=C_hexp( -λ/Ux+ λ/Vy)
§ CONCLUSION
In section 2, we find that how the moving substrate affect the hydrophobicity of droplet and we discuss the analytical solution for Reynolds Equation. Therefore, we would analytically get the contact angle on moving substrate. But we must have some boundary conditions such as h( x, y)|_droplet boundary=H( x, y) , p( x, y)|_droplet boundary=P( x, y) and so on to get the whole solution. In section 3, we find out some promotion and inhibition effects for bouncing question. Finally, we get a completely bouncing criteria Cr for droplet on moving substrate. Some phenomenons could be pridicted by using this criteria. In section 4, we research the effect of extra wind field for droplet. we find that extra wind field could change the final states of droplet. In addition, we get a completely bouncing criteria Cr_Wind for droplet on moving substrate with extra wind by the way that introduce the lift force potential. We also get the splashing criteria β^2_Wind using the R&G Model.
But we don't analytically get a criteria because of the complex viscous dissipation. We just use a simple model to calculate it. We also don't consider the energy dissipation due to the roughness of substrate. It is so important that cound't be ignored in some substrate with big roughness.
Thanks for Shangqian Sun, Hongwang Lu, Jingcheng Hao and Ying Ma 's support for this work.
|
http://arxiv.org/abs/2307.04506v2 | 20230710115703 | Distributed Decisions on Optimal Load Balancing in Loss Networks | [
"Qiong Liu",
"Chehao Wang",
"Ce Zheng"
] | eess.SP | [
"eess.SP"
] |
Distributed Decisions on Optimal Load Balancing
in Loss Networks
Qiong Liu1, Chenhao Wang2, Ce Zheng1
1Télécom Paris, Institut Polytechnique de Paris, France
2Beijing Normal University, China
Email: [email protected], [email protected], [email protected]
==========================================================================================================================================================================================================================
When multiple users share a common link in direct transmission, packet loss and link congestion may occur due to the simultaneous arrival of traffics at the source node. To tackle this problem, users may resort to an indirect path: the packet flows are first relayed through a sidelink to another source node, then transmitted to the destination. This behavior brings the problems of packet routing or load balancing: (1) how to maximize the total traffic in a collaborative way; (2) how self-interested users choose routing strategies to minimize their individual packet loss independently.
In this work, we propose a generalized mathematical framework to tackle the packet and load balancing issue in loss networks. In centralized scenarios with a planner, we provide a polynomial-time algorithm to compute the system optimum point where the total traffic rate is maximized. Conversely, in decentralized settings with autonomous users making distributed decisions, the system converges to an equilibrium where no user can reduce their loss probability through unilateral deviation. We thereby provide a full characterization of Nash equilibrium and examine the efficiency loss stemming from selfish behaviors, both theoretically and empirically. In general, the performance degradation caused by selfish behaviors is not catastrophic; however, this gap is not monotonic and can have extreme values in certain specific scenarios.
load balancing, Nash equilibria, price of anarchy, network congestion, sidelink
§ INTRODUCTION
Since the seminal work of Erlang <cit.>, loss networks have played a crucial role in analyzing and optimizing stochastic systems involving simultaneous resource utilization, and non-backlogging workloads (for an extensive overview, see <cit.>). Meanwhile, in the post-5G era, cloud-enabled networks have emerged as a dominant architecture, where multiple servers collect data from users and relay it to a central hub for final processing. To guarantee network efficacy, that is no server is either overburdened or underutilized, load balancing strategies are well studied, e.g., <cit.>. In this context, loss networks provide valuable mathematical frameworks for comprehending and enhancing load distribution within cloud-enabled networks.
Early load balancing research for cloud-enabled networks focused on centralized scenarios, where a centralized planner scheduled workloads to optimize aspects like performance-energy tradeoffs <cit.> and algorithmic considerations <cit.>. However, due to the stringent latency requirement for real-time decisions and the increasing signaling overhead caused by the large-scale deployment of servers and massive users, distributed decisions become a better solution. In this context, the complexity of the problem increases due to the non-cooperative and competitive behaviors among users within the system.
To address the challenges of load balancing in a distributed way, game theory provides a mathematical framework that describes and analyzes scenarios with interactive decisions <cit.>. Till now, some studies have demonstrated the efficacy of game-theoretic models in addressing load balancing problems. For instance, Mondal et al. <cit.> developed a game-theoretic model for load balancing among competitive cloudlets, while Yi et al. <cit.> investigated a similar problem, incorporating additional considerations of queue-aware strategies. In <cit.>, symmetric loss models where each source has an equal number of users are considered. However, previous studies mostly focused on limited cases of identical user strategies, which may not reflect real-world scenarios, i.e., different users may have different objectives and preferences. Therefore, further research is needed to develop game-theoretic models that can address the challenges of load balancing in a more general and realistic manner.
In this paper, we employ game theory to address load balancing in both distributed and centralized environments, where users have non-identical strategies and the number of users is not evenly distributed. Specifically, we consider the load balancing in a cloud-enabled network consisting of m source nodes (servers) {s_1,…,s_m} and one destination node (central hub) d. Each source s_i has n_i users seeking service, and the traffic originating from each user is assumed to follow an independent Poisson point process with an identical rate. The nodes in the network are connected by two types of communication links, namely sidelinks that connect two sources, and direct links that connect a source and destination. The sidelink has a random identical independent distribution (i.i.d) loss with a fixed probability q, and the direct link has a congestion loss that depends on the arrival rate and service rate of each server.
The user cannot split its traffic, and has to determine how to route all of its traffic from the source node arrived at to the destination node. There are two approaches for the traffic transmission: a direct path (DP) in which the packet goes directly from the source arrived at to the destination, and an indirect path (IP) in which the packet is first relayed to another source node and then takes the direct link from that node to the destination.
We treat packet loss probability as the performance metric in load balancing, instead of additive costs like delay or fees in classical routing games <cit.>, resulting in a non-additive and non-convex optimization process. Each user aims to minimize its own loss probability and engage in a game by strategically selecting its own path. In the end, no user can reduce its loss probability by unilateral deviation and reaches the state of Nash Equilibrium (NE).
§.§ Our Contributions
Our work contributes to the load balancing game in the following aspects: First, we prove two lemmas related to the optimal solution when a centralized planner exists. Based on these lemmas, a low-complexity algorithm that maximizes the total traffic is proposed.
Second, we study the decentralized environment where decisions are made by autonomous and self-interested users. The sufficient and necessary conditions on NE are derived, which depend on the number of users on direct path and each indirect path.
Moreover, since a NE may be suboptimal, we use the price of anarchy (PoA) <cit.> to measure the gap between the NE led by users' selfish behaviors and the system optimum achieved by the centralized planner.
The rest of the paper is structured as follows. The formal model and notations are presented in Section <ref>. In Section <ref>, we provide details to compute the optimal solution that maximizes the total traffic when a centralized planner exists. In Section <ref>, we study the NE in the decentralized decision-making scenarios, and analyzed the efficiency loss stemming from selfish behaviors. In Section <ref>, a fine-grained analysis is performed on the existence of NE in various network configurations for a specific scenario involving two source nodes.
Numerical results are presented and discussed in Section <ref>. Finally, Section <ref> concludes the paper and outlines some future work.
§.§ Other related works
Routing games.
As a special class of congestion games, routing games in a network are problems of routing traffic to achieve
the best possible network performance, and have been studied within various contexts
and within various communities, for example, the mathematics community <cit.>, the telecommunications <cit.>, and theoretical computer science <cit.>. The above references have all in common a cost framework which is additive over links,
such as delays or tolls, and is flow conserving (the amount entering a node equals the amount leaving it). Routing games with non-additive cost in loss networks are studied in <cit.>.
Braess-like paradox in distributed systems.
The Braess-like paradox is said to occur in a network system with distributed behaviors if adding an extra link or adding communication capacity to the system leads to a worse system performance. It widely exists in transportation networks and queuing networks. Bean et al. <cit.> show that it can occur in loss networks.
Kameda et al. <cit.> consider a model similar to ours in that a job (packet) can be processed directly or indirectly; however, they do not consider the loss probability. They identify a Braess-like paradox in which adding capacity to the channel may degrade the system performance on the response time.
Kameda and Pourtallier <cit.> characterize conditions under which such paradoxical behavior occurs, and give examples in which the degradation of performance may increase without bound.
§ MODEL AND PRELIMINARIES
We abstractly model our problem using a graph. Consider a network with m source nodes S={s_1,…,s_m} and one destination node d. For each source node s_i∈ S, let N_i be the set of users arriving at s_i, and n_i=|N_i| be the number of such users. Without loss of generality, we assume n_1 > n_2 >…>n_m. Denote [m]={1,…,m}. There is a total of n=∑_i∈[m]n_i users in the system, who are self-interested players in the game. We say players and users interchangeably throughout this paper. Each user is identified with a flow (or traffic) of packets, which originates from the user and is assumed to form an independent Poisson process with an identical rate ϕ. See Fig. <ref> for illustration.
Each user controls its route that all its packets should follow. For a user associated with s_i∈ S, there are only two types of routes to ship these packets to the destination d: either a direct path (DP) (s_i,d), or an indirect two-hop path (IP) (s_i,s_j,d) for some s_j≠ s_i, in which the packet is first sent to another source s_j by the side link (s_i,s_j), and then passes through the direct link (s_j,d).
Strategies. For every source s_i, each user k∈ N_i decides a one-shot strategy 𝐩_k^(i)=( p_k1^(i),…,p_km^(i))^T∈ [0,1]^m with ∑_j∈[m]p_kj^(i)=1, where p_ki^(i) is the probability of routing all packets through DP, and p_kj^(i) (j∈[m],j≠ i) is the probability of routing all packets through IP (s_i,s_j,d).
When no confusion arises, we simply write the strategy 𝐩_k^(i) as 𝐩_k.
We focus on pure strategies in this paper: a strategy 𝐩_k is pure if 𝐩_k_∞=1, i.e., user k deterministically selects a route with probability 1 (for example, 𝐩_k=(0,1,0,…,0)^T).
Let 𝐩=(𝐩_1,…,𝐩_n) be the strategy profile of all users.
Loss probability and loss rate.
There are two types of losses: (1)
Losses on side links. We assume that a packet originating from node s_i and relayed to node s_j is lost with a fixed probability q for every side link (s_i,s_j), independently of any other loss. Denote by q̅=1-q the probability that a packet is successfully relayed. (2) Congestion losses on direct links. We assume that there is no buffer to restore the backlogged packets, so a packet will be lost when it enters the direct path which is occupied for the transmission of another packet. The transmission time of a packet on a direct link (s_i,d) is a random variable σ following a distribution 𝒳, which is assumed to be an identically independent distribution (i.i.d) for all packets.
Given strategy profile 𝐩, user k∈ N_i continuously sends packets that follow an independent Poisson process with rate p_ki^(i)·ϕ to DP (s_i,d), and an independent Poisson process of packets with rate p_kj^(i)·ϕ to IP (s_i,s_j,d), for any s_j≠ s_i.
Since there is a random
loss on the side link (s_i,s_j), the flow of packets from user k∈ N_i that arrive at the node s_j is also a Poisson process with rate q̅p_kj^(i)ϕ.
Thus, for each source s_i∈ S, the flow over the link (s_i,d) is Poisson distributed with a
traffic rate T_i(𝐩) given by
T_i(𝐩)=∑_k∈ N_ip_ki^(i)ϕ + ∑_j∈[m]\{i}∑_k∈ N_j p_ki^(j)q̅ϕ.
When no confusion arises, we simply write T_i(𝐩) as T_i.
The probability of no congestion loss on the direct link (s_i,d) equals the probability that there is no arrival during a transmission time σ, which is given by
*𝔼_σ∼𝒳e^-T_iσ.
As usual practice, assume 𝒳 is an exponential distribution with a rate parameter μ (service rate) and mean 1/μ. Thus the probability of no congestion loss on (s_i,d) is
*𝔼_σ∼𝒳e^-T_iσ=∫_0^+∞μ e^-μσ e^-T_iσdσ=μ/T_i+μ,
and the loss probability on link (s_i,d) is T_i/T_i+μ.
Given the strategy profile 𝐩, for s_i∈ S and k∈ N_i, the loss rate of user k is defined as
LR_k(𝐩) =[ p_ki^(i)T_i/T_i +μ+( 1 - p_ki^(i)) q +
( 1 - q) ∑_j∈[m]\{i} p_kj^(i)T_j/T_j +μ] ϕ,
and the loss probability of user k is LR_k(𝐩)/ϕ.
Total traffic. Regarding the system efficiency, we measure it by the total traffic rate arriving at the destination d. Given the strategy profile 𝐩,
the total traffic rate TR(𝐩) of the system can be derived in two ways. The first expression is derived as the summation of successful transmission rates on direct links:
TR(𝐩)=∑_i∈ [m]T_i·μ/T_i+μ
=μ[ m-∑_i∈ [m]μ/∑_k∈ N_i p_ki^(i)ϕ +∑_ j∈[m] \{i}∑_k∈ N_jp_ki^(j)q̅ϕ+μ]
where T_i is the traffic rate over link (s_i,d), and μ/T_i+μ is the probability of no congestion loss on (s_i,d).
The second expression is from users' perspective:
TR(𝐩) :=∑_i∈ [m]∑_k∈ N_i(ϕ-LR_k),
where ϕ-LR_k(𝐩) is the traffic rate of user k∈ N_i that successfully arrive at d. It is not hard to see that (<ref>) and (<ref>) are equivalent.
Nash equilibria. A Nash equilibrium (NE) is a strategy profile where no player can decrease its loss probability by unilaterally deviating to any other strategy. Formally, we give a definition.
A strategy profile 𝐩 is a Nash equilibrium, if for any source s_i∈ S and any player k∈ N_i, we have
LR_k(𝐩_k,𝐩_-k)≤ LR_k(𝐩_k',𝐩_-k),
where 𝐩_k' can be any feasible strategy of player k, and 𝐩_-k is the strategy profile of all other players.
We measure the efficiency of NEs by the price of anarchy (PoA) <cit.>, which is defined as the ratio between social efficiencies in an optimal solution and in the worst NE.
Formally, given an instance Γ of this game, we define
PoA(Γ)=TR(opt)/min_𝐩∈ℕ𝔼TR(𝐩).
where opt is an optimal solution of Γ, and ℕ𝔼 is the set of all NEs. The PoA of the whole game is defined as the maximum over all instances, that is, PoA=max_ΓPoA(Γ).
§ CENTRALIZED ANALYSIS
The main technical results of the paper are presented now. We show how to compute an optimal solution that maximizes the total traffic.
Note that the total traffic rate depends on the number of users working on each source by DP or IP, but not the users' identity.
Given a strategy profile 𝐩, let u_i=|{k∈ N_i | p_ki^(i)=1}| be the number of users working with DP (s_i,d), and let v_i=|{k∈ N_j, j ∈ [m] \i| p_ki^(j)=1}| be the users working with IP through link (s_i,d). Define y_i=u_i+v_i as the number of users who choose source s_i (including both DP and IP).
In any optimal solution, for any source s_i, either u_i=n_i or v_i=0 or both hold.
Let 𝐩 be an optimal solution. Suppose for contradiction that u_i<n_i,v_i>0 for some source s_i. Then there exists a user (say, k) in N_i who chooses IP (say, (s_i,s_i',d) for some i'≠ i). Also, since v_i>0, there exist a source s_j≠ s_i and a user l∈ N_j who chooses IP (s_j,s_i,d). The total traffic rate is TR(𝐩)=μ T_i/T_i+μ+∑_w∈ [m]\{i}(μ T_w/T_w+μ).
Now we show that the total traffic rate can be improved by revising 𝐩. Let user k∈ N_i choose DP, and let user l∈ N_j choose IP (s_j,s_i',d). Fixing all others' strategies, denote the new strategy profile by 𝐩', and define u_i',v_i' accordingly. Note that u_i'=u_i+1,v_i'=v_i-1, and T_w(𝐩')=T_w(𝐩) for all source s_w≠ s_i. Since q>0, we have
T_i(𝐩')=(u_i+1)ϕ+(v_i-1)q̅ϕ> u_iϕ+v_iq̅ϕ=T_i(𝐩).
So TR(𝐩')>TR(𝐩), contradicting to the optimality.
Lemma <ref> indicates that if a source (say s_i) provides service to the users of other sources, then all users of s_i choose DP.
In any optimal solution, there must exist ĩ∈[m], such that v_l=0 for all l≤ĩ, and u_j=n_j for all j>ĩ.
Given an optimal solution 𝐩, suppose for contradiction that there exist i,j∈[m] (i<j) such that v_i>0 and u_j<n_j. By Lemma <ref>, we have u_i=n_i and v_j=0. There exists a source s_i' and a user k∈ N_i' selecting IP (s_i',s_i,d). There exists a source s_j' (j'≠ j) and a user k'∈ N_j selecting IP (s_j,s_j',d). Note that when i'=j and j'=i, users k and k' may coincide. The total traffic rate is
TR(𝐩) =μ T_i/T_i+μ+μ T_j/T_j+μ+∑_w∈ [m]\{i,j}(μ T_w/T_w+μ)
=μ(2-1/T_i+μ-1/T_j+μ)+∑_w∈ [m]\{i,j}(μ T_w/T_w+μ).
Now we show that the total traffic rate can be improved by revising 𝐩. Let user k choose IP (s_i',s_j',d) if i'≠ j' and choose DP (s_i',d) if i'=j'. Let user k'∈ N_j choose DP. Fixing all others’ strategies, denote the new strategy profile by 𝐩', and define u_i',v_i' accordingly. Note that v_i' = v_i-1, u_j' = u_j+1, and T_w(𝐩')=T_w(𝐩) for all other sources s_w ≠ s_i,s_j. Since i < j, it follows that n_i ≥ n_j > u_j. Therefore, we have
1/T_i+μ+1/T_j+μ=1/n_iϕ+v_iq̅ϕ+μ+1/u_jϕ+μ
> 1/n_iϕ+v_i'q̅ϕ+μ+1/u_j'ϕ+μ
=1/T_i'+μ+1/T_j'+μ,
which indicates that TR(𝐩)<TR(𝐩'), a contradiction.
Lemma <ref> shows that there exists a threshold ĩ:
1) if i>ĩ, all users from s_i chose DP;
2) if i≤ĩ, portion users chose DP, and portion users chose IP.
Now we are ready to present Algorithm <ref>. The main idea is searching for ĩ in Lemma <ref>. For each candidate of ĩ, let B be the number of users selecting IP, all of whom come from L={s_l | l≤ĩ}, and go to R={s_j | j>ĩ}. For every possible value of B, we compute the best possible way for extracting the B users from L and distributing them over R.
In Algorithm <ref>, step (a) is to make T_l (and thus no congestion probability μ/T_l+μ) as equal as possible for l∈[ĩ]. This can be realized by initializing u_l=n_l, and then removing players one by one from the highest u_l and updating until B players have been removed. The goal of step (b) is to make T_j (and thus μ/T_j+μ) as even as possible. This can be realized by initializing v_j=0, then adding users one by one to v_j'=min_j>ĩn_jϕ+v_jq̅ϕ+μ and updating, until B players have been added. These two steps guarantee that the B loads are distributed in an optimal way to maximize the traffic rate.
Though the output of the algorithm is (u_i^*,v_i^*)_i∈ N, we can easily extend it to a corresponding strategy profile because the situation for each source s_i has been determined. Next we prove that its optimality.
Algorithm <ref> returns an optimal solution for maximizing the total traffic, and runs in O(mn^2) time.
In the first loop, we traverse all indexes in [m] to find the ĩ in Lemma <ref>. In the second loop, we traverse all possible numbers of users who select IP, and given any such a number B, we extract the B users from {s_l | l≤ĩ} and distribute them over {s_j | j>ĩ} in an optimal way to maximize the traffic rate. So all possible optimal solutions have been searched by the algorithm, giving the optimality.
For the time complexity, we have m iterations in the first loop, at most n iterations in the second loop, and the time for each iteration is O(n).
Intuitively, when the transmission loss probability is sufficiently large, all packets should go through DP; when there is no transmission loss, the load of packets should be distributed evenly over all sources. We verify the intuition as follows.
If q=1, the unique optimal solution is that all users choose DP (i.e., u_i=n_i, v_i=0, ∀ i∈ M). If q=0, a strategy profile 𝐩 is optimal if and only if |y_i-y_j|≤ 1 for all i,j∈ [m].
If q=1, TR=∑_i∈[m]μ u_iϕ/u_iϕ+μ is increasing with respect to every u_i. By the monotonicity, the optimum is achieved when u_i=n_i. If q=0, suppose for contradiction that there exist i,j∈ [m] in an optimal solution 𝐩 such that y_i-y_j≥ 2. The total traffic rate is TR=μ(m-∑_k∈ [m]μ/y_kϕ+μ). Consider a new strategy profile 𝐩' with y_i'=y_i-1,y_j'=y_j+1, i.e., a user who chooses source s_i deviates to s_j. Then the total traffic rate becomes TR'=μ(m-∑_k∈[m]\{i,j}μ/y_kϕ+μ-μ/y_i'ϕ+μ-μ/y_j'ϕ+μ)>TR, a contradiction.
§ DECENTRALIZED ANALYSIS
In this section, we study the Nash equilibria in the decentralized decision-making scenario where each user makes a decision on the choice of DP or IP.
§.§ Characterization of NEs
A NE should satisfy that: for a user selecting DP, its loss rate will not decrease if it deviates to any IP; for a user selecting IP, its loss rate will not decrease if it deviates to DP or another IP.
We formalize it as the following characterization.
Given an arbitrary strategy profile 𝐩 with (u_i,v_i)_i∈ [m], let i^*∈min_i∈[m]{u_i+v_iq̅}, and let x_ij∈{0,1} be an indicator where x_ij=1 if there exists at least one user selecting IP (s_i,s_j,d). Then, 𝐩 is a NE, if and only if the following conditions are satisfied:
(i) for all i∈[m] with u_i>0,
we have
q̅(u_i+v_iq̅)≤ u_i^*+v_i^*q̅+q̅+qμ/ϕ;
(ii) for all i,l∈[m] with x_il=1, we have
u_l + v_lq̅≤min{q̅( u_i + 1 + v_iq̅) -qμ/ϕ, u_i^*+ v_i^*q̅+q̅}
Proof sketch: Suppose 𝐩 is a NE. Consider any source s_i∈ S and user k∈ N_i.
Case 1. In its NE, user k selects DP in 𝐩 (denoted as 𝐩_k^(i) where p_ki^(i)=1). If it deviates to IP (s_i,s_j,d) where j ≠ i (denoted as 𝐩'_k^(i) where p_kj^i=1), by Definition <ref>, we have LR_k(𝐩_k^(i),𝐩_-k)≤ LR_k(𝐩'_k^(i),𝐩_-k), equivalent to (<ref>).
Case 2. In its NE, user k selects IP (s_i,s_j,d) in 𝐩. If it deviates to DP, it leads to the first part of (<ref>). If it deviates to another IP, it leads to the second part of (<ref>).
Suppose 𝐩 is a NE. Consider an arbitrary source s_i∈ S and arbitrary user k∈ N_i.
Case 1. User k selects DP in 𝐩 (denoted as 𝐩_k^(i) where p_ki^(i)=1). If it deviates to IP (s_i,s_j,d) where j ≠ i (denoted as 𝐩'_k^(i) where p_kj^i=1), by Definition <ref>, we have LR_k(𝐩_k^(i),𝐩_-k)≤ LR_k(𝐩'_k^(i),𝐩_-k). It is equivalent to
1-μ/u_iϕ+v_iq̅ϕ+μ≤ q+q̅·(1-μ/u_jϕ+(v_j+1)q̅ϕ+μ)
⇔ q̅/u_jϕ+(v_j+1)q̅ϕ+μ≤1/u_iϕ+v_iq̅ϕ+μ
⇔ q̅(u_i+v_iq̅)-qμ/ϕ≤ u_j+(v_j+1)q̅,
The above inequality should hold for all j≠ i, and thus is equivalent to Equation (<ref>).
Case 2. User k selects IP (s_i,s_l,d) in 𝐩. If it deviates to DP (s_i,d), by Definition <ref>, we should have LR_k(𝐩_k^(i),𝐩_-k)≤ LR_k(𝐩'_k^(i),𝐩_-k). It is equivalent to
q+q̅·(1-μ/u_lϕ+v_lq̅ϕ+μ)≤ 1-μ/(u_i+1)ϕ+v_iq̅ϕ+μ
⇔ 1/(u_i+1)ϕ+v_iq̅ϕ+μ≤q̅/u_lϕ+v_lq̅ϕ+μ
⇔ u_l+v_lq̅≤q̅(u_i+1+v_iq̅)-qμ/ϕ.
Moreover, a NE must guarantee that user k will not deviate to another IP (s_i,s_j,0), and thus we should have
1-μ/u_lϕ+v_lq̅ϕ+μ≤ 1-μ/u_jϕ+(v_j+1)q̅ϕ+μ
⇔ u_l+v_lq̅≤ u_j+(v_j+1)q̅.
Note that the above inequality should hold for all j≠ i,l. Therefore, we obtain Equation (<ref>).
§.§ Price of Anarchy
We investigate the price of anarchy in this section, which measures the efficiency of NE. We give an upper bound on the optimal total traffic rate, and a lower bound on the total traffic rate of any NE.
In an optimal solution 𝐩,
the total traffic rate is TR_(𝐩)≤μ m (1-μ/n_1ϕ+μ).
Let i be the index stated in Lemma <ref>. It suffices to show with the proof by contradiction that in the optimal solution 𝐩, u_i+v_iq̅≤ n_i. First, for i=1, we have v_1=0, and thus it satisfies u_1+v_1q̅=u_1≤ n_1. For any i>1, suppose for contradiction that u_i+v_iq̅>n_i. Then v_i>0, and there exists a source s_j and a user k∈ N_j that chooses the IP (s_j,s_i,d), i.e., u_j<n_j. By Lemma <ref>, it must be v_j=0, and thus T_j=u_jϕ. Denote the strategy as 𝐩 with p_ki^(j)=1.
The total traffic rate is
TR(𝐩)=μ( m - μ/T_j + μ - μ/T_i + μ - ∑_w∈[m]\{j}μ/T_w + μ)
We show that the total traffic rate can be improved with user k ∈ N_j deviating from IP (s_j, s_i, d) to DP (s_j,d).
Fixing the strategies of all others, denote by 𝐩' the new strategy profile, and define (u'_w,v'_w,T'_w)_w∈[m] accordingly. Note that u_j'=u_j+1, v_i'=v_i-1, u_i'=u_i, and T_w'=T_w for any w∈[m]\{j}. Since u_i+v_iq̅>n_1≥ n_j≥ u_j, we have
1/T_j+μ+1/T_i+μ=1/u_jϕ+μ+1/u_iϕ+v_iq̅ϕ+μ
> 1/u_j'ϕ+μ+1/u_i'ϕ+v_i'q̅ϕ+μ= 1/T_j'+μ+1/T_i'+μ.
It indicates that TR(𝐩')>TR(𝐩) is a contradiction. Consequently, u_i+v_iq̅≤ n_i ≤n_1. According (<ref>), we have TR_(𝐩)≤μ m(1-μ/n_1ϕ+μ).
Let z=min{n_m,n/4m-q̅-qμ/ϕ}. For every NE 𝐩, the total traffic rate satisfies TR(𝐩)≥μ(m-mμ/zϕ+μ).
Let i^*=min_i∈[m]{u_i+v_iq̅}. Since TR(𝐩)≥μ m(1-μ/(u_i^*+v_i^*q̅)ϕ+μ), it suffices to prove that u_i^*+v_i^*q̅≥ z. If u_i^*+v_i^*q̅≥ n_m, it is done. We only need to consider the case when u_i^*+v_i^*q̅< n_m≤ n_i^*. There exists some users in N_i^* selecting IP. By Equation (<ref>), we have
u_i^*+v_i^*q̅≤q̅(u_i^*+1+v_i^*q̅)-qμ/ϕ.
By Theorem <ref>, for each i∈[m], if u_i>0, then q̅(u_i+v_iq̅)≤ u_i^*+v_i^*q̅+q̅+qμ/ϕ; if v_i>0, then u_i+v_iq̅≤ u_i^*+v_i^*q̅+q̅. In both cases, we obtain u_i+v_i/2≤ 2(u_i^*+v_i^*q̅+q̅+qμ/ϕ).
Summing up over all i∈[m], we have
n/2≤∑_i∈[m](u_i+v_i/2)≤ 2m (u_i^*+v_i^*q̅+q̅+qμ/ϕ),
which implies that u_i^*+v_i^*q̅≥n/4m-q̅-qμ/ϕ.
For any instance with m sources, the price of anarchy is PoA≤ 1+n_1μ/n_1zϕ+zμ, where z=min{n_m,n/4m-q̅-qμ/ϕ}.
Combining the upper bound in Lemma <ref> and the lower bound on TR(𝐩) for any NE 𝐩 in Lemma <ref>, it follows
PoA ≤m - mμ/n_1ϕ + μ/m - mμ/zϕ + μ = n_1(zϕ+μ)/z(n_1ϕ+μ) = 1 + n_1μ/n_1zϕ + zμ.
§ A PARTICULAR CASE: TWO SOURCES
In this section, we focus on the special case of m=2. That is, there are only two sources s_1 and s_2. Assume w.l.o.g. that n_1≥ n_2. For each user k ∈ N_i, there is only one IP. Accordingly, its strategy becomes 𝐩_k^(i) = (p_k1^(i),p_k2^(i)), i = 1, 2.
And we have
n_1 = u_1 + v_2;
n_2 = u_2 + v_1.
The traffic rate T_i(𝐩) in (<ref>) is rephrased as
T_i(𝐩) = ∑_k∈ N_ip_ki^(i)ϕ + ∑_k∈ N_j, j≠ i p_ki^(j)q̅ϕ = u_i ϕ + v_i q̅ϕ.
Given strategy profile 𝐩, the set N is further partitioned into 4 subsets (V_1,V_2,V_3,V_4) where V_1={k∈ N_1 | 𝐩_k^(1) =(1, 0)}, V_2={k∈ N_1 | 𝐩_k^(1) = (0,1)}, V_3={k∈ N_2 | 𝐩_k^(2)=(0,1)} and V_4={k∈ N_2 | 𝐩_k^(2)=(1,0)}.
Clearly, users in V_1 and V_3 choose DP, and users in V_2 and V_4 choose IP.
Suppose 𝐩 is a NE. We study the deviation of users in V_1,V_2,V_3,V_4, respectively.
For user k∈ V_1, the strategy is 𝐩_k^(1) = (p_k1^(1), p_k2^(1))=(1, 0), and the loss rate in (<ref>) is
LR_k( 𝐩 )
=ϕ T_1(𝐩)/T_1(𝐩)+μ = [1 - μ/ϕ/u_1 + v_1q̅ + μ/ϕ] ϕ.
When user k∈ V_1 deviates to IP, the strategy profile becomes 𝐩' = ( 𝐩'_k^(1), 𝐩_-k) where 𝐩_k'^(1) =(0, 1).
The loss rate of user k becomes
LR_i(𝐩') = qϕ+q̅ϕT_2(𝐩')/T_2(𝐩') +μ=[1 -q̅μ/ϕ/u_2 + (v_2 + 1)q̅+μ/ϕ] ϕ.
Since 𝐩 is NE, k has no incentive to deviate, and thus LP_k(𝐩)≤ LR_k(𝐩'),
which is equivalent to
t_1(u_2) : = qμ/ϕ+ u_2(1 +q̅^2) + (n_1 + 1)q̅-n_2q̅^2/2q̅≥ u_1,
where t_1(u_2) is a function with respect to variable u_2.
For user k∈ V_2 with strategy 𝐩_k^(1) =(0, 1), the loss rate is
LR_k(𝐩) = qϕ + q̅ϕT_2(𝐩)/T_2(𝐩) + μ = [1 - q̅μ/ϕ/u_2 + v_2q̅ + μ/ϕ] ϕ.
When user k∈ V_2 deviates to DP, the strategy profile becomes 𝐩' = ( 𝐩'_k^(1), 𝐩_-k) where 𝐩_k'^(1)=(1, 0).
The loss rate of k becomes
LR_k(𝐩') = ϕ T_1(𝐩')/T_1(𝐩') + μ = [ 1 - μ/ϕ/u_1+1+v_1q̅+μ/ϕ]ϕ.
Since 𝐩 is NE, we have LR_k(𝐩)≤ LR_k(𝐩'), that is,
u_1≥qμ/ϕ+ u_2(1 +q̅^2) + (n_1 - 1)q̅- n_2q̅^2/2q̅ = t_1(u_2) - 1.
Symmetrically, for each user k∈ V_3 and k∈ V_4, since 𝐩 is NE, we have
t_2(u_1)-1 ≤ u_2 ≤ t_2(u_1),
where
t_2(u_1) := qμ/ϕ + u_1(1+q̅^2)+(n_2+1)q̅ - n_1q̅^2/2q̅.
Note that Eq. (<ref>) - (<ref>) are the sufficient and necessary conditions for an arbitrary strategy 𝐩 to achieve NE.
Now we are ready to give a characterization of NEs.
Let 𝐩 be an arbitrary strategy profile for the game with two sources. Let u_1 and u_2 be the number of users in N_1 and N_2 who choose DP under 𝐩, respectively. We have
[1] when (a) u_1=n_1,u_2<n_2, or (b) u_1=0,u_2>0, 𝐩 cannot be a NE;
[2] when u_1∈[0,n_1),u_2∈[0,n_2), 𝐩 is NE if and only if u_1≥ t_1(u_2)-1 and u_2≥ t_2(u_1)-1;
[3] when u_1∈(0,n_1),u_2=n_2, 𝐩 is NE if and only if u_1∈[t_1(u_2)-1,t_1(u_2)];
[4] when u_1=n_1,u_2=n_2, 𝐩 is NE if and only if n_1q̅≤ qμ/ϕ+n_2+q̅.
Given 𝐩, let (V_1,V_2,V_3,V_4) be a partition of N as defined above. We discuss the four cases.
Case 1. When (a) u_1=n_1 and u_2<n_2, V_4 is nonempty. If 𝐩 is a NE, it must satisfy t_2(u_1)-1≤ u_2.
However, because q̅u_2≤ n_1,q̅u_2≤ (n_2-1)q̅ and qμ/ϕ>0, it cannot hold.
When (b) u_1=0 and u_2>0, V_2 is nonempty. If 𝐩 is a NE, it must satisfy Eq. (<ref>), that is, u_1≥ t_1(u_2)-1. It follows that 0=2q̅u_1≥ qμ/ϕ+u_2(1+q̅^2)+(n_1-1)q̅-n_2q̅^2≥ qμ/ϕ+1+(n_1-1)q̅-(n_2-1)q̅^2≥ qμ/ϕ+1>0, a contradiction.
Case 2. When u_1∈[0,n_1),u_2∈[0,n_2), V_2 and V_4 are nonempty. It is easy to see that 𝐩 is NE if and only if u_1≥ t_1(u_2)-1 and u_2≥ t_2(u_1)-1 are satisfied simultaneously.
Case 3. When u_2=n_2,u_1∈(0,n_1), V_1,V_2,V_3 are nonempty, and V_4 is empty. 𝐩 is NE if and only if t_1(u_2)-1≤ u_1≤ t_1(u_2) and u_2≤ t_2(u_1) hold simultaneously. Moreover, note that u_2≤ t_2(u_1) is implied by u_1≥ t_1(u_2)-1. Therefore, the sufficient and necessary condition for NE is u_1∈[t_1(u_2)-1,t_1(u_2)].
Case 4. When u_1=n_1,u_2=n_2, V_1,V_3 are nonempty, and V_2,V_4 are empty. 𝐩 is NE if and only if u_1≤ t_1(u_2) and u_2≤ t_2(u_1). It is easy to see that, it is equivalent to n_1q̅≤ qμ/ϕ+n_2+q̅.
Note that every situation of u_1,u_2 is included in the above four cases. So we complete a characterization.
Case 4 can be intuitively explained by considering the sidelink loss probability q over link (s_1,s_2). If q is sufficiently high, no user would prefer the indirect path, and selecting the direct path would be a NE for all users. Conversely, when there is no transmission loss over sidelink (s_1,s_2) (i.e., q=0), every user would prefer to use the source with fewer users. Therefore, the profile of all users selecting DP is a NE only if the user distribution between the two sources is as even as possible, with n_1≤ n_2+1. Based on Theorem <ref>, we give some interesting conclusions.
If a strategy profile with u_1=n_1,u_2=n_2 is optimal, then it is also a NE.
A strategy profile with u_1=0,u_2=0 is a NE, if and only if (a) n_1=n_2+1,q̅=1, or (b) n_1=n_2,n_1(1-q̅^2)≤q̅-qμ/ϕ.
Note that u_1≥ t_1(u_2)-1 and u_2≥ t_2(u_1)-1 cannot hold simultaneously when q>2/n, and u_1≥ t_1(n_2)-1 cannot hold when n_1q̅< qμ/ϕ+n_2+q̅.
When n_1q̅< qμ/ϕ+n_2+q̅ and q>2/n, the unique NE is that all users choose DP, i.e., u_1=n_1,u_2=n_2.
We end this section by proving the existence of NE.
For any game instance with two sources, there exists a NE with u_1>0 and u_2=n_2.
By Theorem <ref> (4), if n_1q̅≤ qμ/ϕ+n_2+q̅, then the strategy profile that all users choose DP (i.e., u_1=n_1,u_2=n_2) is a NE. Otherwise, n_1q̅> qμ/ϕ+n_2+q̅. Let m̃ be an integer in interval [qμ/ϕ+n_2+n_1q̅-q̅/2q̅,qμ/ϕ+n_2+n_1q̅+q̅/2q̅]=[t_1(n_2)-1,t_1(n_2)], which always admits at least one integer. Note that n_1>qμ/ϕ+n_2+n_1q̅+q̅/2q̅≥m̃>0. By Theorem <ref>, a strategy profile with u_1=m̃ and u_2=n_2 is a NE.
§ NUMERICAL EXPERIMENTS
Through numerical simulations, we explore the impact of traffic condition on network performance, i.e., the total traffic rate and PoA.
Recall that the traffic flow originating from each user is Poisson with rate ϕ,
the service rate of each direct link is μ, and the loss probability over each side link is q. Assume ϕ=1 for normalization.
We first present the simulation results in two-source networks. In Fig. <ref>, the PoA and the total traffics are plotted under different q, μ and n_1, showing a PoA of less than 1.08. Such a little gap between the optimal solution and the worst NE suggests that the gain of centralized-decision making over decentralized-decision making is trivial most of the time. As shown in Fig. <ref>, the total traffic decreases with the increase of q, i.e., the increased loss rate on sidelink. On the other hand, the PoA is first increasing from 1 at q=0, implying that the NE and optimal solution are the same with u_1 = n_2 + v_1. That is, we have the equal number of users on (s_1,d) and (s_2,d) in terms of both IP and DP. With the increase of q, the benefit of centralized-decision making is gradually unveiled. However, when p reaches a certain value, the PoA goes down to 1 quickly. An intuitive explanation is that, when q becomes larger than the loss probability on DP, no users will choose IP in NE. And this strategy is optimal as well.
In Fig. <ref>, the traffic rates grow with the increase of μ due to the increased probability of no congestion in (<ref>). This is, a high service rate help clear the collision and relieve congestion on both DP and IP. The PoA curve indicates that either in overloaded or less congested scenarios, there is little improvement of centralized-decision making. In Fig. <ref>, the increased number of users leads to an increase of traffic rate in spite of the rise in loss rate. What is more, PoA tends towards 1 for small and large n_1. As the strategies in opt and NE are much similar for users at source s_1, i.e., DP in less biased scenario and IP severely biased scenario.
In the multi-source network, while the optimal solution can be easily computed by Algorithm <ref>, it is difficult to find all NEs even given Theorem <ref>. Hence, we merely consider a small value of m and n (i.e., m=3). The service rate and traffic arrival rate are fixed as μ=1, ϕ=1. Results are given in Fig. <ref>, which shows similar results in Fig, <ref>. It is obvious that the growth of the total traffic slows down gradually, because given the service rate, an increase of n_1 aggravates the network congestion. Second, the increase of loss rate on sidelink, leads to the increase of loss rate on IP. As a result, more users choose DP instead, which in turn worsens the network congestion.
Figure <ref> plots the performances for a range of q. When q=0 and q=1, the PoA is exactly 1.
The PoA converges to 1 when q goes to 1, because when the service rate is large enough compared with arrival rate, there is a sufficiently small congestion loss and all users like to choose DP.
§ CONCLUSION
In this work, we give a theoretical analysis of a load balancing game in cloud-enabled networks, in which the users want to minimize the loss probability of their packets with suitable routing strategies. In the centralized analysis, an efficient algorithm for maximizing the total traffic rate is proposed, according to Lemma <ref> and Lemma <ref>. In the decentralized analysis, a characterization of Nash equilibrium is given, and the PoA is investigated. Numerical experiments show that the efficiency loss due to selfish behaviors is relatively small in most cases.
There are many future directions that are worth exploring. First, we only focus on pure strategies of players in this work, and an immediate and natural question is how the users act when mixed strategies are allowed. Second, it would be interesting to investigate heterogeneous servers (source nodes) where each s_i serves a different purpose or has a different service rate μ_i. Moreover, while we only consider direct path and one-hop indirect paths, a more general scenario where players can choose multi-hop indirect paths to the destination can be taken into consideration.
IEEEtran
|
http://arxiv.org/abs/2307.05407v2 | 20230711161413 | Weyl's law in Liouville quantum gravity | [
"Nathanaël Berestycki",
"Mo Dick Wong"
] | math.PR | [
"math.PR",
"math-ph",
"math.DG",
"math.MP",
"math.SP",
"nlin.CD"
] |
Enhanced diffusion of tracer particles in non-reciprocal mixtures
Pierre Illien
August 12, 2023
=================================================================
Can you hear the shape of Liouville quantum gravity? We obtain a Weyl law for the
eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows
linearly with n, with the proportionality constant given by the
Liouville area of the domain and a certain deterministic constant c_γ
depending on γ∈ (0, 2). The constant c_γ, initially a complicated function of Sheffield's quantum cone, can be evaluated explicitly and is strictly greater than the equivalent Riemannian constant.
At the heart of the proof we obtain
sharp asymptotics of independent interest for the small-time behaviour of the
on-diagonal heat kernel. Interestingly, we show that the scaled heat
kernel displays nontrivial pointwise fluctuations. Fortunately,
at the level of the heat trace these pointwise fluctuations
cancel each other which leads to the result.
We complement these results with
a number of conjectures on the spectral geometry of Liouville quantum gravity.
§ PROBLEM SETTING AND RESULT
§.§ Weyl's law
Let D ⊂ℝ^2 ≅ℂ be a simply connected[This assumption is probably not necessary but is convenient for some estimates. We have chosen not to make the assumptions on the domain as general possible in order to keep the paper to a reasonable length. With some effort it should be possible to prove the results assuming only that D is a bounded domain with at least one boundary regular point. To avoid any confusion, recall that a point z ∈∂ D is called regular if, for a planar Brownian motion (W_t)_t ≥ 0 starting from z, we have ℙ_z(inf{t > 0: W_t ∉D} = 0) = 1, i.e., W leaves D immediately.
],
bounded domain
and let h(·) be the Gaussian free field on D with Dirichlet boundary condition, i.e. h(·) is a centred Gaussian field on D with covariance kernel given by
𝔼[h(x) h(y)] = G_0^D(x, y) ∀ x, y ∈ D
where G_0^D(x, y) is the Dirichlet-boundary Green's function on D. In other words, for all x y in D we have
G_0^D(x, y) = π∫_0^∞ p_t^D(x,y) dt
where p_t^D(·, ·) is the Dirichlet heat kernel on D, with our time parametrisation chosen such that it represents the transition density of a standard (two-dimensional) Brownian motion (with killing at the boundary). In particular, for any x ∈ D we have
p_t^D(x,x) t → 0+∼ (2π t)^-1 and G_0^D(x, y) y → x= - log|x-y| + 𝒪(1).
Note that there is no factor of two or π in the logarithmic blow-up on the right hand side above, which is a result of our conventions on the Green function and the Gaussian free field (these are consistent with other works on Liouville quantum gravity).
For γ∈ (0, 2), we denote by μ_γ (d·) the Liouville measure (or Gaussian multiplicative chaos measure) associated to h(·), i.e.
μ_γ(dx)
= lim_ϵ→ 0^+ϵ^γ^2/2e^γ h_ϵ(x) dx
= lim_ϵ→ 0^+ R(x; D)^γ^2/2e^γ h_ϵ(x) - γ^2/2𝔼[h_ϵ(x)^2] dx, x ∈ D
where R(x; D) is the conformal radius of D from x. The Liouville measure plays a central role in the emerging theory of Liouville quantum gravity (LQG) <cit.>, or equivalently (but with a slightly different perspective), Liouville conformal field theory <cit.>; see again <cit.> for a survey including a discussion of the physical motivations and references.
In this article we are interested in some fundamental questions pertaining to the geometry of Liouville quantum gravity. The basic problem which motivates us is the following analogue of Mark Kac's celebrated question <cit.>:
Can one hear the shape of Liouville quantum gravity?
In Mark Kac's original question, the setting is the following: we are given a bounded domain D ⊂^d, and the sequence of eigenvalues (λ_n)_n≥ 0 corresponding to -12Δ with Dirichlet boundary conditions in D, and ask if this sequence determines D up to isometry (i.e., up to translation, reflection and rotation). Kac's question has served as a motivation for a remarkable body of work. As is well known since the fundamental work of Weyl <cit.>, the eigenvalues determine at least the volume of D, since if we call
N_0(λ) = ∑_n≥ 01_{λ_n ≤λ}
the eigenvalue counting function, then the celebrated Weyl law asserts that
N_0(λ)/(2λ)^d/2→ω_d/(2π)^dLeb(D)
where ω_d is the volume of the unit ball in ℝ^d. Weyl's law is known to hold in a great degree of generality including Neumann boundary conditions and can be extended to the setting of Riemannian geometry (see e.g. <cit.>). However, it is also known that the answer to Kac's question in general is negative (counterexamples were obtained first by Milnor for five-dimensional surfaces <cit.>, and by Gordon, Webb and Wolpert for concrete bounded planar domains <cit.>).
In this paper we initiate the study of this problem in the context of Liouville quantum gravity, and more generally we begin an investigation of the spectral geometry of LQG, see Figure <ref>. Given a bounded domain D, let (𝐁_t)_t≥ 0 denote the Liouville Brownian motion on D (<cit.>, <cit.>) which we recall is the canonical diffusion in the geometry of LQG. While the infinitesimal generator of this process may not be easily described, the Green measure 𝐆(x, dy) associated to it is rather straightforward, since by construction 𝐁 is a time-change of ordinary Brownian motion. This leads to the expression (<cit.>):
𝐆(x, dy) = G_0^D(x, y) μ_γ(dy).
It is not hard to check that for a fixed x ∈ D, the right hand side is a finite measure on D when γ<2, and this can also be made sense a.s. for all x∈ D simultaneously. The spectral theorem can then be applied (see <cit.> on the torus, and <cit.> for the case of a bounded domain with Dirichlet boundary conditions, which is of interest here; see also <cit.> for the definition of the Liouville Green function). By definition (<cit.>) the eigenvalues λ_n = λ_n(γ) of Liouville Brownian motion are the inverses of the eigenvalues of 𝐆; we also call 𝐟_n(·) = 𝐟_n(·; γ) the corresponding eigenfunctions, normalised to have unit L^2 ( μ_γ) norms. (The eigenvalues and eigenfunctions are fundamentally related to the Liouville heat kernel via a trace formula – see in particular <cit.> and <cit.> for a careful discussion – this will play an important role in our paper but will be discussed later in <Ref>). Equivalently, the eigenpairs (λ_n, 𝐟_n) could be defined from the Dirichlet form associated to Liouville Brownian motion <cit.>: we have
∫_ D (∇ g ·∇𝐟_n) dx = λ_n ∫_D g 𝐟_n μ_γ(dx) ∀ g ∈ L^2(μ_γ) ∩ H_0^1(D).
We are now ready to state our main conjecture concerning the analogue of Kac's question for Liouville quantum gravity:
One can almost surely hear the shape of Liouville quantum gravity. More precisely, the Gaussian free field h is a measurable function of the eigenvalues: that is,
there exists a measurable function ϕ such that
h = ϕ ( (λ_n)_n≥ 0),
almost surely.
In this conjecture the domain D was fixed and assumed to be known. If we do not assume D to be known then it is natural to ask whether the sequence (λ_n)_n≥ 0 determines both the domain D and the Gaussian free field h living on it. However, one quickly realises that if two pairs (D_1,h_1) and (D_2, h_2) are equivalent in the sense of random surfaces (see <cit.>) then they generate the same eigenvalue sequence.
A slightly stronger form of Conjecture <ref> is therefore:
The eigenvalue sequence (λ_n)_n≥ 0 determines the pair (D, h) modulo equivalence of random surfaces.
In fact, it is not hard to see that Conjecture <ref> implies the stronger form Conjecture <ref>. These conjectures are partly motivated by the results of Zelditch <cit.> which show that spectral determination is “generically” possible subject to analyticity conditions on the boundary and some extra symmetries.
In this paper we will not aim to prove this conjecture but instead show that the analogue of Weyl's law for Liouville quantum gravity holds: that is, (λ_n)_n≥ 0 determines at least the LQG volume μ_γ(D) of D. More precisely, our main result is the following. Suppose the eigenvalues (λ_n)_n ≥ 0 are sorted in increasing order, and define the eigenvalue counting function by
𝐍_γ(λ) := ∑_n≥ 0 1_{λ_n ≤λ}.
Let 0< γ <2.
We have
𝐍_γ(λ)/λ c_γμ_γ(D).
Here, the constant c_γ = c_γ(Q-γ), where Q = γ/2 + 2/γ and for m>0, c_γ (m) is defined as follows:
c_γ(m) := 1/π{𝔼[∫_0^∞ℐ(e^γ (B_t - mt))dt]
+𝔼[∫_0^∞ℐ(e^γℬ_t^m)dt]}
where
ℐ(x) := x e^-x, x ∈ℝ,
(B_t)_t ≥ 0 is a standard (1-dimensional) Brownian motion, and (ℬ_t^m)_t ≥ 0 is a Brownian motion with drift m>0 conditioned to be non-negative at all times t ≥ 0.
Readers familiar with Sheffield's theory of quantum cones (<cit.>, see also <cit.>) will recognise the constant c_γ as a somewhat complicated functional of the so-called γ-quantum cone. Perhaps surprisingly, this constant can be evaluated explicitly:
For any γ∈ (0,2), m > 0, we have
c_γ(m) = 1/(πγ m).
In particular,
c_γ= 1/π ( 2 - γ^2/2).
Moreover, lim_γ→ 0^+ c_γ =c_0 := 1/(2π) and c_γ > c_0.
Theorem <ref> corresponds to a Weyl law where the dimension d is taken to be d=2. (Note that taking the limit γ→ 0^+ we recover, at least formally, the classical Weyl's law for Euclidean domains). This corresponds to the fact that the spectral dimension of Liouville quantum gravity is equal to two (see <cit.>, conjectured earlier by Ambjørn <cit.>). At the same time, the fact that c_γ>c_0 shows that one cannot merely naively extrapolate the Riemannian result to LQG. This should probably be viewed as a consequence of the highly disordered, multifractal nature of the geometry in LQG; see Figure <ref>.
Finally, it is known that the Liouville measure μ_γ determines the Gaussian free field h (see <cit.>). This, however, does not imply Conjecture <ref> since we would need to know not only the LQG-mass of the domain D but also that of any (say, open) subset of D in order to entirely determine the measure μ_γ.
§.§ Conjectures and questions on the spectral geometry of LQG
<Ref> shows the growth of the volume-normalised eigenvalue counting function λ↦𝐍_γ(λ) / μ_γ(D) associated to the realisation of GFF in <Ref> and compares it against theoretical predictions from <Ref> as well as Weyl's law for Riemannian manifolds. It is curious to see that the Riemannian prediction provides a better fit for the initial eigenvalues. This may be explained by the fact that the low-frequency eigenpairs computed do not “feel” the roughness of γ-LQG surface (which could be an artefact of the numerical experiment as it involves mollified Gaussian free field on a discretised domain); see <Ref> for a comparison of eigenfunctions.
The simulation is highly suggestive of a number of interesting features which we now briefly discuss.
To begin with, we note that the eigenvalue counting function appears to always stay below the linear function predicted by its Weyl's law.
With probability one, 𝐍_γ(λ) ≤ c_γμ_γ(D) λ for all λ≥ 0.
This is the LQG analogue of a famous conjecture of Pólya <cit.> for Euclidean domains, which is open in general (Pólya proved it for the so-called tiling domains <cit.>, whereas the case for Euclidean balls has been established by Filonov et al. <cit.> only very recently). A closely related result is the Berezin–Li–Yau inequality <cit.> which, informally, says that the conjecture holds for Euclidean domains in a Cesaro sense.
A fascinating question concerns the second order term for the asymptotics of 𝐍_γ(λ) as λ→∞. In the Euclidean world, Weyl famously conjectured that this is of order √(λ) for smooth domains D; more precisely (under our normalisation)
N_0(λ) = c_0 Leb (D) λ - 1/2π |∂ D| √(λ) + o( √(λ))
where |∂ D| denotes the length of the boundary of D. Surprisingly this conjecture is still open in general, as it has been established under an additional geometric assumption by Ivrii <cit.> (essentially, there should not be “too many” periodic geodesics). While this assumption is believed to hold for any smooth domains, it remains to be verified.
In the LQG context, it would be interesting to understand what the correct order of c_γμ_γ(D) λ - 𝐍_γ(λ) should be, and whether one could “hear the perimeter" of the domain. Answers to these questions could be subtle, as it was observed in the literature of random fractals that there could be competitions between boundary corrections and random fluctuations (see e.g. <cit.>). The choice of the Dirichlet variant of GFF here may also affect the subleading order, since the mass distribution with respect to μ_γ has a rapid decay near the boundary ∂ D. In our simulation with γ = 0.5, 𝐍_γ(λ) behaves like c_γμ_γ(D) λ + 𝒪(λ^b) with b being much smaller than 1/2, and the deviation from the best fitting power-law curve appears to follow some central limit theorem, see <Ref>.
Another natural question concerns the behaviour of eigenfunctions in the high energy (semiclassical) limit. As we increase the energy levels _n, do the corresponding eigenfunctions _n typically become delocalised in the sense that their L^2 mass is spread out (as is the case for standard planar Brownian motion, the eigenfunctions of which are akin to sine waves with high frequency), or do they remain localised in some given region (as can happen e.g. in Anderson localisation owing to medium impurities)?
We conjecture that eigenfunctions are typically delocalised, see Figure <ref>. In fact, by analogy with quantum chaos (see e.g. <cit.>) and more precisely the celebrated quantum unique ergodicity conjecture of Rudnick and Sarnak <cit.>, we make the following conjecture:
Fix γ∈ (0,2), and suppose the eigenfunctions _n are normalised to have unit L^2 ( μ_γ)-norm. Then as n →∞,
|_n(x)|^2 μ_γ(dx) ⇒μ_γ(dx)/μ_γ(D)
in the weak-* topology in probability.
Also motivated by the literature on quantum chaos is the question of eigenvalue fluctuations. Following the Bohigas-Giannoni-Schmit conjecture on spectral statistics <cit.> (see also a celebrated conjecture of Sarnak <cit.> for deterministic hyperbolic surfaces), we conjecture that level fluctuations of LQG eigenvalues should resemble those of Gaussian Orthogonal Ensemble (GOE) of random matrices (see e.g. <cit.> for an introduction). For instance, in the concrete example of level spacing distribution of eigenvalues, we conjecture:
For each x ≥ 0,
1/N∑_j=1^N 1_{ c_γμ_γ(D)(λ_j+1 - λ_j) ≤ x} F_GOE(x),
where F_GOE (x) is the GOE level-spacing distribution.
Note that the rescaled eigenvalue gap c_γμ_γ(D)(λ_j+1 - λ_j) is considered above since it is approximately equal to 1 on average in the long run, as established by our Weyl's law (<Ref>).
The spacing distribution F_GOE, also known as Gaudin distrbution (for β = 1) in the literature, may be expressed in terms of a Fredholm determinant involving the Sine kernel <cit.> as well as the Painlevé transcendents <cit.>.
See <Ref> for a comparison between the empirical LQG eigenvalue spacing distribution and our GOE conjecture.
All the conjectures (and results in this paper) above have natural analogues for the eigenvalues of Liouville Brownian motion with Neumann, i.e. reflecting, boundary conditions, both when the underlying GFF itself has Dirichlet or Neumann boundary conditions, as well as for random planar maps. However we do not discuss these variants here in order to keep the paper at a reasonable length.
We end this series of conjectures on the spectral geometry of LQG by asking what (if any) of these results and conjectures become in the critical case γ=2. Note that c_γ= 1/ [π (2 - γ^2/2)] →∞ so it is likely that the Weyl law would require a different way of scaling the eigenvalue counting function compared to Theorem <ref>.
§.§ Short-time heat trace and heat kernel asymptotics
Our main tool for the proof of Theorem <ref> is the study of the short-time asymptotics of the heat kernel of Liouville Brownian motion, for which we establish results which are of independent interest.
For points x,y ∈ D, let 𝐩_t^γ, D(x, y) denote the heat kernel (<cit.>). Recall from <cit.> and <cit.> that there exists a jointly continuous version of the heat kernel in all three arguments (t>0, x ∈ D, y ∈ D) which therefore identifies the function 𝐩_t^γ, D(x, y) uniquely. The heat kernel and spectrum of Liouville Brownian motion are related by the following fundamental trace formula: almost surely, for all t>0 and all x, y ∈ D,
𝐩_t^γ, D(x, y) = ∑_n=1^∞ e^ - λ_n t𝐟_n(x) 𝐟_n(y);
see <cit.>. In particular, setting y = x (which is allowed since this formula holds a.s. simultaneously for all x,y∈ D and t>0), and integrating, we obtain:
∫_D 𝐩_t^γ, D(x, x) μ_γ (dx) = ∑_n=1^∞ e^ - λ_n t.
The integral on the left hand side is known as the heat trace and will be denoted in the following by 𝐒_γ(t;D).
Note that the identity (<ref>) implies that the heat trace 𝐒_γ(t;D) is equal to the Laplace transform of the eigenvalue counting function: in other words,
𝐒_γ(t;D) := ∫_0^∞ e^-t λ d𝐍_γ(λ)
where 𝐍_γ(λ) := ∑_k 1_{λ_k ≤λ}.
As a consequence, using a probabilistic extension of the Hardy–Littlewood Tauberian theorem (see <Ref>),
it is possible to deduce results about eigenvalues of Liouville Brownian motion based on heat-trace asymptotics. Indeed we will obtain Theorem <ref> from the following result:
Let γ∈ (0, 2) and A ⊂ D be any fixed open set. Denoting 𝐒_γ(t) = 𝐒_γ(t; A) := ∫_A _t(x, x) μ_γ(dx), we have
t 𝐒_γ(t;A) → c_γμ_γ(A)
in probability as t → 0^+.
Pointwise asymptotics. Since A was an arbitrary open subset of D, it is natural to wonder if the asymptotics in Theorem <ref> holds pointwise. In other words, if we sample x from the Liouville measure μ_γ and fix it, does 𝐩^γ, D_t (x,x) behave asymptotically (in probability) as c_γ /t ?
It turns out that the small-time behaviour of the heat kernel is much more subtle. We can in fact prove that the answer to the above question is negative and establish the following result:
Let γ∈ (0, 2). Sampling from μ_γ,
J_γ^λ(x) = λ∫_0^∞ e^-λ t t _t (x,x) dt J_γ^∞
in distribution (where the average is over the law of the Gaussian free field h). Here J_γ^∞∈ (0, ∞) is a non-constant random variable with expectation c_γ. More precisely, for any f∈ C_b(D×ℝ_+), we have
𝔼[ ∫_D μ_γ (dx) f (x, J_γ^λ(x) ) ] 𝔼[ ∫_D μ_γ(dx)𝔼 [ f(x, J_γ^∞) ] ]
= ∫_D dx R(x; D)^γ^2/2𝔼 [ f(x, J_γ^∞) ].
By adapting the proof of <Ref>, one could generalise the above to a multiple-point setting and show e.g. for any f ∈ C_b(D×D×ℝ_+ ×ℝ_+),
𝔼[∫_D× Dμ_γ (dx)/μ_γ (D)μ_γ (dy)/μ_γ (D) f (x, y, J_γ^λ(x), J_γ^λ(y) ) ]
𝔼[∫_D× Dμ_γ (dx)/μ_γ (D)μ_γ (dy)/μ_γ (D) f (x, y, J_γ^∞(x), J_γ^∞(y) ) ]
where J_γ^∞(·) are i.i.d. random variables independent of the Gaussian free field h(·).
We now explain why this result rules out that t 𝐩^γ, D_t (x,x) converges to any constant in probability. Suppose by contradiction that
t 𝐩^γ, D_t (x,x) → c
in probability. Then by applying our probabilistic extension of the Hardy–Littlewood Tauberian theorem (see <Ref>) we would then have J^λ_γ(x) converges (in probability) as λ→∞ to c. This would imply that J^∞_γ is the constant random variable equal to c, which is a contradiction.
Note that if the Tauberian theorem (<Ref>) could be extended to cover convergence in distribution, Theorem <ref> would imply that if we sample x from the Liouville measure μ_γ(dx), then the distribution of t 𝐩^γ, D_t (x,x) converges (when averaged with respect to the law of the Gaussian free field h) to a nontrivial random variable. We formulate this as a conjecture:
Let γ∈ (0, 2). Sample x from Liouville measure. Then as t→ 0^+ and we average of the law of the Gaussian free field h,
t 𝐩^γ, D_t (x,x) ξ_γ
for some (non-constant) random variable ξ_γ > 0. In other words,
[ ∫_D μ_γ(dx) f(x, t𝐩^γ, D_t(x,x) ) ] →∫_D [ f(x,ξ_γ) ] R(x;D)^γ^2/2 dx
for any test function f ∈ C_b(D×ℝ_+).
We also believe that if we sample multiple points x_1, …, x_n from the Liouville measure μ_γ then the same convergence holds jointly with the limiting random variables ξ_γ, 1, …, ξ_γ, n being independent of each other as well as the Gaussian free field, similar to what we discussed after Theorem <ref>.
Coming back to quenched heat kernel fluctuations (i.e., when we do not average over the law of the environment) Theorem <ref> suggests that t𝐩^γ,D_t (x,x) has considerable fluctuations. In fact we believe there are nontrivial logarithmic fluctuations in both directions (see in particular <cit.> where an upper bound of this type is proved).
Related works. We point out that results analogous to ours but in the context of random recursive fractals have previously been obtained; starting in particular with the work of Hambly <cit.>. Croydon and Hambly <cit.> obtained similar results for the random fractals given respectively by Aldous' continuous random tree and more generally stable trees. The paper by Charmoy, Croydon and Hambly <cit.> obtained considerable refinements including the second order behaviour. (We thank Takashi Kumagai for drawing our attention to these works). See also earlier works e.g. by Kigami and Lapidus on self-similar (nonrandom) fractals such as the Sierpinski gasket <cit.> where however periodicity phenomena preclude a strict Weyl asymptotics for the eigenvalues.
On the Liouville quantum gravity side, previous results on the short-time asymptotics for the on-diagonal Liouville heat include those by Rhodes and Vargas <cit.> where they proved that 𝐩_t^γ, D (x,x) = t^ -1 + o(1), thereby proving that the spectral dimension is equal to two a.s., a behaviour predicted earlier by Ambjørn et al. in <cit.>. Ours is the first result which gives sharp asymptotics including the identification of the leading order coefficient.
See also <cit.> and <cit.> for the challenging problem of obtaining off-diagonal estimates for Liouville heat kernel.
§.§ Main idea
We briefly explain the main ideas leading to the proof of Theorem <ref>.
Recall that we aim to show
𝐒_γ(t) t → 0∼c_γμ_γ(D) /t
in probability (where a_n ∼ b_n in probability means a_n /b_n → 1 in probability as n →∞). It would be natural to try and use a Tauberian approach directly on 𝐒_γ(t).
It is however very difficult to have a direct handle on 𝐒_γ(t). Our best hope is to use the bridge decomposition (<cit.>, <cit.>; see below for more details) which allows us to relate integrals of 𝐒_γ(t) (with respect to time) to its Euclidean counterpart. However if one naively consider a quantity such as ∫_t^1 𝐒_γ(s) ds, it would appear that the first step would be to prove that this blows up logarithmically as t→ 0 with a proportionality constant dictated by (<ref>) and then try to apply Tauberian theory. However this logarithmic behaviour falls precisely outside the scope of the most classical results in Tauberian theory (one would instead need to appeal to so-called de Haan theory, see e.g. <cit.>).
Luckily, there is a simple solution around this.
Since t ↦𝐒_γ(t) is monotone in t, it suffices (see Lemma <ref> for a proof)
to establish the integrated asymptotics in probability
∫_0^t u 𝐒_γ(u) du ∼ c_γμ_γ(D) t
as t → 0^+,
(note that the multiplication by u in the integral in the left hand side effectively changes the index of regular variation).
Equivalently, by the probabilistic extension of the Tauberian theorem (see Theorem <ref>), it suffices to prove
∫_0^∞ e^-λ u u𝐒_γ(u) du ∼c_γμ_γ(D)/λ as λ→∞
in probability. We now explain how to obtain (<ref>). As already alluded to above, a key tool for this is the following bridge decomposition (<cit.>, <cit.>):
For any measurable f: [0, ∞) → [0, ∞), we have
∫_0^∞ f(t) _t(x, y) dt
= ∫_0^∞xyt[f(F_γ(𝐛))1_{t < τ_D(𝐛)}] p_t(x, y) dt
where
* xyt = law of Brownian bridge (𝐛_s)_s ≤ t of duration t from x to y (without killing); and
* for any process 𝐛 defined on ℝ^2 with starting position 𝐛_0 ∈ D and duration ℓ = ℓ(𝐛):
* τ_D(𝐛) := inf{t > 0: 𝐛_t ∈∂ D},
* F_γ(𝐛) is the Liouville clock associated to 𝐛, i.e. F_γ(𝐛) := ∫_0^ℓ F_γ(ds; 𝐛) with
F_γ(ds; 𝐛) := e^γ h(𝐛_s) - γ^2/2𝔼[h(𝐛_s)^2] R(𝐛_s; D)^γ^2/2 1_{𝐛_s ∈ D}ds,
* p_t(x, y) is the transition density of standard 2-dimensional Brownian motion (in particular p_u(x,x) = 1/ (2π u) for any u > 0 and x ∈ D).
The bridge decomposition as stated in <cit.> has slightly different assumptions (e.g., we need here to restrict to trajectories remaining inside of D, which was not the case in <cit.>, but the proof is straightforward to adapt.
Using Lemma <ref> we now return to the main idea of the proof of Theorem <ref>.
Using the bridge decomposition, the left-hand side of (<ref>) can be rewritten as
∫_0^∞ e^-λ u u𝐒_γ(u) du
= ∫_D μ_γ(dx) ∫_0^∞xxu[F_γ(𝐛) e^-λ F_γ(𝐛)1_{u < τ_D(𝐛)}] p_u(x, x) du.
Thus <Ref> will follow from the following result:
Let A ⊂ D be a fixed open subset of D and γ∈ (0, 2). Then
lim_λ→∞∫_A μ_γ(dx) ∫_0^∞du/2π uxxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛) }] = c_γμ_γ(A)
in the sense of L^1-convergence (and hence convergence in probability).
At a high level, for Theorem <ref> we will draw inspiration from the thick points approach described in <cit.>, however the details are of course much more technical. In particular we develop a general method for handling correlations of possibly different functionals applied to small neighbourhoods in the vicinity of Liouville typical points; see Lemma <ref> for a statement and Section <ref> for a proof of Theorem <ref>.
§.§ Outline of the paper.
We start in Section <ref> with some preliminaries on Gaussian comparison, estimates for the Green's function and decomposition results both for the GFF and Brownian motion, which leads us to the main lemma (Lemma <ref>).
Section <ref> contains the proof of the main results of this paper, namely Theorem <ref>. We start in Proposition <ref> with a quick and simple illustration for how the main lemma is used throughout the paper by computing “one-point estimates” with it. We end with a very brief description of how Theorem <ref> implies both Theorem <ref> and Theorem <ref>.
Section <ref> gives the proof of Theorem <ref> that pertain to the pointwise asymptotics of the heat kernel (as opposed to the heat trace asymptotics which are at the heart of Theorem <ref>, and which involve by definition a spatially averaged heat kernel asymptotics). The identification of the limiting constant in Theorem <ref>, i.e. Theorem <ref>, is also proved in that section.
Finally, Appendix <ref> contains probabilistic extensions of results from asymptotic analysis, namely “asymptotic differentiation under the integral sign” (Lemma <ref>) and Tauberian theorem (Theorem <ref>).
Notations. For the readers' convenience we list a few crucial notations below which are used repeatedly in the main proofs in <Ref> and <Ref>, and provide pointers to their defining equations.
* F_γ(𝐩) and F_γ(ds; 𝐩): Liouville clock associated to the path 𝐩, see (<ref>).
* F_γ^𝒮(𝐩): Liouville clock with insertions in 𝒮, see (<ref>); when 𝒮 = ∅ this coincides with the previous definition.
* 𝒢_I^𝒮(p): `good event' concerning the thickness of Gaussian free field at p ∈ D at dyadic levels in I, see (<ref>); when 𝒮 = ∅ we suppress its dependence in the notation.
* F_γ^𝒮(𝐩; Y): random clock associated to the path 𝐩 with respect to background field Y and insertions in 𝒮, see (<ref>).
Acknowledgement. Part of the work was carried out when both authors were in residence at the Mathematical Sciences Research Institute in Berkeley for the semester programme `The Analysis and Geometry of Random Spaces' in 2022, and we wish to thank the Institute for its hospitality and support (NSF grant 1440140). NB's research is supported by FWF grant P33083 on “Scaling limits in random conformal geometry”.
§ PRELIMINARIES
§.§ Gaussian comparison
Let X(·) and Y(·) be two continuous centred Gaussian field on D, ρ be a Radon measure on D, and P: ℝ_+ →ℝ be a smooth function with at most polynomial growth at infinity. For t ∈ [0, 1], define Z_t(x) := √(t) X(x) + √(1-t) Y(x) and
φ(t):= 𝔼[P(M_t)],
M_t := ∫_D e^Z_t(x) - 1/2𝔼[Z_t(x)^2]ρ(dx).
Then
φ'(t) =
1/2∫_D ∫_D (𝔼[X(x)X(y)] - 𝔼[Y(x) Y(y)])
×𝔼[e^Z_t(x) + Z_t(y) - 1/2𝔼[Z_t(x)^2] - 1/2𝔼[Z_t(y)^2]P”(M_t)] ρ(dx) ρ(dy).
In particular, if there exists some constant C>0 such that
| 𝔼[X(x)X(y)] - 𝔼[Y(x) Y(y)]| ≤ C ∀ x, y ∈ D,
then
|φ(1) - φ(0)|
≤C/2∫_0^1 𝔼[M_t^2 |P”(M_t)|] dt.
Using the same notations as in <Ref>, suppose
𝔼 [ X(x) X(y)] ≤𝔼[Y(x) Y(y)] and P is convex, then φ(1) ≤φ(0), i.e., 𝔼 [ P(M_0)] ≤𝔼[ P(M_1)].
§.§ Estimates for Brownian bridge
Let 𝐛_· = (𝐛_·, 1, 𝐛_·, 2) be a 2-dimensional Brownian bridge with starting position ι(𝐛):= 𝐛_0 and duration ℓ(𝐛) (e.g. ι(𝐛) = x and ℓ(𝐛) = u if 𝐛∼xxu). We recall the following formula for the distribution of running maximum of one-dimensional Brownian bridge:
For i ∈{1, 2} and any k ≥ 0,
00ℓ(max_s ≤ℓ𝐛_s, i≥ k )
= e^-2/ℓk^2 ∀ k ≥ 0.
The exact formula (<ref>) leads to the following inequalities which we shall use repeatedly throughout this article:
For any u > 0,
00ℓ( max_s ≤ℓ |𝐛_s| ≤ u )
≤ 1 ∧2u^2/ℓ
and 00ℓ( max_s ≤ℓ |𝐛_s| ≥ u )
≤ 4 e^-u^2/2l.
The two inequalities follow from
00ℓ( max_s ≤ℓ |𝐛_s| ≤ u )
≤00ℓ( max_s ≤ℓ𝐛_s, 1≤ u )
= 1 - e^-2u^2 / l,
and
00ℓ( max_s ≤ℓ |𝐛_s| ≥ u )
≤ 200ℓ( max_s ≤ℓ |𝐛_s, 1| ≥u/2)
≤ 400ℓ( max_s ≤ℓ𝐛_s, 1≥u/2)
= 2e^-u^2/2l
by <Ref>.
§.§ Estimates for Green's function
Suppose D is a bounded domain with at least one regular point on ∂ D. Then the following estimates hold for our Green's function G_0^D(·, ·).
* For any x, z ∈ D satisfying |x-z| ≤1/3 d(x, ∂ D), we have
| G_0^D(x, z) - [ -log|x-z| + log R(x; D)]| ≤ 6|x-z|/d(x, ∂ D)logR(x; D)/d(x, ∂ D).
* For any x, y, z ∈ D satisfying d(x, z) ≤min(|x-y|, d(x, ∂ D)),
| G_0^D(z, y) - G_0^D(x, y)|
≤ 2[ |x-z|/d(x, ∂ D) + |x-z|/|x-y|].
For the first estimate, it suffices to consider the case where x = 0 and d(x, ∂ D) = 1 by translation and rescaling. But then
|G_0^D(0, z) - [-log|z| + log R(0; D)]|
≤1/π∫_0^2π G_0^D(0, e^iθ) |H_𝔻(z, e^iθ) - H_𝔻(0, e^iθ)|dθ.
Using the fact that
1/2π∫_0^2π G_0^D(0, e^iθ) dθ = log R(0; D)
and the explicit formula for the Poisson kernel on the unit disc 𝔻
H_𝔻(z, e^iθ)
= 1/21-|z|^2/|e^iθ - z|^2, |z| < 1,
we obtain the upper bound (<ref>) by a direct computation.
For the second estimate, we recall the probabilistic representation of the Green's function
G_0^D(·, y) = 𝔼^y [log|W_τ_D - ·|] - log|· - y|
where (W_t)_t ≥ 0 is a (planar) Brownian motion starting from y ∈ D (with respect to the probability measure ℙ^y) and τ_D is its hitting time of ∂ D. Then (<ref>) can be verified directly using the elementary inequality log |1+x| ≤ 2|x| for any |x| ≤1/2.
We state a useful consequence of the above estimate.
Let a, b, x ∈ D be such that max (|x-a|, |x-b|) ≤1/4 d(x, ∂ D). Then
|G_0^D(a, b) - [ -log|a-b| + log R(x; D)]| ≤ 4.
In particular, for any z ∈ B(x, 1/4 d(x, ∂ D)), we have
| log R(z; D) - log R(x; D)| ≤ 4.
Let 𝔼^a be the expectation with respect to a planar Brownian motion (W_t)_t ≥ 0 starting from a ∈ D, and τ_D :={ t > 0: W_t ∈∂ D}. Then
|G_0^D(a, b) - [ -log|a-b| + log R(x; D)]|
= |𝔼^a [ log |W_τ_D - b| ] - log R(x; D) |
≤|𝔼^a [ log |W_τ_D - x| ] - log R(x; D)| + |𝔼^a [ log |W_τ_D - b| ] - 𝔼^a [ log |W_τ_D - x| ]|
= |G_0^D(a, x) - [ -log|a-x| + log R(x; D)]| + |𝔼^a [ log| (W_τ_D - x) + (x - b)/ |W_τ_D - x| | ]|.
Using (<ref>) and Koebe quarter theorem, we have
|G_0^D(a, x) - [ -log|a-x| + log R(x; D)]|
≤ 6 |x-a|/d(x, ∂ D)logR(x; D)/d(x, ∂ D)
≤ 6 1/4log 4 ≤ 3,
whereas the elementary inequality |log |1 + x| | ≤ 2|x| for any |x| ≤12 implies
|𝔼^a [ log| (W_τ_D - x) + (x - b)/ |W_τ_D - x| | ]|
≤ 2 |x-b|/d(x, ∂ D)≤ 1
which gives the desired claim.
For each r > 0, let h_r(·) be the circle average of the Gaussian free field over ∂ B(·, r). Then for any ϵ, δ > 0,
𝔼[ h_ϵ(x) h_δ(y)]
= -log(|x-y| ∨ϵ∨δ) + 𝒪(1)
where the 𝒪(1) error is uniform for all x, y ∈ D bounded away from ∂ D.
§.§ Decomposition of Gaussian free field
Let us mention the following decomposition of Gaussian free field, which will play a crucial role in the proof of <Ref>.
Let κ∈ (0, 1]. Then on some suitable probability space we can construct simultaneously three Gaussian fields h^κ𝔻, X^κ𝔻 and 𝒢^κ𝔻 such that
h^κ𝔻 (·) = X^κ𝔻(·) - Y^κ𝔻(·) on B(0, κ)
where
* h^κ𝔻 is a Gaussian free field on B(0, κ) with Dirichlet boundary condition;
* X^κ𝔻 is the exactly scale invariant field with covariance given by 𝔼[X^κ𝔻(x) X^κ𝔻(y)] = -log|x-y| + logκ on B(0, κ).
* Y^κ𝔻(·) is a Gaussian field on B(0, κ) independent of h, and is uniformly continuous when restricted to compact subset of B(0, κ); moreover Y^κ𝔻(0) = 0.
Since
h^κ𝔻(·) d= h^𝔻(· / κ)
and
X^κ𝔻(·) d= X^𝔻(· / κ)
on B(0, κ), the general result follows from the special case κ = 1 using a scaling argument.
Let us now focus on κ = 1, and view 𝔻⊂ℂ. Recall that
𝔼[X^𝔻(x) X^𝔻(y)]
= - log|x-y|
= -log| x-y/1 - xy̅| - log|1-xy̅|
= G_0^𝔻(x, y) - log|1-xy̅| ∀ x, y ∈𝔻.
We claim that the kernel -log|1-xy̅| is positive definite on 𝔻×𝔻 and therefore could be realised as the covariance kernel of some Gaussian field Y^𝔻: indeed the field can be explicitly constructed by
Y^𝔻(z) := [ ∑_k=1^∞√(2/k)𝒩^ℂ_kz^k], z ∈𝔻
where 𝒩^ℂ_k are i.i.d. standard complex Gaussian random variables. We can then construct a Gaussian free field h^𝔻 independent of Y^𝔻 and set X^𝔻 := h^𝔻 + Y^𝔻 so that (<ref>) holds by definition.
Last but not least, since Y^𝔻(z) is the real part of a random analytic function with radius of convergence equal to 1, it follows immediately that Y^𝔻(z) is uniformly continuous when restricted to any compact subset of 𝔻, and substituting z = 0 into (<ref>) we have Y^𝔻(0) = 0 almost surely, as claimed.
§.§ Williams' path decomposition of Brownian motion
The following result is due to Williams <cit.>; see also <cit.>.
Let (B_t)_t ≥ 0 be a Brownian motion, and for m> 0 write B_t^m := B_t + m t. Fix x > 0 and define
τ_x := inf{t > 0: B_t^m = x}.
Then we have the following equality of path distributions
(x - B_τ_x - t^m)_t ∈ [0, τ_x]d= (ℬ_t^m)_t ∈ [0, L_x]
where (ℬ_t^m)_t ≥ 0 is a Brownian motion with drift m conditioned to stay non-negative, and
L_x := sup{t > 0: ℬ_t^μ = x}.
The following definition will be used in <Ref> of the article: for each m > 0 we define the two-sided process (β_t^m)_t ∈ℝ by
β_t^m =
B_t - mt if t ≥ 0
ℬ_-t^m if t ≤ 0
where (B_t)_t ≥ 0 and (ℬ_t^m)_t ≥ 0 are independent of each other. In particular we can re-express the constant c_γ(m) defined in (<ref>) as
c_γ(m) = 1/π𝔼[ ∫_-∞^∞ℐ(e^γβ_t^m)dt].
Before we proceed, let us explain why the constant c_γ(m) is finite for positive γ and m.
The constant c_γ(m) defined in (<ref>) is finite for any γ, m > 0.
We start with the first expectation in (<ref>), and consider
𝔼[ℐ(e^γ (B_t - mt))]
=
𝔼[e^γ (B_t - mt)exp(-e^γ (B_t - mt))1_{B_t - mt ≤ -1/2mt}]
+
𝔼[e^γ (B_t - mt)exp(-e^γ (B_t - mt))1_{B_t -mt > -1/2mt}]
≤ e^-γ m/2t + ℙ(B_t -mt > -1/2mt)
≤ e^-γ m/2t + e^-1/8 m^2 t.
This shows that
𝔼[∫_0^∞ℐ(e^γ (B_t - mt))dt ]
≤∫_0^∞[ e^-γ m/2t + e^-1/8 m^2 t] dt < ∞.
As for the second expectation in (<ref>), we consider
𝔼[ℐ(e^γℬ_t^m)]
= 𝔼[e^γℬ_t^mexp(-e^γℬ_t^m)1_{ℬ_t^m ≤1/2mt}]
+ 𝔼[e^γℬ_t^mexp(-e^γℬ_t^m)1_{ℬ_t^m > 1/2mt}].
The fact that B_t + mt is stochastically dominated by ℬ_t^m implies that
𝔼[e^γℬ_t^mexp(-e^γℬ_t^m)1_{ℬ_t^m ≤1/2mt}]
≤ℙ(ℬ_t^m ≤1/2 mt )
≤ℙ(B_t + mt ≤1/2 mt )
≤ e^-1/8 m^2 t.
Meanwhile, using the elementary inequality x e^-x≤ 2 e^-x/2 for x ≥ 0 we also obtain
𝔼[e^γℬ_t^mexp(-e^γℬ_t^m)1_{ℬ_t^m > 1/2mt}] ≤ 2 e^-γ m/4t.
Hence,
𝔼[∫_0^∞ℐ(
e^γℬ_t^m)dt]
≤∫_0^∞[ e^-1/8 m^2 t + 2e^-γ m/4t]dt < ∞
and we conclude that c_γ(m) < ∞.
§.§ Main lemma
The following lemma will be used to help us obtain uniform estimates and pointwise limits that are needed for the application of dominated convergence in the main proof. We will be using the following notation: for each γ, m > 0 and function f: [0, ∞) → [0, ∞), define
c_γ(m; f)
:= 1/π𝔼[ ∫_0^∞ f(e^γβ_t^m) dt ]
= 1/π{𝔼[∫_0^∞
f(e^γℬ_t^m)dt]
+ 𝔼[∫_0^∞
f(e^γ (B_t - mt))dt]
}
In particular, if ℐ(x) = xe^-x, then c_γ(m; ℐ) = c_γ(m) as defined in (<ref>).
Consider the following random objects:
* (B_1,t)_t ≥ 0 and (B_2,t)_t ≥ 0 are two independent Brownian motions;
* ℰ_0, ℰ_1, ℰ_2 are non-negative random variables that are independent of (B_1,t)_t ≥ 0 and (B_2, t)_t≥ 0, and 𝔼[ℰ_0] < ∞.
In addition, for each i ∈{1, 2} let m_i, γ_i > 0 and ℐ_i: [0, ∞) → [0, ∞) be such that c_γ_i(m_i; ℐ_i) < ∞ and that ℐ_i(0) = 0. Then the following statements hold.
* For all λ_1, λ_2 > 0,
𝔼[ℰ_0 ∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt]
≤π c_γ_1(m_1; ℐ_1)𝔼[ℰ_0 ]
and 𝔼[ℰ_0 ∏_i=1^2( ∫_0^∞ℐ_i(λ_i ℰ_i e^γ_i (B_i,t - m_it))dt)]
≤[∏_i=1^2 π c_γ_i(m_i; ℐ_i)]𝔼[ℰ_0 ].
* We have
lim_λ_1 →∞𝔼[ℰ_0 ∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt]
= π c_γ_1(m_1; ℐ_1)𝔼[ℰ_0 ]
and lim_λ_1, λ_2 →∞𝔼[ℰ_0 ∏_i=1^2( ∫_0^∞ℐ_i(λ_i ℰ_i e^γ_i (B_i,t - m_it))dt)]
= [∏_i=1^2 π c_γ_i(m_i; ℐ_i)]𝔼[ℰ_0 ].
The random variables ℰ_0, ℰ_1, ℰ_2 need not be independent of each other, and the limit as λ_1, λ_2 go to infinity on the LHS of (<ref>) can be taken in any order/along any subsequence. See also Proposition <ref> for a simple application of Lemma <ref> which gives an idea of how it is applied to the problem of interest.
Let us treat (<ref>) and (<ref>). The assumption on ℐ_1 means that
∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt
= 1_{λ_1 ℰ_1 > 0}∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt a.s.
and so we will analyse the expectation by splitting it into two contributions depending on whether λ_1 ℰ_1 ∈ (0, 1] or λ_1 ℰ_1 > 1. We start with
𝔼[ℰ_0 1_{λ_1 ℰ_1 ∈ (0, 1]}∫_0^∞ℐ(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt ]
= ∑_n ≥ 0𝔼[ℰ_0 1_{λ_1 ℰ_1 ∈ (2^-(n+1), 2^-n]}∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt ]
= ∑_n ≥ 0𝔼[ℰ_0 1_{λ_1 ℰ_1 ∈ (2^-(n+1), 2^-n]}∫_τ_λ_1 ℰ_1^(1)^∞ℐ_1(e^γ_1 (B_1,t - m_1t))dt ]
where
τ_λ_1 ℰ_1^(1) := inf{t ≥ 0: e^γ_1 (B_1,t - m t) = λ_1 ℰ_1}
by strong Markov property. We may control the last expression with the rough upper bound
𝔼[ℰ_0( ∑_n ≥ 01_{λ_1 ℰ_1 ∈ (2^-n, 2^-(n-1)]})∫_0^∞ℐ_1( e^γ_1 (B_1,t - m_1 t)) dt]
= 𝔼[ℰ_0 1_{λ_1 ℰ_1 ∈ (0,1]}]𝔼[∫_0^∞ℐ_1( e^γ_1 (B_1,t - m_1 t)) dt]
≤π c_γ_1(m_1; ℐ_1) 𝔼[ℰ_0 1_{ 0 < λ_1 ℰ_1 ≤ 2}]
which is
* uniformly bounded by π c_γ_1(m_1; ℐ_1) 𝔼[ℰ_0], and
* converging to 0 as λ_1 →∞ by monotone convergence.
Next, we look at the main term
𝔼[ℰ_0 1_{λ_1 ℰ_1 > 1}∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1t))dt ].
Let us introduce a different stopping time
τ_λ_1 ℰ_1^(1) := inf{t > 0: e^γ_1 (B_1,t - m_1 t) = (λ_1 ℰ_1)^-1}
which is strictly positive (and finite) on the event that λ_1 ℰ_1 > 1, where we have
∫_0^∞ℐ_1(λ_1 ℰ_1 e^γ_1 (B_1,t - m_1 t))dt
d=∫_0^∞ℐ_1(exp(γ_1[(B_1,t - m_1t) - (B_1, τ_λ_1 ℰ_1^(1) - m_1τ_λ_1 ℰ_1^(1) ) ])) dt
and the integral on the RHS can be split into two parts:
* t ≥τ_λ_1 ℰ_1^(1). By strong Markov property, the process
[B_1, τ_λ_1 ℰ_1^(1) + t -m_1 ( τ_λ_1 ℰ_1^(1) + t)]
-[B_1, τ_λ_1 ℰ_1^(1)-m_1 τ_λ_1 ℰ_1^(1)], t ≥ 0
is a Brownian motion with negative drift -m_1 independent of (B_1, t - m_1t)_t ≤τ_λ_1 ℰ_1^(1).
* t ≤τ_λ_1 ℰ_1^(1): we apply <Ref> and write
( [B_1, τ_λ_1 ℰ_1^(1) - t - m_1(τ_λ_1 ℰ_1^(1) - t)]- [B_1, τ_λ_1 ℰ_1^(1) - m_1τ_λ_1 ℰ_1^(1)] )_t ∈ [0, τ_λ_1 ℰ_1^(1)]
= (ℬ_1,t^m_1)_t ∈ [0, L_λ_1 ℰ_1^(1)]
where (ℬ_1,t^m_1)_t ≥ 0 is a Brownian motion with drift m_1 conditioned to be non-negative (and independent of ℰ_1), and
L_λ_1 ℰ_1^(1) := sup{t > 0: e^γ_1 ℬ_1,t^m_1 = λ_1 ℰ_1}.
Substituting everything back to the expectation (<ref>), we get
𝔼[ℰ_0 1_{λ_1 ℰ_1 > 1}{∫_0^L_λ_1 ℰ_1^(1)ℐ_1(e^γ_1 ℬ_1,t^m_1)
dt+ ∫_0^∞ℐ_1(e^γ_1 (B_1,t - m_1 t))dt}]
which is
* uniformly bounded by π c_γ_1(m_1; ℐ_1)𝔼[ℰ_0 1_{λ_1 ℰ_1 > 1}], and
* converging to π c_γ_1(m_1; ℐ_1)𝔼[ℰ_0] as λ_1 →∞ by monotone convergence.
This gives (<ref>) and (<ref>). The proof of (<ref>) and (<ref>) is similar and omitted.
§ WEYL'S LAW AND HEAT TRACE ASYMPTOTICS
This section is devoted to the proof of <Ref>. Before we begin, let us mention that we can assume without loss of generality that diam(D) := sup_x, y ∈ D |x-y| < 1/2. This is not a problem because of the scale-invariant nature of the asymptotics in <Ref> (and hence the other results). To simplify notation, we shall also write c_γ = c_γ(Q-γ; ℐ) where ℐ(x) = xe^-x throughout this section.
The following is an outline of our proof of <Ref>, which follows a modified second moment method:
* To avoid any complication arising from the boundary, we perform several pre-processing steps in <Ref> to show that boundary contributions are irrelevant in the limit λ→∞. To certain extent such analysis is a manifestation of Kac's principle of `not feeling the boundary'.
* For γ∈ [1, 2) it is well-known that μ_γ (and related random variables) are not L^2-integrable. Inspired by <cit.>, we introduce a good event on which second moment method can be performed in the entire subcritical phase. We first establish in <Ref> that contribution from the complementary event vanishes as λ→∞, and then provide a roadmap for the remaining analysis.
* Finally, we will evaluate all the second moments by means of dominated convergence and show that they all coincide in the limit as λ→∞.
Note that the last part of the analysis makes heavy use of our Main lemma. To get a flavour of how <Ref> may be applied, it may be instructive to look at the following toy computation.
For γ∈ (0, 2), let μ_γ(dx) := e^γ X^2𝔻(x) - γ^2/2𝔼[X^2𝔻(x)^2] dx be the GMC measure associated to the log-correlated Gaussian field X^2𝔻 with covariance
𝔼[X^2𝔻(x)X^2𝔻(y)] = -log|x-y| + log 2 ∀ x, y ∈ B(0, 2).
Then for any A ⊂ B(0, 1), we have
lim_λ→∞𝔼[∫_A μ_γ(dx) ∫_0^1 du/2π uℐ(λμ_γ(B(x, √(u)))) ]
= c_γ𝔼[μ_γ(A)].
By Fubini and Cameron-Martin theorem, we start by rewriting
𝔼[∫_A μ_γ(dx) ∫_0^1 du/2π uℐ(λμ_γ(B(x, √(u)))) ]
= ∫_A dx 𝔼[ ∫_0^1 du/2π uℐ(λμ_γ(x, √(u))) ]
where
μ_γ(x, √(u))
:= ∫_B(x, √(u))e^γ X^2𝔻(z) - γ^2/2𝔼[X^2𝔻(z)^2]dz/(|x-z|/2)^γ^2.
From exact scale invariance
𝔼[X^2𝔻(x + a√(u))X^2𝔻(x + b√(u))]
= 𝔼[X^2𝔻(a)X^2𝔻(b)] - log√(u) ∀ a, b ∈ B(0, 1),
it follows (with a substitution of variable z ↔ x + √(u) z) that
μ_γ(x, √(u)) d=√(u)^2 - γ^2 e^γ B_t(u) - γ^2/2𝔼[B_t(u)^2]∫_B(0, 1)e^γ X^2𝔻(z) - γ^2/2𝔼[X^2𝔻(z)^2]dz/(|z|/2)^γ^2_=: ℰ_1
where B_t(u)∼𝒩(0, t(u)) is independent of ℰ_1 with t(u) := -log√(u). Thus
μ_γ(x, √(u)) d=ℰ_1 e^γ (B_t(u) - m t(u)) where m = Q-γ with Q = γ/2 + 2/γ.
Using the substitution u = e^-2t we have
∫_A dx 𝔼[ ∫_0^1 du/2π uℐ(λμ_γ(x, √(u))) ]
= ∫_A dx 𝔼[ ∫_0^∞dt/πℐ(λℰ_1 e^γ (B_t - m t) ) ].
If we now apply <Ref> with ℰ_0 := 1/π, then:
* our integrand is uniformly bounded in x ∈ A and λ > 0, and so we can apply dominated convergence when evaluating the limit λ→∞;
* the pointwise limit of our integrand as λ→∞ is given by c_γ = c_γ(m),
i.e. we conclude that
lim_λ→∞∫_D dx 𝔼[ ∫_0^1 du/2π uℐ(λμ_γ(x, √(u))) ]
= c_γ∫_A dx = c_γ𝔼[μ_γ(A)].
§.§ Pre-processing: removal of irrelevant contributions
To avoid any complication when we derive uniform estimates in later steps, we show that contributions from Brownian bridges with high probability of hitting the boundary ∂ D are irrelevant in the following sense.
We have
lim sup_λ→∞𝔼[∫_D μ_γ(dx) ∫_1^∞du/2π uxxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛)}]] = 0.
As ℐ(x) ≤ 1 for all x ≥ 0,
xxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛) }]
≤xxu( 𝐛_s ∈ D ∀ s ≤ u)
≤xxu(max_s ≤ u |𝐛_s - x| ≤ 1)
≤ 1 ∧2/u
by <Ref>, and hence
∫_D μ_γ(dx) ∫_1^∞du/2π uxxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛) }]
≤μ_γ(D) ∫_1^∞du/2π u2/u≤μ_γ(D)
which has finite expectation. On the other hand, since ℐ(x) → 0 as x →∞, we see that xxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛) }] → 0 almost surely for almost every x ∈ D and u ≥ 1. The claim now follows from dominated convergence.
Let us also highlight that boundary contributions are irrelevant in the following sense.
We have
lim sup_κ→ 0^+lim sup_λ→∞𝔼[∫_D 1_{d(x, ∂ D) ≤κ}μ_γ(dx) ∫_0^1du/2π uxxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛) }]] = 0.
In order to prove <Ref>, we first apply Fubini and Cameron-Martin theorem and rewrite (<ref>) as
𝔼[∫_D 1_{d(x, ∂ D) ≤κ}μ_γ(dx) ∫_0^1du/2π uxxu[ ℐ(λ F_γ(𝐛) )1_{u< τ_D(𝐛) }]]
=
∫_D 1_{d(x, ∂ D) ≤κ} R(x; D)^γ^2/2 dx
𝔼[ ∫_0^1du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D }]]
where, for any finite set 𝒮⊂ D and process 𝐩,
F_γ^𝒮(𝐩) :=
∫_0^ℓ(𝐩)
e^γ^2 ∑_z ∈𝒮 G_0^D(z, 𝐩_s)
F_γ(ds; 𝐩).
To proceed further, we need to control the expectation on the RHS of (<ref>) uniformly in λ > 0. We now demonstrate how this can be done by partitioning the probability space according to the range of the Brownian bridge 𝐛, a trick that will be used repeatedly throughout the rest of this article.
For each k ∈ℕ, let
ℋ_k
= ℋ_k(𝐛)
= {max_s≤ℓ(𝐛)|𝐛_s -ι(𝐛)|/√(ℓ(𝐛))∈ [k-1, k) }
where ℓ(𝐛) and ι(𝐛) are the duration and starting point of the Brownian bridge 𝐛 respectively. There exists some C ∈ (0, ∞), possibly dependent on γ but uniformly in x ∈ D, λ > 0 and k ∈ℕ such that
𝔼[ ∫_0^1 1_{ d(x, ∂ D) ≥ 4k√(u)}du/2π uxxu [ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k]]
≤ C 001(ℋ_k).
Let us start by interchanging the order of expectations:
𝔼[ ∫_0^1du/2π u 1_{d(x, ∂ D) ≥ 4k√(u)}xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k]]
= ∫_0^1du/2π u 1_{d(x, ∂ D) ≥ 4k√(u)}xxu[ 𝔼[ ℐ(λ F_γ^{x}(𝐛))]1_ℋ_k].
Applying Cameron–Martin to the inner expectation, we have
𝔼[ℐ(λ F_γ^{x}(𝐛))]
= 𝔼[λ F_γ^{x}(𝐛) e^-λ F_γ^{x}(𝐛)]
= ∫_0^u λ e^γ^2 G_0^D(x, 𝐛_s_1) R(𝐛_s_1; D)^γ^2/2 1_{𝐛_s_1∈ D} ds_1
×𝔼[ exp(-λ∫_0^u e^γ^2 [G_0^D(x, 𝐛_s_1) + G_0^D(𝐛_s_1, 𝐛_s_2)]F_γ(ds_2; 𝐛))].
The rest of the proof may be divided into three steps which we now explain.
Step (i): Gaussian comparison.
On the event ℋ_k, we know that the Brownian bridge (𝐛_s)_s ≤ u stays in the ball B(x, k√(u)). Furthermore, since d(x, ∂ D) ≥ 4k√(u), it follows from <Ref> that
| G_0^D(𝐛_s_1, 𝐛_s_2) - [ - log |𝐛_s_1 - 𝐛_s_2| + log R(x; D) ] | ≤ 4 .
In particular this implies
| G_0^D(x, 𝐛_s_1) - [ - log |x - 𝐛_s_1| + log R(x; D) ] | ≤ 4 (by setting s_2 = 0)
and |log R(x; D) - log R(𝐛_s_1; D)| ≤ 4 (by letting s_2 → s_1)
so that (<ref>) may be upper-bounded by
λ e^6 γ^2 R(x; D)^3γ^2/2∫_0^u 1_{𝐛_s_1∈ D }ds_1/|𝐛_s_1-x|^γ^2
×𝔼[ exp(-λ e^-10γ^2 R(x; D)^5γ^2/2∫_0^u 1_{𝐛_s_2∈ B(x, k√(u)) }e^γ h(𝐛_s_2) - γ^2/2𝔼[h(𝐛_s_2)^2]ds_2/|𝐛_s_2-x|^γ^2 |𝐛_s_1-𝐛_s_2|^γ^2)].
We would like to perform a Gaussian comparison using <Ref> with the convex function P(x) = exp(-x), replacing the Gaussian free field with an exactly scale invariant field X(·) with covariance
𝔼[X(a) X(b)] = -log |a-b| + log R(x; D) + 4 ∀ a, b ∈ B(x, k√(u)).
This field is well-defined because the above kernel is positive definite in a ball of radius at least R(x; D), whereas k√(u)≤ d(x, ∂ D)/4 ≤ R(x; D) where the last inequality follows from Koebe quarter theorem. By construction, we have
𝔼[h(a)h(b)] = G_0^D(a, b)≤𝔼[X(a)X(b)] ∀ a, b ∈ B(x, k√(u)),
and thus (<ref>) may be further upper-bounded by
λ e^6 γ^2 R(x; D)^3γ^2/2∫_0^u 1_{𝐛_s_1∈ B(x, k√(u)) }ds_1/|𝐛_s_1-x|^γ^2
×𝔼[ exp(-λ e^-10γ^2 R(x; D)^5γ^2/2∫_0^u 1_{𝐛_s_2∈ B(x, k√(u)) }e^γ X(𝐛_s_2) - γ^2/2𝔼[X(𝐛_s_2)^2] ds_2/|𝐛_s_2-x|^γ^2 |𝐛_s_1-𝐛_s_2|^γ^2)]
= e^6 γ^2𝔼[λ R(x; D)^3γ^2/2F_γ^{x}(𝐛; X)
exp(-λ e^-14γ^2 R(x; D)^3γ^2/2F_γ^{x}(𝐛; X)
)]
= e^20γ^2𝔼[ℐ(λF_γ^{x}(𝐛; X))] with λ:=λ e^-14γ^2R(x; D)^3γ^2/2
where, for any finite set 𝒮⊂ D,
F_γ^𝒮(𝐩; Y)
:=
∫_0^ℓ(𝐩) e^γ Y(𝐩_s) - γ^2/2𝔼[Y(𝐩_s)^2]ds/∏_z ∈𝒮| 𝐩_s - z |^γ^2.
Step (ii): scale invariance. Under xxu, the rescaled process
(1/√(u)(𝐛_us - x), s ≤ 1)
has the same distribution as a Brownian loop of duration 1 starting from the origin. It follows from (<ref>) and (<ref>) that
xxu[ 𝔼[ ℐ(λ F_γ^{x}(𝐛))]1_ℋ_k]
≤ e^20γ^2xxu[ 𝔼[ ℐ(λF_γ^{x}(𝐛; X)
)]1_ℋ_k]
= e^20γ^2001[ 𝔼[ ℐ(λF_γ^{x}(x + √(u)𝐛_· / u; X)
)]1_ℋ_k]
where
F_γ^{x}(x + √(u)𝐛_· / u; X)
= ∫_0^u 1_{x+√(u)𝐛_s/u∈ B(x, k√(u)) }e^γ X(x+√(u)𝐛_s/u) - γ^2/2𝔼[X(x + √(u)𝐛_s/u)^2]ds/|x + √(u)𝐛_s/u-x|^γ^2
= u ^1 - γ^2/2∫_0^1 1_{𝐛_s∈ B(0, k) }e^γ X(x+√(u)𝐛_s) - γ^2/2𝔼[X(x + √(u)𝐛_s)^2]ds/|𝐛_s|^γ^2.
Let us quickly mention that the presence of the indicator inside the integrands in (<ref>) is not exactly consistent with our definition in (<ref>) but it does not change anything. We are adopting this abuse of notation (here and elsewhere in the article) as a reminder for the reader that the corresponding random variable is analysed on the event ℋ_k.
We now want to proceed by invoking the scale invariance of X(·). For this purpose, let X(·) be a log-correlated Gaussian field on B(0,1) with covariance 𝔼[X(x_1) X(x_2)] = -log|x_1 - x_2| + 4, and B_T_x(u; k) an independent Gaussian random variable with zero mean and variance T_x(u; k) := -log(k√(u)/R(x; D)). (Note that T_x(u;k) ≥ 0 since k√(u)/R(x; D) ≤ k√(u)/d(x, ∂ D) by Koebe quarter theorem and we are working under the condition d(x, ∂ D) ≥ 4k√(u), and thus B_T_x(u;k) is well-defined.) Then
𝔼[X(x_1) X(x_2)] + 𝔼[B_T_x(u; k)^2]
= - log |x_1 - x_2| + 4 - log(k√(u)/R(x; D))
= 𝔼[X(x + k√(u) x_1) X(x + k√(u) x_2)] ∀ x_1, x_2 ∈ B(0, 1),
i.e. we have
X(x+ k√(u) ·) d=X(·) + B_T_x(u;k) on B(0, 1).
Substituting this into F_γ^{x}(x + √(u)𝐛_· / u; X), (<ref>) becomes
u^1-γ^2/2e^γ B_T_x(u; k) - γ^2/2 T_x(u; k)∫_0^1 1_{𝐛_s∈ B(0, k)}e^γX(k^-1𝐛_s) - γ^2/2𝔼[X( k^-1𝐛_s)^2]ds/|𝐛_s|^γ^2_=:F_γ(k^-1𝐛; X)
= e^γ( B_T_x(u; k) - (Q-γ) T_x(u; k))(k/R(x; D))^-(2-γ^2)F_γ(k^-1𝐛; X)
=: e^γ( B_T_x(u; k) - (Q-γ) T_x(u; k))ℰ.
where the law of ℰ = [k/R(x; D)]^-(2-γ^2)F_γ(k^-1𝐛; X) does not depend on u. Summarising all the work we have done from (<ref>) and (<ref>), we have
∫_0^1du/2π u 1_{d(x, ∂ D) ≥ 4k√(u)}xxu[ 𝔼[ ℐ(λ F_γ(𝐛))]1_ℋ_k]
≤ e^20γ^2𝔼⊗001[ ∫_0^1 du/2π u 1_{d(x, ∂ D) ≥ 4k√(u)}ℐ(λF_γ^{x}(x + √(u)𝐛_· / u; X)
)1_ℋ_k]
= e^20γ^2𝔼⊗001[∫_0^1du/2π u 1_{d(x, ∂ D) ≥ 4k√(u)}ℐ(λℰ e^γ( B_T_x(u; k) - (Q-γ) T_x(u; k))) 1_ℋ_k]
≤e^20γ^2/π∫_0^∞ dt 𝔼⊗001[ℐ(λℰ e^γ(B_t - (Q-γ)t)) 1_ℋ_k]
where (B_t)_t ≥ 0 is a Brownian motion. By <Ref>, the last expression is bounded by
e^20 γ^2 c_γ001[1_ ℋ_k]
uniformly in x ∈ D and λ > 0, which concludes the proof.
Observe that
𝔼[ ∫_0^1du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }]]
≤∑_k ≥ 1𝔼[ ∫_0^1du/2π uxxu 1_{ d(x, ∂ D) ≥ 4k√(u)}[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k]]
+ ∑_k ≥ 1𝔼[ ∫_0^1du/2π uxxu 1_{ d(x, ∂ D) ≤ 4k√(u)}[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k]].
We already saw from <Ref> that the first sum is uniformly bounded in x ∈ D and λ > 0. As for the second sum,
∑_k≥ 1𝔼[ ∫_0^1du/2π u 1_{d(x, ∂ D) ≤ 4k√(u)}xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k]]
≤∑_k≥ 1∫_[d(x, ∂ D)/4k]^2^1 du/2π uxxu(ℋ_k)
which may be further bounded, using
xxu(ℋ_k) ≤xxu(max_s ≤ u |𝐛_s - x| ≥ (k-1) √(u))
and <Ref>, by
∑_k ≥ 14/πe^-(k-1)^2/2log4k/d(x, ∂ D)≤ C (1 + log1/d(x, ∂ D))
for some C ∈ (0, ∞) uniformly in λ > 0. In other words, the integrand on the RHS of (<ref>) is bounded by some function independent of λ (and κ) that is integrable with respect to R(x; D)^γ^2/2 dx. The statement of <Ref> now follows from dominated convergence.
Let us also show that
For any fixed κ > 0, we have
lim sup_λ→∞𝔼[∫_D1_{d(x, ∂ D) ≥κ}μ_γ(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛))1_{u ≥τ_D(𝐛) }]] = 0.
Note that for x ∈ D satisfying d(x, ∂ D) ≥κ,
xxu[ ℐ(λ F_γ(𝐛))1_{u > τ_D(𝐛) }]
≤xxu( ∃ s ≤ u: 𝐛_s ∈∂ D )
≤xxu( max_s ≤ u |𝐛_s-x| ≥κ)
≤ 4 e^-κ^2/2u
by <Ref>. Therefore,
∫_D1_{d(x, ∂ D) ≥κ}μ_γ(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ (𝐛))1_{u > τ_D(𝐛) }]
≤μ_γ(D) ∫_0^1 du/u e^-κ^2/2u_<∞
which has finite first moment, and the claim follows from dominated convergence again.
§.§ Part I: L^1-estimates for bad event
We shall denote by h_r(x) the circle average of the field over ∂ B(x, r). Let us introduce the notation
𝒢_I(x)
:= { h_2^-n(x) ≤αlog (2^n) ∀ n ∈ I ∩ℕ}.
As in <cit.>, the key is to be able to work on this good event. The issue is that Gaussian comparison and scale invariance are key to computations of moments, but these do not mix well with good events (essentially, the indicator of the good event cannot be written as some convex function of the mass of the chaos). We will replace this indicator by exponentials in the L^1 computation showing that bad events do not contribute significantly to the expectation, and will need arguments in the subsequent L^2 computation.
Let α > γ. Then
lim_n →∞𝔼[∫_D 1_𝒢_[n, ∞)(x)^cμ_γ(dx) ]
= 0
and lim_n →∞lim sup_λ→∞𝔼[∫_D 1_𝒢_[n, ∞)(x)^cμ_γ(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛))]
]
= 0.
We only treat the second claim since the first one is simpler (and a similar statement was proved in <cit.>). By <Ref>, it suffices to establish the analogous result with the domain of integration in the x-integral replaced by {x: d(x, ∂ D) ≥κ} for any κ > 0.
Let us apply Fubini and Cameron-Martin again and rewrite
𝔼[∫_{d(x, ∂ D) ≥κ} 1_𝒢_[n, ∞)(x)^cμ_γ(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛) )1_ℋ_k]
]
= ∫_{d(x, ∂ D) ≥κ} R(x; D)^γ^2/2dx ∫_0^1 du/2π u𝔼[ 1_𝒢^{x}_[n, ∞)(x)^cxxu[ ℐ(λ F_γ^{x}(𝐛)) 1_ℋ_k]
]
where F_γ^{x}(𝐛) was already defined in (<ref>), and for any finite set 𝒮⊂ D
𝒢_I^𝒮(x) := { h_2^-j(x) + γ∑_z ∈𝒮𝔼[h_2^-j(x) h(z) ] ≤αlog (2^j) ∀ j ∈ I ∩ℕ}.
Since x is bounded away from ∂ D, it follows from <Ref> that there exists some constant C_κ > 0 such that
|𝔼[h_2^-j(x) h_δ(x)] + log (2^-j)| ≤ C_κ ∀δ∈ [0, 2^-j], ∀ j ≥ n.
In particular, for any β > 0 we have
1_𝒢^{x}_[n, ∞)(x)^c ≤∑_j ≥ nexp( β [ h_2^-j(x) + γ𝔼[h_2^-j(x) h(x) ] - αlog (2^j)])
≤ e^(β^2/2 + βγ )C_κ∑_j ≥ n 2^-β/2[2(α - γ ) - β]j e^β h_2^-j(x) - β^2/2𝔼[h_2^-j(x)^2]
and thus
𝔼[ 1_𝒢^{x}_[n, ∞)(x)^cℐ(λ F_γ^{x}(𝐛))]
≤ e^(β^2/2 + βγ )C_κ∑_j ≥ n 2^-β/2[2(α - γ ) - β]j𝔼[e^β h_2^-j(x) - β^2/2𝔼[h_2^-j(x)^2]ℐ(λ F_γ^{x}(𝐛))
]
= e^(β^2/2 + βγ )C_κ∑_j ≥ n 2^-β/2[2(α - γ ) - β]j𝔼[ ℐ(λ F_γ, (j, β)^{x}(𝐛))]
where
F_γ, (j,β)^{x}(𝐛)
:= ∫_0^u e^γ^2 G_0^D(x, 𝐛_s) +γβ𝔼[h_2^-j(x) h(𝐛_s)] F_γ(ds; 𝐛).
Next, let δ∈ (0, κ / 100) and consider
∫_0^1 du/2π u𝔼[
xxu[ ℐ(λ F_γ, (j,β)^{x}(𝐛)) 1_ℋ_k]
]
= ∫_0^δ^2 k^-22^-2jdu/2π u𝔼[
xxu[ ℐ(λ F_γ, (j,β)^{x}(𝐛)) 1_ℋ_k]
]
+∫_δ^2 k^-22^-2j^1 du/2 π u𝔼[
xxu[ ℐ(λ F_γ, (j,β)^{x}(𝐛)) 1_ℋ_k]
].
The second term can be easily bounded by
∫_δ^2 k^-22^-2j^1 du/2π uxxu(ℋ_k) ≤001(ℋ_k) log( k 2^j / δ).
As for the first term, since (by <Ref> again, up to a redefinition of C_κ)
|𝔼[h_2^-j(x) h(z) ] + log(2^-j)| ≤ C_κ ∀ z ∈ B(x, 2^-j)
and 𝐛_·∈ B(x, 2^-j) on the event ℋ_k (under the probability measure xxu with k√(u)≤ 2^-j), one obtains
e^-γβ C_κ F_γ^{x}(𝐛)≤ 2^γβ j F_γ, (j,β)^{x}(𝐛) ≤ e^γβ C_κF_γ^{x}(𝐛)
and hence
∫_0^δ^2 k^-22^-2jdu/2π u𝔼[
xxu[ ℐ(λ F_γ, (j,β)^{x}(𝐛)) 1_ℋ_k]
]
≤∫_0^δ^2 k^-22^-2jdu/2π ue^2γβ C_κ𝔼[
xxu[ ℐ(λ e^-γβ (C_κ + log 2^-j)F_γ^{x}(𝐛)) 1_ℋ_k]
]
≤ e^2γβ C_κ∫_0^δ^2 k^-22^-2jdu/2π u𝔼[xxu[ ℐ(λF_γ^{x}(𝐛))1_ℋ_k]],
λ := λ e^-γβ (C_κ + log 2^-j).
Since 4k√(u)≤ 4δ 2^-j≤κ, the last expression can be bounded by C 001(ℋ_k) for some C ∈ (0, ∞) uniformly in λ> 0 and for all x ∈ D satisfying d(x, ∂ D) ≥κ by <Ref>.
Combining everything together, we have
∫_0^1 du/2π u𝔼[
1_𝒢^{x}_[n, ∞)(x)^cxxu[ ℐ(λ F_γ^{x}(𝐛))]
≤∑_k ≥ 1
e^(β^2/2 + βγ )C_κ∑_j ≥ n 2^-β/2[2(α - γ ) - β]j{∫_0^δ^2 k^-22^-2jdu/2π u𝔼[
xxu[ ℐ(λ F_γ, (j,β)^{x}(𝐛)) 1_ℋ_k]
]
+∫_δ^2 k^-22^-2j^1 du/2 π u𝔼[
xxu[ ℐ(λ F_γ, (j,β)^{x}(𝐛)) 1_ℋ_k]
]
}
≤ (C + logδ^-1) e^(β^2/2 + βγ )C_κ[∑_k ≥ 1 k001(ℋ_k)]
[∑_j ≥ n j 2^-β/2[2(α - γ ) - β]j]
=: C∑_j ≥ n j 2^-β/2[2(α - γ ) - β]j
where C∈ (0, ∞) is independent of n ∈ℕ or λ > 0, uniformly for d(x, ∂ D) ≥κ. Choosing β = α - γ > 0, the above bound is summable and vanishes as n →∞ uniformly. Hence,
lim sup_n →∞lim sup_λ→∞𝔼[∫_{d(x, ∂ D) ≥κ} 1_𝒢_[n, ∞)(x)^cμ_γ(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛) )]
] =0
for any κ > 0, which concludes the proof.
Roadmap for the remaining analysis in <Ref>.
Based on all the estimates that have appeared in the current section, <Ref> can be established if we can show, for any κ > 0 and n_0 = n_0(κ) ∈ℕ sufficiently large that
lim_λ→∞𝔼[
|∫_A μ_γ^κ, n_0(dx) ∫_0^∞du/2π uxxu[ ℐ(λ F_γ(𝐛))] -c_γμ_γ^κ, n_0(A)|^2
] = 0
where μ_γ^κ, n_0(A) := ∫_A μ_γ^κ, n_0(dx) with
μ_γ^κ, n_0(dx) := 1_{d(x, ∂ D) ≥κ} 1_𝒢_[n_0, ∞)(x)μ(dx).
Expanding the second moment on the LHS of (<ref>), it suffices to verify the following claim.
For any κ > 0 and n_0 ∈ℕ such that 2^1-n_0 < κ, we have
lim_λ→∞𝔼[
μ_γ^κ, n_0(A) ∫_A μ_γ^κ, n_0(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛) )]
]
= c_γ𝔼[μ_γ^κ, n_0(A)^2]
and lim_λ→∞𝔼[
( ∫_A μ_γ^κ, n_0(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛) )]
)^2]
= c_γ^2 𝔼[μ_γ^κ, n_0(A)^2].
It is standard to check that the right hand sides of (<ref>) and (<ref>) are finite. Our approach to <Ref> will be based on a dominated convergence argument. More specifically, we shall apply Fubini/Cameron-Martin to rewrite the LHS's of (<ref>) and (<ref>) as some integrals over A × A, and then provide uniform estimates and evaluate pointwise limits for the integrands in order to conclude the desired results. The analysis of the cross term (<ref>) will be performed in <Ref>, and that of the diagonal term (<ref>) in the subsequent <Ref>.
§.§ Part II: analysis of cross term (<ref>)
As explained just now, our proof of (<ref>) starts with an application of Fubini and Cameron-Martin theorem: we have
𝔼[
∫_A × Aμ_γ^κ, n_0(dy) μ_γ^κ, n_0(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛) )]
]
= ∫_A × A 1_{d(x, ∂ D) ≥κ}1_{d(y, ∂ D) ≥κ}R(x; D)^γ^2/2R(y; D)^γ^2/2e^γ^2 G_0^D(x, y) dxdy
×𝔼[ 1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
]
where (recalling (<ref>) and (<ref>))
F_γ^{x, y}(𝐩)
= ∫_0^ℓ(𝐩) e^γ^2 [G_0^D(x, 𝐩_s) + G_0^D(y, 𝐩_s)] F_γ(ds; 𝐩)
and 𝒢_I^{x, y}(·)
= { h_2^-k(·) + γ𝔼[ h_2^-k(·) (h(x)+ h(y))]≤αlog (2^k) ∀ k ∈ I ∩ℕ}.
In order to apply dominated convergence to (<ref>) and (<ref>), we have to establish integrable upper bounds (with respect to e^γ^2 G_0^D(x, y)≍ |x-y|^-γ^2) as well as pointwise limits (as λ→∞) of the expectation on the RHS of (<ref>).
§.§.§ Uniform estimate for the cross term
Recall the assumption that diam(D) < 1/2, which in particular implies that -log|x-y| > 0 for any distinct x, y ∈ D.
Let β > 0 and n_0 ∈ℕ satisfying 2^1-n_0 < κ. Then there exists some constant C = C(κ, n_0, γ, α, β) ∈ (0, ∞) such that
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
]
≤ C (1 - log|x-y|) |x-y|^(2γ - α)β- β^2/2
uniformly in λ > 0 and x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ.
Observe that the bound (<ref>) is integrable if one chooses α sufficiently close to γ∈ (0, √(2d)) and β = 2γ - α such that (2γ - α)^2 / 2 < d.
Similar to the proof of <Ref>, we will consider
xxu[ ℐ(λ F_γ^{x, y}(𝐛))]
= ∑_k ≥ 1xxu[ ℐ(λ F_γ^{x, y}(𝐛)) 1_ℋ_k]
and split our analysis into two cases, depending on the distance between x and y.
Case 1: |x-y| ≥ 2^-n_0. Using the observation that
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_(k2^n_0+1)^-2^1du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛)) 1_ℋ_k]
]
≤∫_(k2^n_0+1)^-2^1du/2π uxxu(ℋ_k)
≤001(ℋ_k) log(k2^n_0+1)
which is summable in k, it suffices to show that the sum
∑_k ≥ 1𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^(k2^n_0+1)^-2du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛)) 1_ℋ_k]
]
is bounded with the desired uniformity in the statement of <Ref>.
Recall on the event ℋ_k (and under the probability measure xxu) that 𝐛_·∈ B(x, k√(u)) ⊂ B(x, 2^-(n_0+1)). By the continuity of the Green's function away from the diagonal, there exists some C_D(n_0) < ∞ such that
|G_0^D(y, 𝐛_s)| ≤ C_D(n_0) ∀ s ≤ u ≤ (k2^n_0+1)^-2
since |y-𝐛_s| ≥ |x-y| - |x - 𝐛_s| ≥ 2^-(n_0+1). In particular, for any u ∈ [0, (k2^n_0+1)^-2] we have
e^-γ^2 C_D(n_0) F_γ^{x}(𝐛)
≤ F_γ^{x, y}(𝐛)
≤ e^γ^2 C_D(n_0)F_γ^{x}(𝐛)
and hence
ℐ(λ F_γ^{x, y}(𝐛)) ≤ e^2γ^2 C_D(n_0)ℐ( λ F_γ^{x}(𝐛))
with λ := λ e^-γ^2 C_D(n_0). Therefore, the sum (<ref>) can be upper bounded by
e^2γ^2 C_D(n_0)∑_k ≥ 1𝔼[∫_0^(k2^n_0+1)^-2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛) ) 1_ℋ_k]
].
This may be further bounded uniformly in λ > 0 with <Ref>, which is applicable since
u ≤ (k2^n_0+1)^-2 ⇒
4k√(u)≤ 2^1-n_0 < κ≤ d(x, ∂ D).
Case 2: |x-y| < 2^-n_0. Using <Ref>, there exists some constant C_κ∈ (0, ∞) such that for any ϵ, δ > 0,
|𝔼[h_ϵ(a) h_δ(b)] + log( |a-b| ∨ϵ∨δ) | ≤ C_κ
uniformly for all a, b ∈ D bounded away from ∂ D by at least a distance of κ / 2. If we let n_0 ≤ n ∈ℕ satisfy 2^-(n+1)≤ |x-y| < 2^-n, then
𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)
⊂{ h_2^-n(x) + γ𝔼[ h_2^-n(x) (h(x) + h(y))]≤αlog (2^n_0)}
⊂{ h_2^-n(x) ≤ (α - 2γ) log (2^n) + 2 C_κ}.
In particular, for any β > 0 we have
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y) ≤exp{-β[h_2^-n(x) - (α - 2γ) log (2^n) -2 C_κ]}
= e^2β C_κ e^β (α - 2γ) log(2^n) + β^2/2𝔼[h_2^-n(x)^2] e^-β h_2^-n(x) - β^2/2𝔼[h_2^-n(x)^2]
≤C |x-y|^(2γ - α)β- β^2/2 e^-β h_2^-n(x) - β^2/2𝔼[h_2^-n(x)^2]
for some constant C = C(κ, γ, α, β) ∈ (0, ∞). Substituting this into the LHS of (<ref>) and applying Cameron-Martin theorem, we see that
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
]
≤C |x-y|^(2γ - α)β- β^2/2𝔼[∫_0^1 du/2π uxxu[ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛) )]
]
where
F_γ, (n, -β)^{x, y}(𝐛)
= ∫_0^ℓ(𝐛) e^γ^2 [G_0^D(x, 𝐛_s) + G_0^D(y, 𝐛_s)] - βγ𝔼[h(𝐛_s) h_2^-n(x)] F_γ(ds; 𝐛).
Let us consider
𝔼[∫_0^1 du/2π uxxu[ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛) )]
]
≤∑_k ≥ 1𝔼[∫_( |x-y|/4k)^2^1 du/2π uxxu[ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛) )1_ℋ_k]
]
+ ∑_k ≥ 1𝔼[∫_0^( |x-y|/4k)^2du/2π uxxu[ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛) )1_ℋ_k]
]
and show that they are bounded with the desired uniformity, from which we can conclude the proof. The first sum on the RHS is easily bounded by
∑_k ≥ 1𝔼[∫_( |x-y|/4k)^2^1 du/2π uxxu[1_ℋ_k]
]
≤∑_k ≥ 1[ -log |x-y| + log(4k) ]001(ℋ_k)
and when multiplied by |x-y|^(2γ - α)β- β^2/2 satisfies a bound of the form (<ref>). As for the second sum, note that
u ≤(|x-y|/4k)^2 ⇒ 4k√(u)≤ |x-y| < 2^-n_0 < 1/2κ≤ d(x, ∂ D),
and we would like to follow arguments similar to those in Case 1 and apply <Ref>. To do so, first observe on the event ℋ_k that
𝐛_s ∈ B(x, k√(u)) ⊂ B(x, |x-y|/ 4)
and in particular d(𝐛_s, ∂ D) ≥κ /2 for all s ≥ 0. The estimate (<ref>) then implies
|G_0^D(y, 𝐛_s) + log|y-𝐛_s| | ≤ C_κ
and |𝔼[h(𝐛_s) h_2^-n(x) ] + log (2^-n)| ≤ C_κ
for the entire duration of the Brownian bridge 𝐛. Since there exists some absolute constant C > 0 such that
max{| log |y-𝐛_s| - log |x-y| |, | log(2^-n) - log |x-y| | }≤ C,
we see (from (<ref>)) that there exists some constant C = C(κ, β, γ) ∈ (0, ∞) such that
C^-1
F_γ^{x}(𝐛)
≤ |x-y|^-γ(β-γ) F_γ, (n, -β)^{x, y}(𝐛)
≤C F_γ^{x}(𝐛).
Gathering all the work so far, we arrive at
∑_k ≥ 1𝔼[∫_0^( |x-y|/4k)^2du/2π uxxu[ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛) )1_ℋ_k]
]
≤C^2∑_k ≥ 1𝔼[∫_0^( |x-y|/4k)^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛) )1_ℋ_k]
]
where λ := λC^-1 |x-y|^γ (β-γ). This expression is uniformly bounded in λ > 0 by <Ref> and we are done.
§.§.§ Pointwise limit of the cross term
We now argue that
For any fixed n_0 ∈ℕ satisfying 2^1-n_0 < κ,
lim_λ→∞𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ℐ(λ F_γ^{x, y}(𝐛) ) ]
]
= c_γℙ(𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y))
for any distinct points x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ and -log_2|x-y| ∉ℕ.
The proof of the above lemma relies on a similar claim with an extra cutoff:
Under the same setting as <Ref>, for any integer m > 3 + max(n_0, -log_2 |x-y|) sufficiently large,
lim_λ→∞𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)∫_0^1 du/2π uxxu[ℐ(λ F_γ^{x, y}(𝐛) ) ]
]
= c_γℙ(𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)).
Let us fix some δ∈ (0, 2^-m) sufficiently small, and for each k ∈ℕ define
I_k := 𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)∫_0^(δ/k)^2du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛)) 1_ℋ_k]
],
and
I_k^c := 𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)∫_(δ/k)^2^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛)) 1_ℋ_k]
].
Our goal is to show that
lim_λ→∞∑_k ≥ 1 I_k^c = 0
and lim_λ→∞∑_k ≥ 1 I_k = c_γℙ(𝒢_[n_0, m)^{x, y}(x) ∩𝒢_[n_0, m)^{x, y}(y)).
Bounding the residual terms I_k^c. Using <Ref>,
I_k^c ≤∫_(δ/k)^2^1 du/2π uxxu(ℋ_k)
≤ -2e^-1/2(k-1)^2log(δ/k)
which is summable in k ∈ℕ uniformly in λ > 0. Arguing as before using the fact that 1 ≥ℐ(λ F_γ^{x, y}(𝐛)) → 0 as λ→∞, we obtain lim_λ→∞ I_k^c = 0 and lim_λ→∞∑_k ≥ 1 I_k^c = 0 by two applications of dominated convergence.
Gaussian comparison.
We now treat the main term I_k. By a change of variable, recall
F_γ^{x, y}(𝐛)
= ∫_0^u e^γ^2 [G_0^D(x, 𝐛_s) + G_0^D(y, 𝐛_s)] F_γ(ds; 𝐛)
= u ∫_0^1 e^γ^2 [G_0^D(x, 𝐛_s/u) + G_0^D(y, 𝐛_s/u)] F_γ(ds; 𝐛_· / u)
= uF_γ^{x, y}(𝐛_· / u).
Writing everything in terms of standardised Brownian bridge, we have
xxu[ ℐ(λ F_γ^{x, y}(𝐛) ) 1_ℋ_k]
= 001[ℐ(λ u F_γ^{x, y}(x+√(u)𝐛)) 1_ℋ_k]
and hence
I_k := 𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k∫_0^(δ/k)^2du/2π uℐ(λ u F_γ^{x, y}(x+√(u)𝐛))
].
Set η = 4·2^-m< |x-y|/2∧κ/2 so that the balls B(x, η), B(y, η) are disjoint and contained in our domain D. Since 0 < -log_2|x-y| ∉ℕ, there exists some d_x, y∈ℕ such that 2^-d_x, y < |x-y| < 2^-d_x, y+1, and it is possible to pick m sufficiently large so that
|x-y| - η > 2^-d_x, y and |x-y| + η < 2^-d_x, y + 1.
We apply the domain Markov property of Gaussian free field on B(x, η) ∪ B(y,η) and perform the decomposition
h(·) = h(·) + h^x, η(·) + h^y, η(·)
where
* h^x, η and h^y, η are Gaussian free fields on B(x, η) and B(y, η) respectively with Dirichlet boundary conditions,
* h(·) is the harmonic extension of h to B(x, η) ∪ B(y, η),
and all these three objects are independent of each other. Let us further perform a radial-lateral decomposition of the Gaussian free field
h^x, η(·) = h^x, rad(·)+ h^x, lat(·)
where
𝔼[h^x, rad(a) h^x, rad(b)]
= -log|a-x| ∨ |b-x|/η,
𝔼[h^x, lat(a) h^x, lat(b)]
= G_0^𝔻(a-x/η, b-x/η) - 𝔼[h^x, rad(a) h^x, rad(b)].
We now clarify the choice of δ∈ (0, 2^-m), assuming that it is sufficiently small such that
|G_0^𝔻(a-x/η, b-x/η) +log|a-b/η| |
≤δ ∀ a, b ∈ B(x, δ)
as well as
| G_0^D(x, y) - G_0^D(z, y) | ≤δ and | log R(x; D) - log R(z; D) | ≤δ
for all z ∈ B(x, δ) (this is possible by <Ref>). If we write
ℰ_x(δ):= sup_z ∈ B(x, δ) | h(z) - h(x)|,
e_x(δ):= sup_z ∈ B(x, δ) | 𝔼[h(z)^2 - h(x)^2]|,
then for any √(u)≤δ/ k we have
F_γ^{x, y}(x+√(u)𝐛)
≤ e^5 γ^2/2δ + γℰ_x(δ) + γ^2/2 e_x(δ) R(x; D)^3γ^2/2 e^γ^2 G_0^D(x, y) e^γh(x) - γ^2/2𝔼[h(x)^2]
×∫_0^1 e^γ h^x, η(x + √(u)𝐛_s) - γ^2/2𝔼[h^x, η(x + √(u)𝐛_s)^2]ds/|√(u)𝐛_s|^γ^2,
≥[e^5 γ^2/2δ + γℰ_x(δ) + γ^2/2 e_x(δ)]^-1 R(x; D)^3γ^2/2 e^γ^2 G_0^D(x, y) e^γh(x) - γ^2/2𝔼[h(x)^2]
×∫_0^1 e^γ h^x, η(x + √(u)𝐛_s) - γ^2/2𝔼[h^x, η(x + √(u)𝐛_s)^2]ds/|√(u)𝐛_s|^γ^2
and thus
ℐ(λ u F_γ^{x, y}(x+√(u)𝐛))
≤ E_x(δ)^-2ℐ(λ E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, η(·) + h(x)) )
where
λ := λ R(x; D)^3γ^2/2
e^γ^2 G_0^D(x, y),
E_x(δ)
:= [e^5 γ^2/2δ + γℰ_x(δ) + γ^2/2 e_x(δ)]^-1,
and F_γ^{x}(·; ·) was defined in (<ref>). Substituting everything back into (<ref>), we obtain
I_k ≤∫_0^(δ / k)^2du/2π u𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k
E_x(δ)^-2
×ℐ(λ E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, η(·) + h(x)) )
].
We shall perform a (conditional) Gaussian comparison, replacing the lateral field h^x, lat associated with h^x, η
by the field
𝔼[X(z_1) X(z_2)] = log|z_1-x| ∨ |z_2-x|/|z_1-z_2| ∀ z_1, z_2 ∈ B(x, δ).
Note that this replacement is possible because h^x, lat is independent of 𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y). To see why this is the case, let us go back to the decomposition (<ref>) and consider
h_r(x) = h_r(x) + h^x, η_r(x) + h_r^y, η(x)
where the subscript r refers to averaging over the circle ∂ B(x, r):
* Given the condition (<ref>) on our choice of m and η, we have ∂ B(x, 2^-j) ∩ B(y, η) = ∅ for all j ∈ℕ (see <Ref>). This means h_2^-j^y, η(x) = 0 for all j ∈ [n_0, ∞) ∩ℕ. On the other hand, h_r^x, η(x) = h^x, rad(x + r) is independent of h^x, lat by the definition of radial-lateral decomposition. Hence h_2^-j(x) = h_2^-j(x) + h^x, rad(x + 2^-j) for any j ∈ℕ, i.e. h^x, lat is independent of 𝒢^{x, y}_[n_0, m)(x).
* Similarly, ∂ B(y, 2^-j) ∩ B(x, η) = ∅ for all j ∈ℕ means that h^x, η (and in particular h^x, lat) is independent of the circle average of h centred at y at all dyadic scales, and is therefore independent of 𝒢^{x, y}_[n_0, m)(y).
We also have (by <Ref>)
|𝔼[h^x, lat(z_1) h^x, lat(z_2)] - 𝔼[X(z_1) X(z_2)] |
= |G_0^𝔻(z_1-x/η, z_2-x/η) +log|z_1-z_2/η| |
≤ 20√(u)
for all z_1, z_2 ∈ B(x, k√(u)) with u ∈ [0, (δ/k)^2] for δ sufficiently small. As a result, if we consider
F_γ^{x}(x+√(u)𝐛; h^x, rad + X + h(x))
:= e^γh(x) - γ^2/2𝔼[h(x)^2]
×∫_0^1 e^γ h^x, rad(x + √(u)𝐛_s) - γ^2/2𝔼[h^x, rad(x + √(u)𝐛_s)^2]e^γX(x+√(u)𝐛_s) - γ^2/2𝔼[X(x+√(u)𝐛_s)^2]ds/| √(u)𝐛_s|^γ^2,
then <Ref> combined with the fact that
|x^2 ∂^2/∂ x^2ℐ(λ x)|
≤ e^-λ x[2(λ x)^2 + |λ x|^3] ≤ 40 ∀λ, x ≥ 0,
implies
| ∫_0^(δ/k)^2du/2π u𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k
E_x(δ)^-2
×ℐ(λ E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, η(·) + h(x)) )
]
- ∫_0^(δ/k)^2du/2π u𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k
E_x(δ)^-2
×ℐ(λ E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, rad + X + h(x)) )
]|
≤𝔼[E_x(δ)^-2]001(ℋ_k)∫_0^(δ/k)^2du/2π u20 √(u)/2· 40 ≤ 400 𝔼[E_x(δ)^-2] δ/k e^-1/2(k-1)^2
which is summable in k uniformly in λ > 0. This gives rise to a negligible contribution as we send δ→ 0 towards the end of the proof.
Uniform control and identifying the limit.
Let us examine the Gaussian fields appearing in the definition of F_γ^{x}(x+√(u)𝐛; h^x, rad + X + h(x)). Observe that
h^x, rad(x + δ e^-t) - h^x, rad(x + δ), t ≥ 0
is a Brownian motion independent of h^x, rad(x + δ). In particular, the field
h(z):= [h^x, rad(z) - h^x, rad(x + δ)] + X(z), z ∈ B(x, δ)
is independent of 𝒢_[n_0, m]^{x, y}(x) and 𝒢_[n_0, m]^{x, y}(y), and is furthermore exactly scale invariant with covariance
𝔼[h(z_1) h(z_2)]
= -log|z_1-z_2| +log (δ)
= -log|z_1-z_2/k√(u)| - log(k√(u) / δ)
∀ z_1, z_2 ∈ B(x, k√(u)).
We can then apply spatial rescaling and obtain
∫_0^1 e^γ h^x, rad(x + √(u)𝐛_s) - γ^2/2𝔼[h^x, rad(x + √(u)𝐛_s)^2]e^γX(x+√(u)𝐛_s) - γ^2/2𝔼[X(x+√(u)𝐛_s)^2]ds/| 𝐛_s|^γ^2
= e^γ h^x, rad(x+δ) - γ^2/2𝔼[h^x, rad(x+δ)^2]∫_0^1 e^γh(x + √(u)𝐛_s) - γ^2/2𝔼[h(x + √(u)𝐛_s)^2]ds/|𝐛_s|^γ^2
d=e^γ h^x, rad(x+δ) - γ^2/2𝔼[h^x, rad(x+δ)^2] e^γ B_T - γ^2/2 TF_γ^{0}(k^-1𝐛; X^𝔻)
where
* F_γ^{0}(k^-1𝐛; X^𝔻) = ∫_0^1 |𝐛_s|^-γ^2 e^γ X^𝔻(k^-1𝐛_s) - γ^2/2𝔼[X^𝔻(k^-1𝐛_s)^2]ds, with X^𝔻 being the Gaussian field on the unit disc 𝔻 satisfying 𝔼[X^𝔻(z_1) X^𝔻(z_2)] = -log|z_1 - z_2|;
* T = T(u; k, δ) = -log (k√(u)/δ) and B_T is an independent 𝒩(0, T) random variable.
Using the fact that h(x) + h^x, rad(x+δ) = h_δ(x) + h_δ^x, η(x) = h_δ(x), we have
u F_γ^{x}(x+√(u)𝐛; h^x, rad + X + h(x))
d= (δ / k)^2 - γ^2 (k√(u)/δ)^2-γ^2e^γh(x) - γ^2/2𝔼[h(x)^2]
× e^γ h^x, rad(x+δ) - γ^2/2𝔼[h^x, rad(x+δ)^2]
e^γ B_T - γ^2/2 TF_γ^{0}(k^-1𝐛; X^𝔻)
= e^γ (B_T - (Q-γ)T)
(δ / k)^2 - γ^2 e^γ h_δ(x) - γ^2/2𝔼[h_δ(x)^2]F_γ^{0}(k^-1𝐛; X^𝔻)
=: e^γ (B_T - (Q-γ)T)ℛ_x.
Substituting everything back to our main expression, and doing the change of variable k√(u) / δ = e^-t, we obtain
∫_0^(δ/k)^2du/2π u𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k
E_x(δ)^-2
×ℐ(λ E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, rad + X + h(x)) )
]
= 1/π∫_0^∞𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k
E_x(δ)^-2ℐ(λ E_x(δ)ℛ_xe^γ (B_t - (Q-γ)t))
] dt
where (B_t)_t ≥ 0 is a Brownian motion independent of everything else.
Using <Ref>, we see that (<ref>) is uniformly bounded by
c_γ𝔼⊗001[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
1_ℋ_k
E_x(δ)^-2]
= π c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^-2]
001( ℋ_k)
≤ C e^-1/2(k-1)^2
for some C ∈ (0, ∞) independent of k ∈ℕ, and this is summable in k. Moreover, the same lemma suggests that (<ref>) converges, as λ→∞, to
c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^-2]001( ℋ_k).
Combining these with (<ref>) and (<ref>), we have
lim inf_λ→∞∑_k ≥ 1 I_k
≤∑_k ≥ 1 c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^-2]001( ℋ_k)
+ ∑_k ≥ 1 400 𝔼[E_x(δ)^-2] δ/k e^-1/2(k-1)^2
= c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^-2] + 𝒪(δ).
Now, recall that E_x(δ)^-2 is non-negative, non-increasing in δ, has finite moments and E_x(δ) 1 almost surely. Since δ > 0 is arbitrary in our analysis, it follows from monotone convergence that
lim inf_λ→∞𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛))]]
≤lim_δ→ 0^+ c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^-2]
= c_γℙ(𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)).
A matching lower bound can be obtained in a similar fashion, by noting that
ℐ(λ F_γ^{x, y}(𝐛))
≥ E_x(δ)^2ℐ(λ E_x(δ)^-1
u F_γ^{x}(x+√(u)𝐛; h^x, η(·) + h(x)) )
(cf. (<ref>)) so that
lim sup_λ→∞∑_k ≥ 1 I_k
≥ c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^2] + 𝒪(δ).
and therefore
lim sup_λ→∞𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛))]
]
≥lim_δ→ 0^+ c_γ𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^2]
=c_γℙ(𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)).
This completes the proof of <Ref>.
In order to obtain the desired result, we need to send the cutoff parameter m →∞ in (<ref>). In particular, it suffices to show that
lim_m →∞lim sup_λ→∞𝔼[
1_𝒢^{x, y}_[m, ∞)(p)^c∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛))]
]
= 0
for p ∈{x, y} and any fixed and distinct x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ.
Our first step is to establish a bound on the indicator function 1_𝒢^{x, y}_[m, ∞)(p) by adapting the argument in (<ref>). Let us assume without loss of generality that m ∈ℕ is sufficiently large so that 2^-m < |x-y|. Since x and y are bounded away from ∂ D, there exists some constant C_κ > 0 such that
|𝔼[h_2^-n(p) h_δ(p')] - log(2^-n∨ |p-p'|)| ≤ C_κ ∀δ∈ [0, 2^-n], ∀ n ≥ m
for any p, p' ∈{x, y} by <Ref>. Recalling
𝒢^{x, y}_[m, ∞)(p)
:= { h_2^-n(p) + γ𝔼[h_2^-n(p) (h(x) + h(y))] ≤αlog(2^n) ∀ n ∈ [m, ∞) ∩ℕ},
it holds for any β > 0 that
1_𝒢^{x, y}_[m, ∞)(p)^c ≤∑_n ≥ mexp( β[h_2^-n(p) + γ𝔼[h_2^-n(p) (h(x) + h(y))] - αlog (2^n)])
≤e^(β^2/2 + 2βγ )C_κ/|x-y|^γ∑_n ≥ m 2^-β/2[2(α - γ ) - β]n e^β h_2^-n(p) - β^2/2𝔼[h_2^-n(p)^2]
Using this bound, we obtain by Cameron-Martin theorem that
𝔼[
1_𝒢^{x, y}_[m, ∞)(p)^c∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛))]
]
≤e^(β^2/2 + 2βγ )C_κ/|x-y|^γ∑_n ≥ m 2^-β/2[2(α - γ ) - β]n𝔼[
e^β h_2^-n(p) - β^2/2𝔼[h_2^-n(p)^2]∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛))]
]
= e^(β^2/2 + 2βγ )C_κ/|x-y|^γ∑_n ≥ m 2^-β/2[2(α - γ ) - β]n𝔼[
∫_0^1 du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛))]
]
where
F_γ, (p, n, β)^{x, y}(𝐛)
:= ∫_0^u e^γ^2 [G_0^D(x, 𝐛_s) + G_0^D(y, 𝐛_s)] +γβ𝔼[h_2^-n(p) h(𝐛_s)] F_γ(ds; 𝐛).
Let us now fix δ∈ (0, 1/8min(|x-y|, κ)), and split the sum in (<ref>) (without the prefactor) into
∑_n ≥ m 2^-β/2[2(α - γ ) - β]n∑_k ≥ 1𝔼[
∫_(2^-nδ/k)^2^1 du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛))1_ℋ_k]
]
+ ∑_n ≥ m 2^-β/2[2(α - γ ) - β]n∑_k ≥ 1𝔼[
∫_0^(2^-nδ/k)^2du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛))1_ℋ_k]
].
The first double sum is easily bounded by
∑_n ≥ m 2^-β/2[2(α - γ ) - β]n∑_k ≥ 1001(ℋ_k) log (2^n k / δ) ≲ m 2^-β/2[2(α - γ ) - β]m
uniformly in λ > 0 and vanishes as m →∞ provided that β∈ (0, 2(α - γ)). To treat the remaining double sum in (<ref>), recall for each k, n ∈ℕ and on the event ℋ_k that the Brownian bridge 𝐛 satisfies (under the probability measure xxu)
𝐛_s ∈ B(x, k√(u)) ⊂ B(x, 2^-nδ)
and in particular |𝐛_s - y| ≥δ and d(𝐛_s, ∂ D) ≥κ/2 for all s ≤ u ≤ (2^-nδ / k)^2.
and observe that B(x, δ) ∩ B(y, δ) = ∅ by our choice of δ. We may therefore assume (up to a re-definition) that the constant C_κ in (<ref>) also satisfies
|𝔼[h_2^-n(x) h(𝐛_s)] - log(2^-n)| ≤ C_κ and |𝔼[h_2^-n(y) h(𝐛_s)] | ≤ C_κ
for all n ≥ m and any s. This means, in particular, that
e^-γ(γ+β)C_κ≤F_γ, (x,n, β)^{x, y}(𝐛)/2^γβ n F_γ^{x}(𝐛)≤ e^γ(γ+β)C_κ and
e^-γ(γ+β)C_κ≤F_γ, (y,n, β)^{x, y}(𝐛)/F_γ^{x}(𝐛)≤ e^γ(γ+β)C_κ
and hence
𝔼[
∫_0^(2^-nδ/k)^2du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛))1_ℋ_k]
]
≤
e^2γ(γ+β)C_κ×𝔼[
∫_0^(2^-nδ/k)^2du/2π uxxu[ ℐ(λ e^-γ(γ+β)C_κ 2^γβ n F_γ^{x}(𝐛))1_ℋ_k]
] for p=x,
𝔼[
∫_0^(2^-nδ/k)^2du/2π uxxu[ ℐ(λ e^-γ(γ+β)C_κ F_γ^{x}(𝐛))1_ℋ_k]
] for p=y.
In either case this can be further upper bounded by C 001(ℋ_k) uniformly in n, k ∈ℕ and λ > 0 by <Ref> (as d(x, ∂ D) ≥κ≥ 4k √(u) is automatically satisfied). Substituting this back to the second sum in(<ref>), we see that
∑_n ≥ m 2^-β/2[2(α - γ ) - β]n∑_k ≥ 1𝔼[
∫_0^(2^-nδ/k)^2du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛))1_ℋ_k]
]
≲∑_n ≥ m 2^-β/2[2(α - γ ) - β]n∑_k ≥ 1001(ℋ_k)
≲ 2^-β/2[2(α - γ ) - β]m 0
which concludes our proof of (<ref>).
§.§ Part III: analysis of the diagonal term (<ref>)
We now consider the diagonal term
𝔼[
( ∫_A μ_γ^κ, n_0(dx) ∫_0^1 du/2π uxxu[ ℐ(λ F_γ(𝐛) )]
)^2]
= ∫_A × A 1_{d(x, ∂ D) ≥κ}1_{d(y, ∂ D) ≥κ}R(x; D)^γ^2/2R(y; D)^γ^2/2e^γ^2 G_0^D(x, y) dxdy
×𝔼[ 1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
∫_0^1 dv/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃) )]
]
where 𝐛 and 𝐛̃ are two independent Brownian bridges distributed according to xxu and yyv respectively.
§.§.§ Uniform estimates for the diagonal term
Let β > 0 and n_0 ∈ℕ satisfying 2^1-n_0 < κ. Then there exists some constant C = C(κ, n_0, γ, α, β) ∈ (0, ∞) such that
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
∫_0^1 dv/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃)) ]
]
≤ C [1 - log |x-y|]^2 |x-y|^(2γ - α)β- β^2/2
uniformly in λ > 0 and x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ.
As we saw earlier, the proof of <Ref> relies on <Ref>. The “two-point" analogue of this estimate is as follows.
Denote by c(x,y) = c(x,y; κ):= 1/8min(|x-y|, κ). There exists some C = C(γ, κ) ∈ (0, ∞) such that
𝔼[ (∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ_1 F_γ^{x}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x,y)^2dv/2π vyyv [ ℐ(λ_2 F_γ^{y}(𝐛))1_ℋ_k(𝐛)])
]
≤ C 001(ℋ_j)001(ℋ_k)
uniformly in λ_1, λ_2 > 0, j, k ∈ℕ, and x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ.
By Fubini, we rewrite the LHS of (<ref>) as
∫_0^j^-2 c(x, y)^2∫_0^k^-2 c(x,y)^2du/2π udv/2π v
×xxu⊗yyv[𝔼[ℐ(λ_1 F_γ^{x}(𝐛))ℐ(λ_2 F_γ^{y}(𝐛̃))]
1_ℋ_j(𝐛) 1_ℋ_k(𝐛̃)]
where 𝐛 and 𝐛̃ are two independent Brownian bridges distributed according to xxu and yyv respectively. The rest of our analysis will be divided into two steps, mirroring the structure of the proof of <Ref>.
Step (i): Gaussian comparison. We want to derive a two-point analogue of the bound (<ref>), i.e. for max(j√(u), k√(v)) ≤ c(x,y) and on the event ℋ_j(𝐛) ∩ℋ_k(𝐛̃), we aim to establish an inequality of the form
𝔼[ℐ(λ_1 F_γ^{x}(𝐛))ℐ(λ_2 F_γ^{y}(𝐛̃))]
≤ C 𝔼[ℐ(λ_1
F_γ^{x}(𝐛; X))ℐ(λ_2 F_γ^{y}(𝐛̃; X))]
where X is some Gaussian field which shall be defined in (<ref>), and λ_1, λ_2 > 0 and C ∈ (0, ∞) will be suitably chosen in (<ref>) and (<ref>) respectively. For now, we just emphasise that the constant C on the RHS of (<ref>) will be independent of λ_1, λ_2 and satisfy the desired uniformity in j, k ∈ℕ and x, y ∈ D as described in the statement of <Ref>.
To establish an inequality of the form (<ref>), we begin by applying Cameron-Martin to the LHS of (<ref>) and rewrite
𝔼[ℐ(λ_1 F_γ^{x}(𝐛))ℐ(λ_2 F_γ^{y}(𝐛̃))]
= 𝔼[λ_1 λ_2 F_γ^{x}(𝐛)F_γ^{y}(𝐛̃) e^-λ_1F_γ^{x}(𝐛) - λ_2F_γ^{y}(𝐛̃)]
= λ_1 λ_2 ∫_0^u ds_1 ∫_0^v dt_1 e^γ^2 [G_0^D(x, 𝐛_s_1) + G_0^D(y, 𝐛̃_t_1) + G_0^D(𝐛_s_1, 𝐛̃_t_1)] R(𝐛_s_1; D)^γ^2/2
R(𝐛̃_t_1; D)^γ^2/2
×𝔼[
exp(-λ_1 ∫_0^u e^γ^2 [G_0^D(x, 𝐛_s_2) + G_0^D(𝐛_s_1, 𝐛_s_2)
+ G_0^D(𝐛̃_t_1, 𝐛_s_2)]F_γ(ds_2; 𝐛)
- λ_2 ∫_0^v e^γ^2 [G_0^D(y, 𝐛̃_t_2) + G_0^D(𝐛̃_t_1, 𝐛̃_t_2)
+ G_0^D(𝐛_s_1, 𝐛̃_t_2)]F_γ(dt_2; 𝐛̃)
)
].
Since max(j√(u), k √(v)) ≤ c(x,y), we have 𝐛_s ∈ B(x, j√(u)) ⊂ B(x, c(x,y)) and 𝐛̃_t ∈ B(y, k√(v)) ⊂ B(y, c(x,y)) on the event ℋ_j(𝐛) ∩ℋ_k(𝐛̃). Moreover:
* The following estimates apply for all s_1, s_2 ≤ u and t_1, t_2 ≤ v from <Ref>:
| G_0^D(𝐛_s_1, 𝐛_s_2) - [ - log |𝐛_s_1 - 𝐛_s_2| + log R(x; D) ] | ≤ 4,
| G_0^D(𝐛̃_t_1, 𝐛̃_t_2) - [ - log |𝐛̃_t_1 - 𝐛̃_t_2| + log R(y; D) ] | ≤ 4.
Let us recall again that these inequalities above imply, in particular, that
[ | G_0^D(x, 𝐛_s_1) - [ - log |x - 𝐛_s_1| + log R(x; D) ] | ≤ 4; | G_0^D(y, 𝐛̃_t_1) - [ - log |y - 𝐛̃_t_1| + log R(y; D) ] |≤ 4 ] (by setting s_2, t_2 = 0)
as well as
[ |log R(𝐛_s_1; D) - log R(x; D)|≤ 4; |log R(𝐛̃_t_1; D) - log R(y; D)|≤ 4 ] (by letting s_2 → s_1, t_2 → t_1)
for all s_1 ≤ u and t_1 ≤ v.
* We have d(a, ∂ D) ∧ d(b, ∂ D) ≥κ/2 for all a ∈ B(x, j√(u)) and b ∈ B(x, k√(v)). This allows us to apply the estimate (<ref>) several times below; in particular,
| G_0^D(𝐛_s, 𝐛̃_t) + log |𝐛_s - 𝐛̃_t| | ≤ C_κ ∀ s ≤ u, t ≤ v.
* By definition, we also have
c(x, y) ≤|x-y|/8≤ |𝐛_s - 𝐛̃_t| ≤ 2|x-y| ∀ s ≤ u, t ≤ v.
Combining all these estimates, we can upper bound (<ref>) with
λ_1 λ_2 ∫_0^u ds_1 ∫_0^v dt_1 e^(12 + C_κ) γ^2 R(x; D)^3γ^2/2 R(y; D)^3γ^2/2/|x - 𝐛_s_1|^γ^2|y - 𝐛̃_t_1|^γ^2 c(x, y)^γ^2
×𝔼[
exp(-λ_1 2^-γ^2 e^-(10 + C_κ) γ^2R(x; D)^5γ^2/2/|x-y|^γ^2∫_0^u e^γ h(𝐛_s_2) - γ^2/2𝔼[h(𝐛_s_2)^2]ds_2/|x-𝐛_s_2|^γ^2 |𝐛_s_1 - 𝐛_s_2|^γ^2
-λ_2 2^-γ^2 e^-(10 + C_κ) γ^2R(y; D)^5γ^2/2/|x-y|^γ^2∫_0^v e^γ h(𝐛̃_t_2) - γ^2/2𝔼[h(𝐛̃_t_2)^2]dt_2/|y-𝐛̃_t_2|^γ^2 |𝐛̃_t_1 - 𝐛̃_t_2|^γ^2)
].
Let us now introduce a new (centred) Gaussian field X(·) = X(·; κ) on B(x, c(x,y)) ∪ B(y, c(x, y)) with covariance
𝔼[X(a) X(b)]
= 𝔼[X_x(a) X_x(b)] 1_{a, b ∈ B(x, c(x, y))}
+ 𝔼[X_y(a) X_y(b)]1_{a, b ∈ B(y, c(x,y))}
+ 𝔼[N_x, y^2]
where
* X_x (·) and X_y(·) are two independent exactly scale invariant Gaussian fields on the two balls B(x, c(x,y)) and B(y, c(x,y)) respectively, and both of their covariance kernels are of the form
(a, b) ↦ -log |a-b| + log c(x,y);
* N_x, y is an independent Gaussian random variable with zero mean and variance equal to C_κ - log c(x, y).
(The fact that X_x and X_y exist follows from the fact that the kernel (a, b) ↦ -log |a-b| is positive definite on the unit ball in dimension 2.) By construction, we see that
𝔼[X(a) X(b)]
=
- log |a-b| + C_κ if a, b belong to the same ball
- log c(|x-y|) + C_κ otherwise
≥ - log|a-b| + C_κ
≥ G_0^D(a, b)
∀ a, b ∈ B(x, c(x, y) ∪ B(y, c(x,y))
where the last inequality follows from the definition of C_κ in (<ref>) (sending ϵ, δ to 0). Therefore, by Gaussian comparison we further upper bound (<ref>) by
λ_1 λ_2 ∫_0^u ds_1 ∫_0^v dt_1 e^(12 + C_κ) γ^2 R(x; D)^3γ^2/2 R(y; D)^3γ^2/2/|x - 𝐛_s_1|^γ^2|y - 𝐛̃_t_1|^γ^2 c(x,y)^γ^2
×𝔼[
exp(-λ_1 2^-γ^2 e^-(10 + C_κ) γ^2R(x; D)^5γ^2/2/|x-y|^γ^2∫_0^u e^γ X(𝐛_s_2) - γ^2/2𝔼[X(𝐛_s_2)^2]ds_2/|x-𝐛_s_2|^γ^2 |𝐛_s_1 - 𝐛_s_2|^γ^2
-λ_2 2^-γ^2 e^-(10 + C_κ) γ^2R(y; D)^5γ^2/2/|x-y|^γ^2∫_0^v e^γ X(𝐛̃_t_2) - γ^2/2𝔼[X(𝐛̃_t_2)^2]dt_2/|y-𝐛̃_t_2|^γ^2 |𝐛̃_t_1 - 𝐛̃_t_2|^γ^2)
].
Finally, recall the RHS of (<ref>): by Cameron-Martin we have
C 𝔼[ℐ(λ_1
F_γ^{x}(𝐛; X))ℐ(λ_2 F_γ^{y}(𝐛̃; X))]
= C λ_1λ_2
∫_0^u ds_1 ∫_0^v dt_1 e^γ^2C_κ/|x-𝐛_s_1|^γ^2|y-𝐛̃_t_1|^γ^2c(x,y)^γ^2
×𝔼[
exp(-λ_1 e^2γ^2 C_κ/c(x,y)^γ^2∫_0^u e^γ X(𝐛_s_2) - γ^2/2𝔼[X(𝐛_s_2)^2]ds_2/|x-𝐛_s_2|^γ^2 |𝐛_s_1 - 𝐛_s_2|^γ^2
-λ_2 e^2γ^2 C_κ/c(x,y)^γ^2∫_0^v e^γ X(𝐛̃_t_2) - γ^2/2𝔼[X(𝐛̃_t_2)^2]dt_2/|y-𝐛̃_t_2|^γ^2 |𝐛̃_t_1 - 𝐛̃_t_2|^γ^2)
].
Comparing (<ref>) and (<ref>), we can now choose
λ_1
= λ_1 [ e^-10 - 3C_κc(x, y)/2|x-y|]^γ^2 R(x; D)^5γ^2/2,
λ_2
= λ_2 [ e^-10 - 3C_κc(x, y)/2|x-y|]^γ^2 R(y; D)^5γ^2/2,
and
C = [ 2|x-y|/c(x, y)]^2γ^2 e^(32 + 6C_κ)γ^2 R(x; D)^-γ^2R(y; D)^-γ^2
which can be bounded uniformly in x, y satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ. This concludes Step (i) of the proof.
Step (ii): scale invariance. For any λ_1, λ_2 > 0 and j√(u), k√(v)≤ c(x,y), we aim to establish an identity of the form
xxu⊗yyv⊗𝔼[
ℐ(λ_1
F_γ^{x}(𝐛; X))ℐ(λ_2 F_γ^{y}(𝐛̃; X))1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)]
=
001^⊗ 2⊗𝔼[
ℐ(λ_1 ℰ_x(𝐛) e^γ (B_1, T_1(u) - (Q-γ) T_1(u)))
ℐ(λ_2 ℰ_y(𝐛̃)e^γ (B_2, T_2(v) - (Q-γ) T_2(v)))1_ℋ_j(𝐛)∩ℋ_k(𝐛̃)]
where B_1, T_1(u)∼𝒩(0, T_1(u)) and B_2, T_2(v)∼𝒩(0, T_2(v)) are two random variables independent of each other and everything else (including the random variables ℰ_x(𝐛) and ℰ_y(𝐛̃) which will be specified later), with
T_1(u) := -log(j √(u)/ c(x, y))
and
T_2(v) := -log(k √(v)/c(x,y))
which are non-negative for the range of values of (u, v) under consideration.
To commence with, let us recall the definition of the field X(·) in (<ref>). On the event ℋ_j(𝐛)∩ℋ_k(𝐛̃), we have
F_γ^{x}(𝐛; X)
= F_γ^{x}(𝐛; X_x(·) + N_x,y)
and F_γ^{y}(𝐛̃; X)
= F_γ^{y}(𝐛̃; X_y(·) + N_x,y).
Let us standardise our Brownian loops just like what was done in (<ref>); in other words, we rewrite
xxu⊗yyv⊗𝔼[
ℐ(λ_1
F_γ^{x}(𝐛; X))ℐ(λ_2 F_γ^{y}(𝐛̃; X))1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)]
= 001^⊗ 2⊗𝔼[
ℐ(λ_1F_γ^{x}(x + √(u)𝐛_· / u; X_x(·) + N_x,y))
×ℐ(λ_2 F_γ^{y}(y + √(v)𝐛̃_· / v; X_y(·) + N_x,y))1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)]
where (based on the same argument in (<ref>))
[ F_γ^{x}(x + √(u)𝐛_· / u; X_x(·) + N_x,y); F_γ^{y}(y + √(v)𝐛̃_· / v; X_y(·) + N_x,y) ]
= e^γ N_x, y - γ^2/2𝔼[N_x, y^2]×[ u^1-γ^2/2∫_0^1 1_{𝐛_s ∈ B(0, j)}e^γ X_x(x + √(u)𝐛_s)- γ^2/2𝔼[X_x(x + √(u)𝐛_s)^2]ds/|𝐛_s|^γ^2; v^1-γ^2/2∫_0^1 1_{𝐛̃_t ∈ B(0, k)}e^γ X_y(y + √(v)𝐛̃_t)- γ^2/2𝔼[X_y(y + √(v)𝐛̃_t)^2]dt/|𝐛̃_t|^γ^2 ].
Now, let X_x, X_y be two independent Gaussian fields on the unit ball with covariance kernels (a, b) ↦ -log |a-b| and recall (<ref>). Then for any a, b ∈ B(0, 1), one has the following equivalence in covariance:
𝔼[X_x(x+ j√(u) a)X_x(x+ j√(u) b)]
= 𝔼[X_x(a)X_x(b)] + 𝔼[B_1, T_1(u)^2]
and 𝔼[X_y(y+ k√(v) a)X_y(y+ k√(v) b)]
= 𝔼[X_y(a)X_y(b)] +𝔼[B_2,T_2(v)^2]
and thus
[ F_γ^{x}(x + √(u)𝐛_· / u; X_x(·) + N_x,y); F_γ^{y}(y + √(v)𝐛̃_· / v; X_y(·) + N_x,y) ]
d=
e^γ N_x, y - γ^2/2𝔼[N_x, y^2]×[ u^1-γ^2/2 e^γ B_1, T_1(u) - γ^2/2 T_1(u)∫_0^1 1_{𝐛_s ∈ B(0, j)}e^γX_x(𝐛_s)- γ^2/2𝔼[X_x(𝐛_s)^2]ds/|𝐛_s|^γ^2; v^1-γ^2/2 e^γ B_2, T_2(v) - γ^2/2 T_2(v)∫_0^1 1_{𝐛̃_t ∈ B(0, k)}e^γX_y(𝐛̃_t)- γ^2/2𝔼[X_y(𝐛̃_t)^2]dt/|𝐛̃_t|^γ^2 ]
=
e^γ N_x, y - γ^2/2𝔼[N_x, y^2]×[ F_γ^{x}(𝐛; X_x) [c(x,y)/j]^2-γ^2exp(γ [B_1, T_1(u) - (Q-γ) T_1(u)] ); F_γ^{y}(𝐛̃; X_y) [c(x,y)/k]^2-γ^2exp(γ [B_2, T_2(v) - (Q-γ) T_2(v)] ) ].
Substituting this into (<ref>), we conclude that (<ref>) holds with
[ ℰ_x(𝐛); ℰ_y(𝐛̃) ]
:= e^γ N_x, y - γ^2/2𝔼[N_x, y^2]×[ F_γ^{x}(𝐛; X_x) [c(x,y)/j]^2-γ^2; F_γ^{y}(𝐛̃; X_y) [c(x,y)/k]^2-γ^2 ].
Concluding the proof of <Ref>. Combining the two claims (<ref>) and (<ref>), we see that (<ref>) is upper-bounded by
∫_0^j^-2 c(x,y)^2∫_0^k^-2 c(x,y)^2du/2π udv/2π v001^⊗ 2⊗𝔼[
ℐ(λ_1 ℰ_x(𝐛) e^γ [B_1, T_1(u) - (Q-γ) T_1(u)])
×ℐ(λ_2 ℰ_y(𝐛̃)e^γ [B_2, T_2(v) - (Q-γ) T_2(v)])1_ℋ_j(𝐛)∩ℋ_k(𝐛̃)]
up to a multiplicative constant C ∈ (0, ∞) inherited from the RHS of (<ref>).
Note that by definition, the distributions of ℰ_x(𝐛) and ℰ_y(𝐛̃) do not depend on the value of u and v. If we now consider the substitution s = T_1(u) and t = T_2(v), then (<ref>) can be further rewritten as
∫_0^∞∫_0^∞ds dt/π^2001^⊗ 2⊗𝔼[
ℐ(λ_1 ℰ_x(𝐛) e^γ B_1, s^-(Q-γ))
ℐ(λ_2 ℰ_y(𝐛̃)e^γ B_2, t^-(Q-γ))1_ℋ_j(𝐛)∩ℋ_k(𝐛̃)]
= 1/π^2001^⊗ 2⊗𝔼[
( ∫_0^∞ℐ(λ_1 ℰ_x(𝐛) e^γ B_1, s^-(Q-γ)) ds)
×(∫_0^∞ℐ(λ_2 ℰ_y(𝐛̃)e^γ B_2, t^-(Q-γ))dt)
1_ℋ_j(𝐛)∩ℋ_k(𝐛̃)]
where (B_i, t^-(Q-γ))_t ≥ 0 are two independent Brownian motions with drift -(Q-γ) < 0 that are independent of everything else. By <Ref> (or more precisely the estimate (<ref>)), we see that this expectation is bounded uniformly in λ_1, λ_2 > 0 by
[π c_γ]^2 001^⊗ 2⊗𝔼 [1_ℋ_j(𝐛)∩ℋ_k(𝐛̃)]
= [π c_γ]^2 001(ℋ_j)001(ℋ_k)
which is our desired claim (<ref>).
Recall c(x, y):= 1/8min(|x-y|, κ), and consider
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
∫_0^1 dv/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃)) ]
]
≤∑_k ≥ 1𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_k^-2 c(x,y)^2^1dv/2π vyyv(ℋ_k))
×(∫_0^1du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
]
+ ∑_j≥ 1𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_j^-2 c(x, y)^2^1du/2π uxxu(ℋ_j))
×(∫_0^1dv/2π vyyv [ ℐ(λ F_γ^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
+ ∑_j, k ≥ 1𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x, y)^2dv/2π vyyv [ ℐ(λ F_γ^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
].
The first sum on the RHS is upper bounded by
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_0^1du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
]
∑_k ≥ 1(logk/c(x, y)) 001(ℋ_k)
≲𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_0^1du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
] [1 - log c(x, y)].
Since the remaining expectation can be controlled by <Ref>, it follows that the first sum indeed satisfies a bound of the form (<ref>). The same argument applies to the second sum in (<ref>).
To conclude the proof we must show that the third sum in (<ref>) satisfies a similar bound. We now consider two cases, following arguments similar to that of the proof of <Ref>.
Case 1: |x-y| ≥ 2^-n_0.
Our goal here is to show that the third sum in (<ref>) is bounded uniformly in λ > 0. (This is enough to conclude the estimate (<ref>) as |x-y| is bounded away from 0.)
For any max(j√(u), k√(v)) ≤ c(x, y), we have on the event ℋ_j(𝐛) ∩ℋ_k(𝐛̃) that
𝐛_·∈ B(x, j √(u)) ⊂ B(x, c(x, y))
and 𝐛̃_·∈ B(y, k √(u)) ⊂ B(y, c(x,y)).
Based on the definition of c(x, y), we know that the two balls B(x, c(x, y)) and B(y, c(x,y)) are at least |x-y| / 2 ≥ 2^-(n_0 + 1) apart from each other. By the continuity of the Green's function away from the diagonal, there exists some constant C_D(n_0) < ∞ such that
max( |G_0^D(y, 𝐛_s)|, |G_0^D(x, 𝐛̃_t)|) ≤ C_D(n_0)
∀ s ≤ u, t ≤ v
and hence
ℐ(λ F_γ^{x, y}(𝐛)) ≤ e^2γ^2 C_D(n_0)ℐ( λ F_γ^{x}(𝐛))
and ℐ(λ F_γ^{x, y}(𝐛̃)) ≤ e^2γ^2 C_D(n_0)ℐ( λ F_γ^{y}(𝐛̃))
for λ := λ e^-γ^2 C_D(n_0). Putting everything back together, we have
∑_j, k ≥ 1𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x,y)^2dv/2π vyyv [ ℐ(λ F_γ^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
≤ e^4γ^2 C_D(n_0)∑_j, k ≥ 1𝔼[
(∫_0^j^-2 c(x,y)^2du/2π uxxu [ ℐ(λ F_γ^{x}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x,y)^2dv/2π vyyv [ ℐ(λ F_γ^{y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
which is bounded uniformly in λ > 0 (and hence λ > 0) by <Ref>.
Case 2: |x-y| < 2^-n_0.
Recall (<ref>) where n ≥ n_0 is chosen to be the integer satisfying 2^-(n+1)≤ |x-y| < 2^-n. We have
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)(∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x, y)^2dv/2π vyyv [ ℐ(λ F_γ^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
≲ |x-y|^(2γ - α)β- β^2/2𝔼[ e^-β h_2^-n(x) - β^2/2𝔼[h_2^-n(x)^2](∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ F_γ^{x, y}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x, y)^2dv/2π vyyv [ ℐ(λ F_γ^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
= |x-y|^(2γ - α)β- β^2/2𝔼[
(∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x,y)^2dv/2π vyyv [ ℐ(λ F_γ, (n, -β)^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
where the notation F_γ, (n, -β)^{x, y}(·) was defined in (<ref>).
By definition, on the event ℋ_j(𝐛) ∩ℋ_k(𝐛̃) we have
max(|𝐛_s - x|, |𝐛̃_t - y|)
≤ c(x, y)
≤1/8|x-y|
< 2^-n_0-2 < κ/4,
and in particular d(𝐛_s, ∂ D) ∧ d(𝐛̃_t, ∂ D) ≥κ/2 for any s ≤ℓ(𝐛), t ≤ℓ(𝐛̃). By (<ref>) we have
|G_0^D(y, 𝐛_s) + log|y-𝐛_s| | ≤ C_κ,
|G_0^D(x, 𝐛̃_t) + log|x-𝐛̃_t| |≤ C_κ,
|𝔼[h(𝐛_s) h_2^-n(x) ] + log (2^-n)| ≤ C_κ,
|𝔼[h(𝐛̃_t) h_2^-n(x) ] + log (2^-n)| ≤ C_κ.
Combining these estimates with the fact that
max{| log |y-𝐛_s| - log |x-y| |,
| log |x - 𝐛̃_t| - log |x-y| |,
| log(2^-n) - log |x-y| | }≤ C
for some absolute constant C>0 (say C = log 2), we obtain both (<ref>) and
C^-1
F_γ^{y}(𝐛)
≤ |x-y|^-γ(β-γ) F_γ, (n, -β)^{x, y}(𝐛)
≤C F_γ^{y}(𝐛)
where C = C(κ, β, γ) ∈ (0, ∞). This means (<ref>) can be upper-bounded by
C^4 |x-y|^(2γ - α)β- β^2/2𝔼[
(∫_0^j^-2 c(x, y)^2du/2π uxxu [ ℐ(λ F_γ^{x}(𝐛))1_ℋ_j(𝐛)])
×(∫_0^k^-2 c(x, y)^2dv/2π vyyv [ ℐ(λ F_γ^{y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
with λ := λC^-1 |x-y|^γ(β - γ). This expression can now be controlled uniformly in λ > 0 and j, k ∈ℕ by <Ref> and we are done after taking the sum over j, k ≥ 1. This concludes the proof of <Ref>.
§.§.§ Pointwise limit of the diagonal term
We now state the pointwise limit for our diagonal term.
For any fixed n_0 ∈ℕ satisfying 2^1-n_0 < κ,
𝔼[
1_𝒢^{x, y}_[n_0, ∞)(x) ∩𝒢^{x, y}_[n_0, ∞)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
∫_0^1 dv/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃)) ]
]
= c_γ^2
ℙ(𝒢_[n, ∞)(x) ∩𝒢_[n, ∞)(y))
for any distinct points x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ and -log_2|x-y| ∉ℕ.
The analysis of diagonal term is very similar to that of the cross term performed in <Ref>, so we only sketch the arguments here.
Step (i). We need a “two-point" analogue of <Ref>, i.e. we first show that for any m > 3 + max(n, -log_2 |x-y|) sufficiently large,
lim_λ→∞𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )]
∫_0^1 dv/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃)) ]
]
= c_γ^2 ℙ(𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)).
Let us fix some δ∈ (0, 2^-m) as before, and define for each j, k ∈ℕ
I_j,k := 𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y) ∫_0^(δ/j)^2du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛) )1_ℋ_j(𝐛)]
×∫_0^(δ/k)^2dv/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)]
].
In order to establish (<ref>), it suffices to show
lim_λ→∞∑_j, k ≥ 1 I_j,k = c_γ^2 ℙ(𝒢_[n_0, m)^{x, y}(x) ∩𝒢_[n_0, m)^{x, y}(y))
using a similar dominated convergence approach. As in the proof of <Ref> we just highlight the steps for the upper bound of I_j, k.
* By considering the domain Markov property of Gaussian free field (<ref>) and performing a radial-lateral decomposition of the two independent Gaussian fields
h^x, η(·) = h^x, rad(·)+ h^x, lat(·)
and
h^y, η(·) = h^y, rad(·)+ h^y, lat(·),
one obtains the following analogue of (<ref>): we have
I_j, k≤ ∫_0^(δ/j)^2du/2π u∫_0^(δ/k)^2dv/2π v
×𝔼⊗001^⊗ 2[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y) 1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)E_x(δ)^-2E_y(δ)^-2
×ℐ(λ_x E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, η(·) + h(x)) )
×ℐ(λ_y E_y(δ)
v F_γ^{y}(y+√(v)𝐛̃; h^y, η(·) + h(y)) )
]
where, for p ∈{x, y},
λ_p
:= λ R(p; D)^3γ^2/2
e^γ^2 G_0^D(x, y),
E_p(δ)
:= [e^5 γ^2/2δ + γℰ_p(δ) + γ^2/2 e_p(δ)]^-1,
with
ℰ_p(δ):= sup_z ∈ B(p, δ) | h(z) - h(p)|,
e_p(δ):= sup_z ∈ B(p, δ) | 𝔼[h(z)^2 - h(p)^2]|.
* We need two (conditional) Gaussian comparisons to replace h^p, lat with the field
𝔼[X^p(z_1) X^p(z_2)] = log|z_1-p| ∨ |z_2-p|/|z_1-z_2| ∀ z_1, z_2 ∈ B(p, δ)
for each p ∈{x, y}. One can show (with a computation similar to that in (<ref>)) that these replacements would yield an error that is summable in j, k ∈ℕ uniformly in λ > 0, and negligible as δ→ 0^+. In other words, we just need to study
∫_0^(δ/j)^2 du/2π u∫_0^(δ/k)^2dv/2π v
×𝔼⊗001^⊗ 2[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y) 1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)E_x(δ)^-2E_y(δ)^-2
×ℐ(λ_x E_x(δ)
u F_γ^{x}(x+√(u)𝐛; h^x, rad + X^x + h(x)) ) )
×ℐ(λ_y E_y(δ)
v F_γ^{y}(y+√(v)𝐛̃; h^y, rad(·) +X^y + h(y)) )
].
* Following the same scaling argument as in (<ref>), one can show that (<ref>) is equal to (cf. (<ref>))
∫_0^∞∫_0^∞dsdt/π^2𝔼⊗001^⊗ 2[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y) 1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)E_x(δ)^-2E_y(δ)^-2
×ℐ(λ_x E_x(δ)
ℛ_x e^γ (B_x, s - (Q-γ)s) ) )
ℐ(λ_y E_y(δ)
ℛ_y e^γ (B_y, t - (Q-γ)t) ) )
]
where (B_x, s)_s ≥ 0 and (B_y, t)_t ≥ 0 are two standard Brownian motions independent of each other and everything else, and we are ready to apply <Ref> to obtain a uniform bound (summable over j, k ≥ 1) as well as the limiting value as λ→∞.
Summarising all the analysis above, one obtains by dominated convergence
lim sup_λ→∞∑_j, k ≥ 1 I_j, k
≤lim sup_δ→ 0^+∑_j, k ≥ 1lim_λ→∞∫_0^∞∫_0^∞dsdt/π^2𝔼⊗001^⊗ 2[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y) 1_ℋ_j(𝐛) ∩ℋ_k(𝐛̃)
×
E_x(δ)^-2E_y(δ)^-2ℐ(λ_x E_x(δ)
ℛ_x e^γ (B_x, s - (Q-γ)s) ) )
ℐ(λ_y E_y(δ)
ℛ_y e^γ (B_y, t - (Q-γ)t) ) )
]
= lim sup_δ→ 0^+∑_j, k ≥ 1 c_γ^2 𝔼[
1_𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)
E_x(δ)^-2E_y(δ)^-2]001( ℋ_j)001( ℋ_k)
= c_γ^2 ℙ(𝒢^{x, y}_[n_0, m)(x) ∩𝒢^{x, y}_[n_0, m)(y)),
and when combined with an analogous lower bound this concludes the proof of (<ref>).
Step (ii). We want to establish a “two-point" analogue of (<ref>), i.e.
lim_m →∞lim sup_λ→∞𝔼[
1_𝒢^{x, y}_[m, ∞)(p)^c (∫_0^1 du/2π uxxu[ ℐ(λ F_γ^{x, y}(𝐛))])
×(∫_0^1 du/2π vyyv[ ℐ(λ F_γ^{x, y}(𝐛̃))])
] = 0
for p ∈{x, y} and any fixed and distinct x, y ∈ D satisfying d(x, ∂ D) ∧ d(y, ∂ D) ≥κ.
To do so, we first use (<ref>) and follow the argument in (<ref>) to bound the expectation in (<ref>) by
e^(β^2/2 + 2βγ )C_κ/|x-y|^γ∑_n ≥ m 2^-β/2[2(α - γ ) - β]n𝔼[ (∫_0^1 du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛))])
×(∫_0^1 dv/2π vyyv[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛̃))])
]
where F_γ, (p, n, β)^{x, y}(·) was defined in (<ref>), and β∈ (0, 2(α - γ)) is fixed.
Recall c(x, y) := 1/8min(|x-y|, κ). Based on a splitting analysis similar to that in (<ref>), the proof is complete if we can show, for some δ∈ (0, c(x, y)), that
lim sup_m →∞lim sup_λ→∞∑_n ≥ m 2^-β/2[2(α - γ ) - β]n
×∑_j, k ≥ 1𝔼[
(∫_0^(2^-nδ / j)^2du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛)) 1_ℋ_j(𝐛)])
×(∫_0^(2^-nδ / k)^2dv/2π vyyv[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]=0.
But by (<ref>), one can check easily that
𝔼[(∫_0^(2^-nδ / j)^2du/2π uxxu[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛)) 1_ℋ_j(𝐛)])
×(∫_0^(2^-nδ / k)^2dv/2π vyyv[ ℐ(λ F_γ, (p, n, β)^{x, y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
≲𝔼[(∫_0^(2^-nδ / j)^2du/2π uxxu[ ℐ(λ_x, p F_γ^{x}(𝐛)) 1_ℋ_j(𝐛)])
×(∫_0^(2^-nδ / k)^2dv/2π vyyv[ ℐ(λ_y, p F_γ^{y}(𝐛̃))1_ℋ_k(𝐛̃)])
]
for some suitable λ_x, p, λ_y, p > 0 (cf. (<ref>)), and the above inequality is ≲001(ℋ_j)001(ℋ_k) by <Ref>. Thus (<ref>) is upper bounded (up to a multiplicative factor) by
lim sup_m →∞lim sup_λ→∞∑_n ≥ m 2^-β/2[2(α - γ ) - β]n∑_j, k ≥ 1001(ℋ_j)001(ℋ_k)
≲lim sup_m →∞
2^-β/2[2(α - γ ) - β]m = 0
and this concludes the proof of Lemma <ref>.
Combining with the other estimates in this section, this also concludes the proof of Theorem <ref>.
§.§ Proof of Theorems <ref> and <ref>.
Given Theorem <ref>, the proof of Theorem <ref> proceeds as explained in Section <ref>. In short, Theorem <ref> and the bridge decomposition establish that
∫_0^∞ e^-λ u u𝐒_γ(u) du ∼c_γμ_γ(D)/λ as λ→∞
which is (<ref>). By an application of the Tauberian theorem (Theorem <ref>) this implies
∫_0^t u 𝐒_γ(u) du ∼ c_γμ_γ(D) t
as t → 0^+,
which is (<ref>). Lemma <ref> implies that
t 𝐒_γ(t) →c_γμ_γ(D)
in probability, as desired for Theorem <ref>.
Since 𝐒_γ(t) is the Laplace transform of the eigenvalue counting function 𝐍_γ(λ), Theorem <ref> follows again from an application of the probabilistic Tauberian theorem (Theorem <ref>).
§ POINTWISE HEAT KERNEL ASYMPTOTICS
§.§ Proof of <Ref>
Based on a similar scaling argument as before, let us assume that diam(D) < 1/2, and we shall continue to write c_γ = c_γ(Q-γ; ℐ) throughout <Ref> without risk of confusion. By standard approximation argument, it suffices to establish <Ref> for test functions f that are uniformly bounded and Lipschitz, and without loss of generality suppose
sup_x ∈D, u ∈_+ |f(x, u)| + sup_x ∈D[ sup_u, v ∈_+| f(x, u) - f(x, v)/u-v|] ≤ 1.
To begin with, we apply the bridge decomposition and rewrite the LHS of (<ref>) as
𝔼[ ∫_D μ_γ (dx) f ( x, J_γ^λ(x) ) ]
= 𝔼[ ∫_D μ_γ (dx) f (x, ∫_0^∞du/2π uxxu[ ℐ(λ F_γ(𝐛))1_{u< τ_D(𝐛) }] ) ]
= ∫_D R(x; D)^γ^2/2 dx 𝔼[ f (x, ∫_0^∞du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ) ].
Since f is uniformly bounded, the expectation in the integrand above is bounded, and by dominated convergence we just need to show that
lim_λ→∞𝔼[ f (x, ∫_0^∞du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ) ]
= 𝔼 [ f(x, J_γ^∞) ]
for any κ > 0 and x ∈ D satisfying d(x, ∂ D) ≥ 2κ (see <Ref> for the definition of J_γ^∞).
§.§.§ Step 1: truncating the time integral
Let δ_1 ∈ (0, 1) be some fixed but arbitrary number (possibly dependent on x). Similar to our proof of <Ref> we would first like to truncate the u-integral:
Let x ∈ D satisfying d(x, ∂ D) ≥ 2κ. We have
lim sup_λ→∞| 𝔼[ f (x, ∫_0^∞du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ) ]
- 𝔼[ f (x, ∫_0^δ_1^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ) ]| = 0.
Thanks to the Lipschitz control (<ref>), the LHS of (<ref>) (before taking the limit λ→∞) is bounded by
[ ∫_δ_1^2^∞du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ].
Since (·) ≤ 1, we know from <Ref> (with the assumption diam(D) < 1/2) that
xxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }]
≤xxu( 𝐛_s ∈ D ∀ s ≤ u)
≤xxu( |𝐛_s - x | ≤ 1 ∀ s ≤ u)
≤ 1 ∧2/u
which is integrable with respect to du/2π u on [δ_1^2, ∞). As ℐ(λ F_γ^{x}(𝐛)) 0 almost surely, it follows from dominated convergence that
lim_λ→∞[ ∫_δ_1^2^∞du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ] = 0
which leads to the desired claim (<ref>).
§.§.§ Step 2: restricting the range of Brownian bridge
The next step would be to restrict the range of our Brownian bridge 𝐛. Unlike the proof of <Ref> where we needed to partition the probability space, here we introduce a cutoff parameter n ∈ℕ and assume from now that δ_1 is small enough such that 4n δ_1 <κ.
Let
ℋ_n
= ℋ_n(𝐛)
= {max_s≤ℓ(𝐛)|𝐛_s -ι(𝐛)|/√(ℓ(𝐛)) < n }
= ⋃_k=1^n ℋ_k.
Then for any x ∈ D satisfying d(x, ∂ D) ≥ 2κ, we have
lim sup_n →∞lim sup_δ_1 → 0^+lim sup_λ→∞ | 𝔼[ f (x, ∫_0^δ_1^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }] ) ]
-𝔼[ f (x, ∫_0^δ_1^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_n] ) ] | = 0.
The LHS of (<ref>) (before taking any of the limit) is bounded by
𝔼[ ∫_0^δ_1^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_{u< τ_D(𝐛) }∩ℋ_n^c] ]
≤∑_k ≥ n+1∫_δ_1^2 k^-2^δ_1^2du/2π u𝔼[ xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k] ]
+ ∑_k ≥ n+1∫_0^δ_1^2 k^-2du/2π u𝔼[ xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k] ].
where ℋ_k was defined in (<ref>). The first sum is upper bounded by
∑_k ≥ n+1∫_δ_1^2 k^-2^δ_1^2du/2π u𝔼[ xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k] ]
≤∑_k ≥ n+1∫_δ_1^2 k^-2^δ_1^2du/2π uxxu( ℋ_k)
≤∑_k ≥ n+1∫_δ_1^2 k^-2^δ_1^2du/2π u· 4 e^-(k-1)^2/2
≤∑_k ≥ n+1 2 e^-(k-1)^2/2log k
where the second last inequality follows from <Ref>. This vanishes as n →∞ uniformly in λ and δ_1.
Let us look at the second sum. Since 4k√(u)≤ 4k √(δ_1^2 k^-2) = 4 δ_1 ≤κ≤ d(x, ∂ D) for u ∈ [0, δ_1^2k^-2], we obtain
∑_k ≥ n+1∫_0^δ_1^2 k^-2du/2π u𝔼[ xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k] ]
≤∑_k ≥ n+1∫_0^1du/2π u 1_{d(x, ∂ D) ≥ 4k √(u)}𝔼[ xxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_k] ]
≤ C ∑_k ≥ n+1001(ℋ_k) = C 001(ℋ_n^c)
where the last inequality follows from <Ref> with C > 0 independent of λ. This bound again vanishes uniformly in λ and δ_1 as n→∞, and this concludes the proof of (<ref>).
§.§.§ Step 3: decomposition of Gaussian free field
We now need to argue that the Gaussian free field h(·) locally behaves like an exactly scale invariant field. In the proof of <Ref>, this was achieved by Gaussian interpolation/comparison. It is not clear how this method could be adapted to the analysis here, though, since we are dealing with arbitrary test functions f. We shall therefore pursue a different strategy based on the decomposition of Gaussian fields.
Applying the domain Markov property of Gaussian free field similar to that in (<ref>), we can write
h(·) = h(·) + h^x, η(·) + h^y, η(·)
but here we choose η∈ (κ/2, κ) (and in particular δ_2 := nδ_1 < η). Since the random variable F_γ^{x}(𝐛) (recall (<ref>)) only depends on h(·) on B(x, δ_2) on the event H_n when we restrict u ∈ [0, δ_1^2] and (<ref>) can be rewritten as
F_γ(ds; 𝐛) :=
e^γh(𝐛_s) - γ^2/2𝔼[h(𝐛_s)^2]
e^γ h^x, η(𝐛_s) - γ^2/2𝔼[h^x, η(𝐛_s)^2]
R(𝐛_s; D)^γ^2/2 1_{𝐛_s ∈B(x, δ_2)}ds.
We shall perform further decomposition with the help of <Ref>, and write
h^p, η(·) = X^p, η(·) - Y^p, η(·) on B(p, η)
for p ∈{x, y}, where X^p, η(·) d= X^η𝔻(· - p) and Y^p, η(·) d= Y^η𝔻(· - p) in the notation of (<ref>).
We claim that when δ_1 (and hence δ_2) is small, F_γ^{x}(𝐛) is approximately equal to R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]F_γ^{x}(𝐛; X^x, η) where (recalling (<ref>))
F_γ^{x}(𝐛; X^x, η)
:=
∫_0^ℓ(𝐛) e^γ X^x, η(𝐛_s) - γ^2/2𝔼[X^x, η(𝐛_s)^2]1_{𝐛_s ∈B(x, δ_2)}ds/| 𝐛_s - x |^γ^2.
For any x ∈ D satisfying d(x, ∂ D) ≥ 2κ, we have
lim sup_δ_1 → 0^+lim sup_λ→∞| 𝔼[ f (x, ∫_0^δ_1^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_n] ) .
-. f (x, ∫_0^δ_1^2du/2π uxxu[ ℐ(λ
R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]F_γ^{x}(𝐛; X^x, η)
)1_ℋ_n] ) ] | = 0.
Fix ϵ∈ (0, 1), and suppose δ_1 > 0 (and hence δ_2 := n δ_1 > 0) is sufficiently small such that
(1+ϵ)^-1 R(x; D) ≤ R(w; D) ≤ (1+ϵ) R(x; D) ∀ w ∈B(x, δ_2)
as well as
| G_0^D(x, w) - [ -log|x-w| + log R(x; D)]| ≤ϵ ∀ w ∈B(x, δ_2)
which is possible by <Ref>. We also introduce the event
𝒪_ϵ(x, δ_2)
:= {|(γh(w) - γ^2/2𝔼[h(w)^2]) - (γh(x) - γ^2/2𝔼[h(x)^2] )| ≤ϵ ∀ w ∈B(x, δ_2) }
∩{|γ Y^x, η(w) - γ^2/2𝔼[Y^x, η(w)^2] | ≤ϵ ∀ w ∈B(x, δ_2) }
and bound the LHS of (<ref>) by
ℙ(O_ϵ(x, δ_2)^c)
+ 𝔼{ 1_O_ϵ(x, δ_2)| ∫_0^δ_1^2du/2π uxxu[ ℐ(λ F_γ^{x}(𝐛))1_ℋ_n]
-
∫_0^δ_1^2du/2π uxxu[ ℐ(λ
R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]F_γ^{x}(𝐛; X^x, η)
)1_ℋ_n]
|}.
Let us further rewrite (<ref>) (on the event 𝒪_ϵ(x, δ_2) and ℋ_n) as
F_γ(ds; 𝐛)
:= e^γh(𝐛_s) - γ^2/2𝔼[h(𝐛_s)^2][e^γ Y^x, η(𝐛_s) - γ^2/2𝔼[Y^x, η(𝐛_s)^2]]^-1
×
e^γ X^x, η(𝐛_s) - γ^2/2𝔼[X^x, η(𝐛_s)^2]
R(𝐛_s; D)^γ^2/2 1_{𝐛_s ∈B(x, δ_2)}ds.
Then based on the definition of ϵ as well as the event 𝒪_ϵ(x, δ_2), it is straightforward to verify that
C(ϵ)^-1≤F_γ^{x}(𝐛) /R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]F_γ^{x}(𝐛; X^x, η)≤ C(ϵ)
where C(ϵ) = (1+ϵ)^γ^2/2 e^(γ^2 +2) ϵ. Combining this two-sided control with the fact that
|ℐ(u) - ℐ(v)|
= | ue^-u - v e^-v|
≤∫_v^u |(1-s) e^-s|ds
≤ 2(u-v) e^-v/2
for any u ≥ v ≥ 0, one can check that
| ℐ(λ F_γ^{x}(𝐛))
- ℐ(λ
R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]F_γ^{x}(𝐛; X^x, η)
)
|
≤
4[C(ϵ) - 1] C(ϵ) ℐ(
λ/2C(ϵ) R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]F_γ^{x}(𝐛; X^x, η)
).
Summarising everything so far, the estimate (<ref>) can be bounded by
ℙ(O_ϵ(x, δ_2)^c)
+ 4[C(ϵ) - 1] C(ϵ) 𝔼[ ∫_0^δ_1^2du/2π uxxu[
ℐ(
λ C_x(ϵ) F_γ^{x}(𝐛; X^x, η)
)
1_ℋ_n]]
with C_x(ϵ) := 1/2C(ϵ) R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2].
We now perform a space-time rescaling of the Brownian bridge (<ref>), and write
𝔼⊗xxu[
ℐ(
λ C_x(ϵ)F_γ^{x}(𝐛; X^x, η)
)
1_ℋ_n]
= 𝔼⊗001[
ℐ(
λ C_x(ϵ)
F_γ^{x}(x + √(u)𝐛_·/u; X^x, η)
)
1_ℋ_n]
= 𝔼⊗001[
ℐ(
λ C_x(ϵ)
F_γ^{0}(√(u)𝐛_·/u; X^η𝔻)
)
1_ℋ_n]
where
F_γ^{0}(√(u)𝐛_·/u; X^η𝔻)
= ∫_0^u e^γ X^η𝔻(√(u)𝐛_s/u) - γ^2/2𝔼[X^η𝔻( √(u)𝐛_s/u)^2]ds/| √(u)𝐛_s/u|^γ^2
= u ^1 - γ^2/2∫_0^1 e^γ X^η𝔻(√(u)𝐛_s) - γ^2/2𝔼[X^η𝔻(√(u)𝐛_s)^2]ds/|𝐛_s|^γ^2.
Since
𝔼[X^η𝔻( n√(u) x_1)X^η𝔻(n√(u) x_2) ]
= -log|x_1 - x_2| - logn√(u)/η
= 𝔼[ X^𝔻(x_1) X^𝔻(x_2)] + 𝔼[B_T(u, n)^2]
∀ x_1, x_2 ∈𝔻
where T(u, n) := -logn√(u)/η > 0 (as 2n√(u)≤ 2nδ_1 < κ/2 < η) and B_T(u, n)∼𝒩(0, T(u, n)) is independent of X^𝔻, we see that (<ref>) (on the event H_n) is equal in distribution to
u^1-γ^2/2 e^γ B_T(u, n) - γ^2/2T(u, n)∫_0^1 e^γ X^𝔻(n^-1𝐛_s) - γ^2/2𝔼[X^η𝔻(n^-1𝐛_s)^2]ds/|𝐛_s|^γ^2
= u^1-γ^2/2 e^γ B_T(u, n) - γ^2/2T(u, n) n^-γ^2F_γ^{0}(n^-1𝐛; X^𝔻)
= e^γ (B_T(u, n) - (Q-γ)T(u, n)) (n/η)^-(2-γ^2)n^-γ^2F_γ^{0}(n^-1𝐛; X^𝔻).
Setting ℰ:= C_x(ϵ) (n/η)^-(2-γ^2)n^-γ^2F_γ^{x}(𝐛; X^𝔻), we obtain
𝔼[ ∫_0^δ_1^2du/2π uxxu[
ℐ(
λ C_x(ϵ) F_γ^{x}(𝐛; X^x, η)
)
1_ℋ_n]]
=
𝔼⊗001[ ∫_0^δ_1^2du/2π uℐ(
λℰ e^γ (B_T(u; n) - (Q-γ) T(u; n)))
1_ℋ_n]
≤𝔼⊗001[ ∫_0^∞dt/πℐ(
λℰ e^γ (B_t - (Q-γ) t))
1_ℋ_n]
≤ c_γ
where the last inequality follows from (<ref>) of <Ref>. Therefore, (<ref>) is uniformly bounded in λ→∞ by
ℙ(O_ϵ(x, δ_2)^c)
+ 4[C(ϵ) - 1] C(ϵ) · c_γ.
As δ_1 → 0^+ (and hence δ_2 → 0^+), we have ℙ(O_ϵ(x, δ_2)^c) → 0 by the continuity of the Gaussian fields h(·) and Y^x, η(·) in a neighbourhood of x. Since ϵ > 0 is arbitrary, we can send ϵ→ 0^+ and conclude that (<ref>) holds.
§.§.§ Step 4: identifying the limiting random variable J_γ^∞
All that remains to be done is to establish the pointwise limit.
Let C_x:=R(x; D)^3γ^2/2e^γh(x) - γ^2/2𝔼[h(x)^2]. For any x ∈ D satisfying d(x, ∂ D) ≥ 2κ, we have
lim_n →∞lim_δ_1 → 0^+lim_λ→∞𝔼[f(x, ∫_0^δ_1^2du/2π uxxu[ ℐ(
λ C_x F_γ^{x}(𝐛; X^x, η)
)1_ℋ_n]) ]
= 𝔼[ f(x, J_γ^∞)]
with
J_γ^∞
:=
∫_-∞^∞dt/π001[ ℐ(
∫_0^1 e^-γβ_t - log|𝐛_s|^Q-γe^γX(e^-t𝐛_s) - γ^2/2𝔼[X(e^-t𝐛_s)^2]ds/|𝐛_s|^2)]
where
* X(·) is a scale-invariant Gaussian field defined on ℝ^2 ≅ℂ with covariance kernel
𝔼[X(x_1) X(x_2)]= log|x_1| ∨ |x_2|/|x_1 - x_2|;
* (β_t^Q-γ)_t ∈ℝ is the γ-quantum cone, i.e. the two-sided stochastic process defined in (<ref>) with m = Q-γ.
We begin by standardising our Brownian bridge like (<ref>), i.e.
𝔼[f(x, ∫_0^δ_1^2du/2π uxxu[ ℐ(
λ C_x F_γ^{x}(𝐛; X^x, η) )1_ℋ_n]) ]
= 𝔼[f(x, ∫_0^δ_1^2du/2π u001[ ℐ(
λ C_x F_γ^{0}( √(u)𝐛_·/u; X^η𝔻)1_ℋ_n]) ].
Unlike the proof of the last lemma where the exact scaling relation of X^η𝔻 was used, we have to proceed with the radial-lateral decomposition here: for x_1, x_2 ∈ B(0, η) recall
𝔼[X^η𝔻(x_1)X^η𝔻(x_2) ]
= -log| x_1/η| ∨| x_2/η|
+ log|x_1| ∨ |x_2|/|x_1 - x_2|
= 𝔼[B_T(x_1)B_T(x_2)] + 𝔼[X(x_1) X(x_2)]
where T(·) = -log |·/η| and (B_t)_t ≥ 0 is a Brownian motion independent of X(·). Then (<ref>) is equal to
F_γ^{0}(√(u)𝐛_·/u; X^η𝔻)
= η^2-γ^2∫_0^1
|√(u)𝐛_s / η|^2-γ^2e^γ X^η𝔻(√(u)𝐛_s) - γ^2/2𝔼[X^η𝔻(√(u)𝐛_s)^2]ds/|𝐛_s|^2
=η^2-γ^2∫_0^1 e^γ[B_T(√(u)𝐛_s) -(Q-γ) T(√(u)𝐛_s)]e^γX(√(u)𝐛_s) - γ^2/2𝔼[X(√(u)𝐛_s)^2]ds/|𝐛_s|^2
and thus
∫_0^δ_1^2du/2π u001[ ℐ(
λ C_x F_γ^{0}( √(u)𝐛_·/u; X^η𝔻)1_ℋ_n]
= ∫_-logδ_1^∞dt/π001[ ℐ(
λ C_x η^2-γ^2
×∫_0^1 e^γ[B_t - log|𝐛_s / η| -(Q-γ)(t - log|𝐛_s / η|)]e^γX(e^-t𝐛_s) - γ^2/2𝔼[X(e^-t𝐛_s)^2]ds/|𝐛_s|^2)
1_ℋ_n]
=
∫_-log(δ_1 / η) - τ^∞dt/π001[ ℐ(
∫_0^1 e^γ[B_t - log|𝐛_s|+τ -(Q-γ)(t - log|𝐛_s| + τ)]
-γ[B_τ -(Q-γ)τ]
×e^γX(η e^τ e^-t𝐛_s) - γ^2/2𝔼[X(η e^τ e^-t𝐛_s)^2]ds/|𝐛_s|^2)
1_ℋ_n]
with
τ
:= τ_λ C_x η^2-γ^2
:= inf{u > 0: e^γ [B_u - (Q-γ)u] = (λ C_x η^2-γ^2)^-1}.
Since η e^τ is independent of the scale invariant field X, we see that (<ref>) has the same distribution as
∫_-log(δ_1 / η) - L^∞dt/π001[ ℐ(
∫_0^1 e^-γβ_t - log|𝐛_s|^Q-γe^γX(e^-t𝐛_s) - γ^2/2𝔼[X(e^-t𝐛_s)^2]ds/|𝐛_s|^2)
1_ℋ_n]
where L := L_λ C_x η^2-γ^2 := sup{u > 0: β_-u^Q-γ = λ C_x η^2-γ^2} by <Ref>. As everything inside 001[·] in (<ref>) is non-negative and independent of λ C_x, and L∞ a.s., it follows from monotone convergence that (<ref>) converges as λ→∞ to
∫_-∞^∞dt/π001[ ℐ(
∫_0^1 e^-γβ_t - log|𝐛_s|^Q-γe^γX(e^-t𝐛_s) - γ^2/2𝔼[X(e^-t𝐛_s)^2]ds/|𝐛_s|^2)
1_ℋ_n].
Now that the above expression is independent of δ_1 > 0, we may first send δ_1 → 0^+ and then n →∞ (so that the condition 4nδ_1 < κ remains satisfied) to conclude the proof by monotone convergence and continuous mapping theorem.
Combining all the analysis from Step 1–4 above, we are only left with the final task of verifying 𝔼[J_γ^∞] = c_γ. A direct computation would not be straightforward, and we shall proceed instead by reversing the sequence of arguments in the proof of <Ref> and making use of
𝔼[J_γ^∞]
= lim_n →∞lim_λ→∞∫_0^δ_1^2du/2π u𝔼⊗001[ ℐ(
λ C_x F_γ^{0}( √(u)𝐛_·/u; X^η𝔻)1_ℋ_n]
as a result of monotone convergence. Note that the evaluation needed on the RHS is independent of the choice of δ_1 (which is allowed to depend on n), and in particular we may take δ_1 = η / n to ensure that 𝐛_·∈ B(0, η) = η𝔻 on the event H_n. We then follow the strategy in <Ref> and invoke the scaling behaviour of X^η𝔻, leading us to
F_γ^{0}(√(u)𝐛_·/u; X^η𝔻)
d=
u^1 - γ^2/2 e^γ B_T - γ^2/2 T∫_0^1 e^γ X^η𝔻(η/n𝐛_s) - γ^2/2𝔼[X^η𝔻(η/n𝐛_s)^2]ds/|𝐛_s|^γ^2
d=
e^γ(B_T - (Q-γ)T)(η/n)^2-γ^2∫_0^1 e^γ X^𝔻(n^-1𝐛_s) - γ^2/2𝔼[X^𝔻(n^-1𝐛_s)^2]ds/|𝐛_s|^γ^2_=:ℰ_x, n.
where B_T ∼𝒩(0, T) with T = T(u; n, η) := -log(n √(u)/η) is independent of everything else. Substituting this back to (<ref>), we obtain
𝔼[J_γ^∞]
= lim_n →∞lim_λ→∞∫_0^∞dt/π𝔼⊗001[ ℐ(
λ C_x e^γ(B_t - (Q-γ)t)ℰ_x, n)1_ℋ_n]
= lim_n →∞ c_γ𝔼⊗001[1_ℋ_n]
= c_γ
by <Ref>, and the proof of <Ref> is now complete.
§.§ Evaluating the constant c_γ(m): proof of <Ref>
Recall from <Ref> that c_γ(m) defined by the probabilistic representation (<ref>) or equivalently (<ref>) is finite for any γ, m > 0. Moreover, from <Ref> we may write
π c_γ(m)
= lim_λ→∞𝔼[ ∫_0^∞ℐ(λ e^γ (B_t - mt))dt]
= lim_λ→∞∫_0^∞ dt ∫_0^∞λ ue^-λ uℙ(e^γ (B_t - mt)∈ du)
= lim_λ→∞λ∫_0^∞ ue^-λ u[ ∫_0^∞1/u γ√(2π t)exp(-1/2γ^2 t (log u +γ m t)^2) dt ]_(*) du
where (*) is integrable for any u > 0. By the standard Hardy-Littlewood-Karamata Tauberian Theorem (i.e. <Ref> in the deterministic setting), we also have
π c_γ(m)
= lim_λ→∞λ∫_0^1/λ u [ ∫_0^∞1/u γ√(2π t)exp(-1/2γ^2 t (log u +γ m t)^2) dt ]_(*) du
= lim_λ→∞𝔼[ ∫_0^∞ℐ(λ e^γ (B_t - mt))dt]
with ℐ(x) := x 1_{x ≤ 1} and in particular ℐ(x) = 0 for x > 1. Introducing the stopping time τ_λ := inf{t > 0: e^γ (B_t - mt) = 1/λ}, we have for any λ > 0 that
𝔼[ ∫_0^∞ℐ(λ e^γ (B_t - mt))dt]
= 𝔼[ ∫_τ_λ^∞ℐ(λ e^γ (B_t - mt))dt]
= 𝔼[ ∫_τ_λ^∞ℐ(e^γ [(B_t - mt) - (B_τ_λ - mτ_λ )])dt]
= 𝔼[ ∫_0^∞ℐ(e^γ (B_t - mt))dt]
by the strong Markov property. If we denote by Φ(·) the cumulative distribution function of standard Gaussian random variables, then
c_γ(m)
= 1/π∫_0^∞ e^(γ^2/2 - γ m) t𝔼[ e^γ B_t - γ^2/2t 1_{B_t - mt ≤ 0}]dt
= 1/π∫_0^∞ e^(γ^2/2 - γ m) tΦ( (m - γ)√(t)) dt
= 2/πγ(γ - 2m){[ e^(γ^2/2 - γ m) tΦ( (m - γ)√(t)) ]_0^∞ - ∫_0^∞ e^(γ^2/2 - γ m) t∂_t Φ( (m - γ)√(t)) dt}
= 1/πγ(γ - 2m)[- 1 - 2(m-γ)∫_0^∞ e^(γ^2/2 - γ m) s^2 e^-(m-γ)^2 s^2/2ds/√(2π)]
= 1/πγ(γ - 2m)[- 1 - m-γ/m]
= 1/πγ m
which is our desired result.
§ PROBABILISTIC ASYMPTOTICS
This appendix collects some probabilistic generalisations of common asymptotic results that are suitable in the context of convergence in probability. The first one concerns “asymptotic differentiations".
Let α, β > 0 be fixed, and φ(u): ℝ_+ ↦ℝ_+ a random non-increasing function. Suppose there exists some a.s. positive random variable C such that
t^-β∫_0^t u^α - 1φ(u) du
C,
then
t^α - βφ(t) β C.
Without loss of generality suppose C = 1 almost surely. We start with the upper bound, i.e. we would like to establish
lim_t → 0^+ℙ( t^α - βφ(t) - β > ϵ) = 0 ∀ϵ > 0.
For this, consider, for fixed b > 1, the deterministic inequality
∫_b^-1t^t u^α - 1φ(u) du
≥φ(t) ∫_b^-1t^t u^α-1 du
= t^αφ(t) 1 -b^-α/α.
Then for any ϵ' > 0, we have
lim_t → 0^+ℙ( t^α - βφ(t) - β > ϵ)
≤lim_t → 0^+ℙ((α/1-b^-α) t^-β∫_b^-1t^t u^α - 1φ(u) du - β≥ϵ)
≤lim_t → 0^+ℙ(|t^-β∫_0^t u^α-1φ(u) du - 1| > ϵ' )
+lim_t → 0^+ℙ( |(b^-1t)^-β∫_0^b^-1t u^α-1φ(u) du - 1| > ϵ' )
+ 1{(α/1-b^-α) [ (1+ϵ') - b^-β(1-ϵ')] - β > ϵ}
= 1{α(1-b^-β)/1-b^-α - β + α(1+b^-β)/1-b^-αϵ'> ϵ}.
Given that
lim_b → 1α(1-b^-β)/1-b^-α - β = 0,
we can choose b sufficiently close to 1 and then ϵ' > 0 sufficiently small such that
| α(1-b^-β)/1-b^-α - β| < ϵ/2 and α(1+b^-β)/1-b^-αϵ'
< ϵ/2,
in which case the indicator function in (<ref>) is always evaluated to 0. By a similar argument, one may obtain the lower bound
lim_t → 0^+ℙ( t^α - βφ(t) - β < ϵ) = 0
by considering the integral ∫_t^bt u^α - 1φ(u) du. This concludes the proof.
The next result is a probabilistic generalisation of the Hardy–Littlewood Tauberian theorem. The version we are stating is slightly more general than what is needed here as it could be of independent interest. Recall that a function L: (0, ∞) → (0, ∞) is slowly varying at zero if lim_t → 0^+ L(xt)/L(t) = 1 for any x > 0.[One can also talk about slow variation at infinity by considering the analogous ratio limit as t →∞.]
Let ν(d ·) be a non-negative random measure on ℝ_+, ν(t) := ∫_0^t ν(ds), and suppose the Laplace transform
ν̂(λ) := ∫_0^∞ e^-λ sν(ds)
exists almost surely for some (and hence all) λ > 0. If
* ρ∈ [0, ∞) is fixed;
* L: (0, ∞) → (0, ∞) is a deterministic slowly varying function at 0; and
* C_ν is some non-negative (finite) random variable,
then we have:
λ^ρ/L(λ^-1)ν̂(λ) C_ν ⇒ t^-ρ/L(t)ν(t) C_ν/Γ(1+ρ).
The same implication also holds when one considers the asymptotics as λ→ 0^+ and t →∞ in (<ref>) (but with L being slowly varying at infinity) instead.
Following <cit.> as well as <cit.>, our proof of <Ref> is based on adapting Karamata's argument to the probabilistic setting, and the main ingredient is the following deterministic approximation lemma.
For each α≥ 0 and ϵ∈ (0, 1/2e), there exist some constant C = C(α) < ∞ independent of ϵ and polynomials _±(·) without constant terms (i.e. _±(0) = 0) such that _-(x) ≤ 1_[e^-1, 1](x) ≤_+(x) for any x ∈ [0,1] and
∫_0^1 |_±(x) - 1_[e^-1, 1](x) | α(log1/x)^α-1dx/x≤ C(α) ϵ.
We focus on the construction of _+ since the other one is similar. To begin with, define a continuous function h: [0, 1] →_+ by
h(x) =
0 if x ∈ [0, e^-1 - ϵ],
ϵ^-1 [x - (e^-1 - ϵ)] if x ∈ [e^-1 - ϵ, e^-1],
1 if x ∈ [e^-1, 1].
It is straightforward to see that h(x) ≥ 1_[e^-1, 1](x) for all x ∈ [0,1] and
∫_0^1 [h(x) - 1_[e^-1, 1](x)] α(log1/x)^α-1dx/x ≤∫_e^-1 - ϵ^e^-1 [x - (e^-1 - ϵ)]^2 α(log1/x)^α-1dx/x
≤α e (log (2e))^αϵ^2.
Next, using Weierstrass theorem, there exists some polynomial (·) such that
|(x) - (h(x)/x + ϵ)| ≤ϵ ∀ x ∈ [0, 1].
This means in particular that _+(x) := x (x) (which is a polynomial without constant term) satisfies _+(x) ≥ h(x) ≥ 1_[e^-1, 1](x) for all x ∈ [0,1] and
∫_0^1 [_+(x) - h(x)] α(log1/x)^α-1dx/x≤ 2 ϵα∫_0^1 (log 1/x)^α-1 dx
= 2Γ(α + 1) ϵ.
Combining everything, we arrive at
∫_0^1 |_±(x) - 1_[e^-1, 1](x) | α(log1/x)^α-1dx/x≤[ α e (log (2e))^α + 2Γ(α+1)] ϵ
which concludes the proof.
We shall focus on the claim (<ref>), as the other case (i.e. the same implication but with λ→ 0^+ and t →∞) follows from the arguments below ad verbatim. To begin with, observe that for each k ∈ℕ,
t^-ρ/L(t)∫_0^∞ e^-k/tsν(ds)
= k^-ρL(t/k)/L(t)[ (k/t)^ρ/L(t/k)ν̂(k/t)]
k^-ρ C_ν
= C_ν/Γ(1+ρ)∫_0^∞ e^-ks d(s^ρ).
Let us fix some ϵ > 0 to be chosen later, and find a polynomial _+(x) = ∑_k=1^m p_k x^k satisfying the conditions in <Ref>. Since m = m(ϵ) > 0 is finite, (<ref>) combined with a simple union bound argument suggests that
t^-ρ/L(t)∫_0^∞_+(e^-s/t) ν(ds)
= t^-ρ/L(t)∑_k=1^m p_k ∫_0^∞ e^-k/tsν(ds)
C_ν/Γ(1+ρ)∑_k=1^m p_k ∫_0^∞ e^-ks d(s^ρ)
= C_ν/Γ(1+ρ)∫_0^∞_+(e^-s) d(s^ρ).
On the other hand,
ν(t) = ∫_0^∞ 1_[e^-1, 1](e^-s/t) ν(ds)
≤∫_0^∞_+(e^-s/t) ν(ds).
Thus for any δ > 0, we have
lim sup_t → 0^+ℙ( t^-ρ/L(t)ν(t) - C_ν/Γ(1+ρ) > δ)
≤lim sup_t → 0^+ℙ( t^-ρ/L(t)∫_0^∞_+(e^-s/t)ν(ds) - C_ν/Γ(1+ρ) > δ)
≤lim sup_t → 0^+ℙ( t^-ρ/L(t)∫_0^∞_+(e^-s/t)ν(ds) - C_ν/Γ(1+ρ)∫_0^∞_+(e^-s) d(s^ρ) > δ/2)
+ ℙ( C_ν/Γ(1+ρ)[ ∫_0^∞_+(e^-s) d(s^ρ)- 1 ]> δ/2)
= ℙ( C_ν/Γ(1+ρ)∫_0^∞[_+(e^-s) - 1_[e^-1, 1](e^-s)]d(s^ρ)> δ/2)
≤ℙ( C_ν/Γ(1+ρ)· C(ρ) ϵ > δ/2)
where C(ρ) ϵ comes from the deterministic bound (<ref>). Since ϵ > 0 is arbitrary, we can send ϵ→ 0^+ and obtain
lim sup_t → 0^+ℙ( t^-ρ/L(t)ν(t) - C_ν/Γ(1+ρ) > δ) = 0.
Similarly, using the polynomial approximation _-(·) we can also obtain
lim sup_t → 0^+ℙ(C_ν/Γ(1+ρ) - t^-ρ/L(t)ν(t) > δ) = 0
and the proof is complete.
alpha
|
http://arxiv.org/abs/2307.04618v1 | 20230710150416 | Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation | [
"Hamed Bouzari Nezhad",
"Syksy Rasanen"
] | gr-qc | [
"gr-qc"
] |
=1
|
http://arxiv.org/abs/2307.04030v1 | 20230708184619 | Adaptive Force-Based Control of Dynamic Legged Locomotion over Uneven Terrain | [
"Mohsen Sombolestan",
"Quan Nguyen"
] | cs.RO | [
"cs.RO"
] |
Adaptive Force-Based Control of Dynamic Legged Locomotion over Uneven Terrain
Mohsen Sombolestan and Quan Nguyen
M. Sombolestan and Q. Nguyen are with the Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, email: [email protected], [email protected].
=========================================================================================================================================================================================================================================
Agile-legged robots have proven to be highly effective in navigating and performing tasks in complex and challenging environments, including disaster zones and industrial settings.
However, these applications normally require the capability of carrying heavy loads while maintaining dynamic motion.
Therefore, this paper presents a novel methodology for incorporating adaptive control into a force-based control system.
Recent advancements in the control of quadruped robots show that force control can effectively realize dynamic locomotion over rough terrain.
By integrating adaptive control into the force-based controller, our proposed approach can maintain the advantages of the baseline framework while adapting to significant model uncertainties and unknown terrain impact models.
Experimental validation was successfully conducted on the Unitree A1 robot.
With our approach, the robot can carry heavy loads (up to 50% of its weight) while performing dynamic gaits such as fast trotting and bounding across uneven terrains.
Adaptive control, Model predictive control (MPC), Quadruped robots, Unknown impact model.
§ INTRODUCTION
Legged robots have numerous potential uses, from search and rescue operations to autonomous construction. To perform these tasks effectively, it is important for the robot to have an accurate understanding of the environment it will be operating in. However, due to the complexity of the robot and the environment, the model of the robot itself might contain a significant level of uncertainty and affect the robot's stability, particularly when performing agile movements. To overcome these challenges, there is a need for the development of a control framework that can effectively compensate for these uncertainties in real-time.
The utilization of convex model predictive control (MPC) with the single rigid body (SRB) model in legged robots <cit.> has greatly enhanced the real-time implementation of diverse walking gaits. Unlike the balance controller based on quadratic programming <cit.>, MPC offers the capability to perform agile motions like jumping <cit.> and high-speed bounding <cit.> for quadruped robots. Additionally, MPC exhibits robustness in traversing rough and uneven terrains. However, it is important to note that MPC assumes perfect knowledge of the dynamic model.
To enhance trajectory tracking in the presence of unknown and changing disturbances, researchers have explored the combination of MPC with adaptive control techniques <cit.>. Additionally, parameter estimation techniques have been employed to further improve the robustness of the control system <cit.>. These approaches aim to adapt the controller and estimate system parameters to effectively compensate for uncertainties and disturbances, leading to improved trajectory tracking performance. It is worth noting that all of these studies were conducted using a position-based controller model.
In this work, we tackle the legged robot locomotion issue in real-world scenarios with a significant level of uncertainty. The uncertainty can come from both the robot model and the environment. Since our proposed method is based on a force controller, it retains the advantage of robustness to uneven terrain. Thanks to MPC as our baseline controller, our framework can be extended to different locomotion gaits and trajectories without adjusting the controller parameters. Additionally, by incorporating the adaptive controller, our control system can handle significant model uncertainty. As a result, our approach enables legged robots to move across different terrains with unknown impact models.
§.§ Related Works
§.§.§ Offline Learning
The offline learner can either leverage a model-based control approach or learn the control system from scratch. Using a model-based method, researchers mainly target learning the dynamic to improve the controller performance <cit.>. One example of this approach is the integration of deep learning with MPC, in which the proposed model tries to learn the cost or dynamic terms of an MPC <cit.>. This hybrid method shows considerable improvement for the aerial robot <cit.> when learning the dynamic model from experimental data.
The major limitation of this method is that it is restricted to the dynamic model learned during the training phase. However, the dynamic model is prone to frequent changes in real-world scenarios due to environmental uncertainties and external disturbances.
To overcome the limitations of previous approaches, there has been growing interest in utilizing reinforcement learning (RL) to train models from scratch. The key advantage of RL models is their ability to adapt swiftly to changes in real-world environments due to being trained in diverse environments with varying properties. In the case of quadruped robots, an RL model can directly predict appropriate joint torques for traversing different types of terrain, as demonstrated by Chen et al. <cit.>. Additionally, Bellegarda et al. <cit.> enable quadrupeds to run quickly while carrying unknown loads by training the model to learn foot positions. However, these methods heavily rely on domain randomization during training to generalize well to challenging environments. Yang et al. <cit.> also propose an end-to-end RL method that utilizes proprioceptive states and visual feedback to predict environmental changes.
§.§.§ Online Learning
To address inaccuracies in model-based controllers, researchers have explored an alternative approach using online learning, particularly supervised learning methods <cit.>. In this approach, the focus is on learning disturbances online <cit.>, and in some cases, researchers also aim to learn the dynamics of the system itself <cit.>. Furthermore, this approach has been successfully applied for online calibration of kinematic parameters in legged robots <cit.>.
In addition to that, in a recent study, a Lipschitz network method has been developed to bridge the model-reality gap in real-time <cit.>. The online learning method shares a close relationship with adaptive control, and numerous studies have explored the combination of these two approaches <cit.>. This combination aims to leverage the advantages of both methods, allowing for dynamic adaptation and continuous learning from real-time data to improve control system performance.
Perhaps closest to our work in terms of online adaption is the learning method presented in <cit.> for legged robots. The authors correct the model behind the controller using a supervised learner while the robot is walking in an unknown environment. The data is collected during the robot's operation to learn a linear residual model which can compensate for system errors. However, in the transition from simulation to experiment, the acceleration estimators make noisy data required for training the model. As a result, the method is only applied to estimate the linear terms since the angular terms data proved to be too noisy to be helpful in the model.
§.§.§ Adaptive Control
The goal of adaptive control is to tune the controller's variables online during deployment <cit.>. Adaptive control has been applied for manipulation tasks to robotic arms <cit.>, mobile robots <cit.>, and quadruped robots <cit.>.
The conventional Model Reference Adaptive Control (MRAC) architecture was originally designed for controlling linear systems in the presence of parametric uncertainties <cit.>. However, it lacks the ability to characterize the input/output performance of the system during the transient phase. To address this limitation and improve the transient performance of adaptive controllers, the L_1 adaptive control offers several advantages over traditional MRAC, such as decoupling adaptation and robustness within a control framework <cit.>.
In addition, by incorporating a low-pass filter in adaptation law, the L_1 adaptive control can provide stability <cit.> and transient performance <cit.>. Therefore, the L_1 adaptive control technique guarantees robustness with fast adaptation <cit.>, an essential criterion in dynamic robotics applications. Recently, by integrating L_1 adaptive controller and Bayesian learner, researchers leverage the fast adaption performance of the L_1 adaptive controllers and introduce a safe simultaneous control and learning framework <cit.>.
For legged robots, the adaptive controller has also been employed to find the value and location of the center of mass <cit.>. Our work on L_1 adaptive control for bipedal robots <cit.> considers a Control Lyapunov Function (CLF)-based controller as a closed-loop nonlinear reference model for the L_1 adaptive controller. It was validated for the robot's walking <cit.> and running <cit.>. However, the control framework in this prior work is based on Hybrid Zero Dynamics <cit.>, which uses joint position control to track the desired trajectory from optimization for each robot joint.
Moreover, in <cit.>, an adaptive control based on a CLF is designed for quadrupeds to interact with unknown objects. Then, they combined the criteria derived by adaptive control as a constraint in an MPC framework. However, adding more inequality constraints to MPC makes the controller more complex in terms of computation. In our approach, we compute a residual vector for compensating dynamic uncertainty, which makes the controller more time-efficient. Additionally, by employing our method, the robot is able to adapt to terrains with unknown impact models.
§.§ Contributions
A preliminary version of this research previously appeared in <cit.>; however, this paper presents several novel contributions to the prior work. This work incorporates the L_1 adaptive controller into the model predictive control (MPC).
The proposed control system leverages MPC due to its robustness to uneven terrain, contact constraint, and generalization to different locomotion gaits. Moreover, by integrating adaptive control into MPC, the proposed model can compensate for significant model uncertainty. In the previous work <cit.>, the robot can only perform quasi-static walking; however, in this work, the robot can perform dynamic motions thanks to MPC. Finally, the authors present new hardware experiments to demonstrate the effectiveness of the proposed adaptive MPC (as illustrated in fig: first fig).
The main contributions of the paper are as follows:
* We introduce a novel control system that combines the L_1 adaptive control into the force-based control system, designed to address the challenges posed by model uncertainty in real-world applications.
* Thanks to MPC, our approach offers greater versatility as it can be adapted to a wide range of locomotion gaits and trajectories. Moreover, our method can handle terrain uncertainty, allowing the robot to navigate rough terrains, such as grass and gravel, as well as high-sloped terrain.
* By integrating the adaptive control into MPC, it is possible for quadruped robots to carry an unknown heavy load (up to 50% of the robot's weight) across challenging terrains, with the capability of executing dynamic gaits such as fast trotting and bounding. This is a significant improvement compared to our previous work, which only allowed the robot to perform quasi-static walking.
* The combination of using MPC for both the reference model and the real model in the adaptive controller makes the control system computationally expensive, leading to potential delays in computation. To ensure real-time performance, we have developed an update frequency scheme for the control system, which allows for the optimized allocation of processing resources to each control component.
* Our proposed approach enables the control system to adapt to terrains with unknown impact models, such as soft terrain. Traversing soft terrain is a challenging task for quadruped robots. The A1 robot can walk on double-foam terrain in different directions using our method. In comparison, the robot cannot maintain its balance using the baseline controller, resulting in a collapse.
The remainder of the paper is organized as follows.
sec: background presents the baseline control architecture for quadruped robots and provides some knowledge on force-based controllers.
In sec: control overview, we will briefly present an overview of our control approach. Then, our proposed adaptive force-based controller using balance controller and MPC will be elaborated in sec: adaptive control and sec: adaptive MPC, respectively.
Furthermore, the numerical and experimental validation are shown in sec: Results. Finally, sec: conclusion provides concluding remarks.
§ PRELIMINARIES
In this section, we present the background on the control architecture of quadruped robots and describe each control component. According to <cit.>, the robot's control system consists of several modules, including a high-level controller, low-level controller, state estimation, and gait scheduler as presented in fig: ControlOverview.
A reference trajectory can be generated for high-level control from user input and state estimation. The gait scheduler defines the gait timing and sequence to switch between each leg's swing and stance phases. The high-level part controls the position of the swing legs and optimal ground reaction force for stance legs based on the user commands and gait timing. As the baseline for the stance leg controller, we will use two common approaches: 1) quadratic program (QP) based balancing controller <cit.> and 2) model predictive control (MPC) <cit.>.
The low-level leg control converts the command generated by high-level control into joint torques for each motor. These modules of the control architecture will be described briefly in the following subsections. More details can be found in <cit.>.
§.§ Gait Scheduler
The A1's gait is defined by a finite state machine using a leg-independent phase variable to schedule contact and swing phases for each leg <cit.>. The gait scheduler utilizes independent boolean variables to define contact states scheduled s_ϕ∈{1 = contact, 0 = swing} and switch each leg between swing and stance phases. Based on the contact schedule, the controller will execute either position control during swing or force control during stance for each leg.
In our previous work <cit.>, we focus on the application of load-carrying tasks, where the load is unknown to the robot or the control system. Having more legs on the ground during walking could also mean that the robot could produce a larger total ground reaction force to support the heavy load. Therefore, we used a quasi-static walking gait to maximize the number of legs on the ground during walking (i.e., 3 stance legs and 1 swing leg throughout the gait).
However, in this paper, our framework is not limited by any specific gait. Similar to the baseline MPC control approach <cit.>, the approach can work for different gaits by only changing the gait definition in the gait scheduler.
§.§ Desired Trajectory
The desired trajectory is generated based on the robot's velocity command. The robot operator commands xy-velocity and yaw rate, then xy-position and yaw are determined by integrating the corresponding velocity. z position contains a constant value of 0.3 m, and the remaining states (roll, roll rate, pitch, pitch rate, and z-velocity) are always zero.
§.§ Single Rigid Body (SRB) Model of Robot
Due to the complexity of the legged robot, a simplified rigid-body model has been used to present the system's dynamic. This model enables us to calculate the ground reaction forces (GRFs) in real-time. A few assumptions have been made to achieve simplified robot dynamics<cit.>:
Assumption 1: The robot has low inertia legs, so their effect is negligible.
Assumption 2: For small values of roll (ϕ) and pitch (θ), the rotation matrix R which transforms from the body to world coordinates, can be approximated as the rotation matrix corresponding to the yaw angle (ψ):
R≅R_z(ψ) = [[ cos(ψ) -sin(ψ) 0; sin(ψ) cos(ψ) 0; 0 0 1 ]]
Therefore, by defining the robot's orientation as a vector of Z-Y-X Euler angles Θ = [ϕ, θ, ψ]^T, the rate of change of the robot's orientation can be approximated as <cit.>:
Θ̇≅R_z(ψ) ω_b
where ω_b is the robot's angular velocity in the world frame.
Assumption 3: For small angular velocity, the following approximation can be made:
d/dt(I_G _b) = I_G _b + _b × (I_G _b) ≈I_G _b
where I_G∈ℝ^3 × 3 is the moment of inertia in the world frame.
Based on the above assumptions, the state representation of the system is as follows <cit.>:
[[ ṗ_c; Θ̇; p̈_c; _b ]] = [[ 0_3 0_3 1_3 0_3; 0_3 0_3 0_3 R_z(ψ); 0_3 0_3 0_3 0_3; 0_3 0_3 0_3 0_3 ]]_D∈ℝ^12 × 12[[ p_c; Θ; ṗ_c; _b ]]_X∈ℝ^12 +
[[ 0_6 × 12; M^-1A ]]_H∈ℝ^12 × 12F + [[ 0_6 × 1; G ]]
with
M = [[ m 1_3 0_3; 0_3 I_G ]] ∈ℝ^6 × 6
A = [[ 1_3 … 1_3; [p_1 - p_c] × … [p_4 - p_c] × ]] ∈ℝ^6 × 12
G = [[ g; 0_3 × 1 ]] ∈ℝ^6
where m is the robot's mass, g∈ℝ^3 is the gravity vector, p_c∈ℝ^3 is the position of the center of mass (COM), p_i∈ℝ^3 (i ∈{1,2,3,4}) are the positions of the feet, p̈_c∈ℝ^3 is body’s linear acceleration, _b∈ℝ^3 is angular acceleration, and F = [F_1^T, F_2^T, F_3^T, F_4^T]^T ∈ℝ^12 are the ground reaction forces acting on each of the robot’s four feet. The term [p_i - p_c] × is the skew-symmetric matrix representing the cross product (p_i - p_c) ×F_i.
Note that p_i and F_i are presented in the world frame. Therefore, the state representation of the system can be rewritten in the compact form:
Ẋ = DX + HF + [[ 0_6 × 1; G ]]
§.§ Balance Controller
One of the baseline control approach for calculating GRFs for quadruped robots is the balance controller presented in <cit.> based on quadratic program (QP) solver. Based on the assumptions presented in sec: simplified robot dynamic, the approximated dynamic model between the body acceleration and GRFs is as follows:
[[ 1_3 … 1_3; [p_1 - p_c] × … [p_4 - p_c] × ]]_A∈ℝ^6 × 12F = [[ m (p̈_c +g); I_G _b ]]_b∈ℝ^6
and the vector b in (<ref>) can be rewritten as:
b = M ([[ p̈_c; _b ]] + G).
Since the model (<ref>) is linear, the controller can naturally be formulated as the following QP problem <cit.>, which can be solved in real-time at 1 kHz:
F^* = F∈ℝ^12argmin (AF - b_d)^T S (AF - b_d)
+ γ_1 F^2 + γ_2 F - F_prev^* ^2
d≤CF≤d̅
F_swing^z=0
where b_d is the desired dynamics. The idea of designing b_d will be elaborated in sec: closed_loop. The cost function in (<ref>) includes terms that consider three goals, including (1) driving the COM position and orientation to the desired trajectories; (2) minimizing the force commands; and (3) minimizing the change of the current solution F^* with respect to the solution from the previous time-step, F^*_prev.
The priority of each goal in the cost function is defined by the weight parameters S∈ℝ^6 × 6, γ_1, γ_2 respectively.
The constraints in the QP formulation enforce friction constraints, input saturation, and contact constraints.
The constraint d≤CF≤d̅ ensures that the optimized forces lie inside the friction pyramid and the normal forces stay within a feasible range. More details can be found in <cit.>.
Besides the friction constraint, we will enforce the force constraints for the swing legs, F_swing=0. The swing legs are then kept in the posing position until it switches to the stance phase. More details on swing leg control are provided in sec: swing leg.
§.§ SRB-based Convex MPC
The calculation of GRFs in quadruped robots is often approached through Model Predictive Control (MPC) <cit.>. This method determines the optimal sequence of inputs over a finite-time horizon, taking into account any constraints within the dynamic model. Every time MPC is executed in the control system, only the first computed control input from the MPC cycle is applied. The inputs determined over the finite time horizon are only used for the optimization problem and are not directly applied in the control system.
To have the dynamic equation in the convenient state-space form, gravity should be added to the state. So, the system can represent as:
Ẋ^c = D^c X^c + H^c F
where
X^c = [[ p_c; Θ; ṗ_c; _b; ||g|| ]] ∈ℝ^13
D^c = [[ 0_3 0_3 1_3 0_3 0_3 × 1; 0_3 0_3 0_3 R_z(ψ) 0_3 × 1; 0_3 0_3 0_3 0_3 g/||g||; 0_3 0_3 0_3 0_3 0_3 × 1; 0_1 × 3 0_1 × 3 0_1 × 3 0_1 × 3 0 ]] ∈ℝ^13 × 13
H^c = [[ 0_6 × 12; M^-1A; 0_1 × 12 ]] ∈ℝ^13 × 12
We consider a linear MPC problem with horizon length k as follows:
min_F_i ∑_i=0^k-1e_i+1^T Q_i e_i+1 + F_i^T R_i F_i
s.t. X^c_i+1 = D_t,iX^c_i + H_t,iF_i
d≤CF_i≤d̅
where
F_i is the computed ground reaction forces at time step i, Q_i and R_i are diagonal positive semi-definite matrices, D_t,i and H_t,i are discrete time system dynamics matrices. The e_i+1 is the system state error at time step i define as e = [e_p, ė_p]^T ∈ℝ^12, with
e_p = [[ p_c-p_c,d; log(R_d R^T) ]]∈ℝ^6, ė_p = [[ ṗ_c-ṗ_c,d; _b -_b,d ]]∈ℝ^6,
where p_c,d∈ℝ^3 is the desired position of COM, ṗ_c,d∈ℝ^3 is the desired body's linear velocity, and _b,d∈ℝ^3 is the desired body's angular velocity. The desired and actual body orientations are described using rotation matrices R_d∈ℝ^3 × 3 and R∈ℝ^3 × 3, respectively. The orientation error is obtained using the exponential map representation of rotations <cit.>, where the log(.):ℝ^3 × 3→ℝ^3 is a mapping from a rotation matrix to the associated rotation vector <cit.>.
The constraint d≤CF_i ≤d̅ is equivalent to the constraint in equation (<ref>) at time step i.
§.§ Swing Leg Control
For the swing legs, the final footstep location for each leg is calculated from the corresponding hip location using a linear combination of Raibert heuristic <cit.>, and a feedback term from the capture point formulation <cit.>. The final footstep locations (p_f,i) are projected on an assumed ground plane and are calculated by:
p_f,i = p_h,i + T_c_ϕ/2ṗ_c,d + √(z_0/g)(ṗ_c - ṗ_c,d)
where T_c_ϕ is the stance time scheduled, z_0 is the height of locomotion and p_h,i∈ℝ^3 is the position of the corresponding hip i. A Beizer curve calculates the desired swing trajectory (including desired position p_d,i and velocity v_d,i) for swing legs which starts from the initial lift-off position p_0,i and ends at the final touch-down location p_f,i.
§.§ Low-level Control
The low-level leg control can generate joint torque commands from the high-level controller. For low-level force control, the controller transforms the force vector to the hip frame by rotation matrix R. Then, joint torques are calculated as follows:
τ_stance, i = -J(q_i)^TR^TF_i
where J(q_i)∈ℝ^3 × 3 is the leg Jacobian matrix and q_i is the joints angle of leg i-th.
To track the desired swing trajectory for each foot, a PD controller with a feedforward term is used to compute joint torques <cit.>:
τ_swing, i = J(q_i)^T[K_p,p(p_d,i - p_i)+K_d,p(v_d,i-v_i)]
where p_d,i and v_d,i are desired foot position and velocity, respectively, p_i and v_i are actual foot position and velocity in the robot's frame, K_p,p∈ℝ^3 × 3 and K_d,p∈ℝ^3 × 3 are the diagonal matrices of the proportional and derivative gains.
§ OVERVIEW OF THE PROPOSED APPROACH
This section will present an overview of our novel control architecture to incorporate adaptive control into the force control framework. While our approach is not limited to any specific adaptive control approach, we decide to use L_1 adaptive control <cit.> thanks to its advancement in guaranteeing fast adaptation and smooth control signals. Note that our proposed control system is designed for the stance leg control part in the control architecture of the quadruped robot (see fig: ControlOverview).
Our prior work <cit.> introduced an adaptive control based on Hybrid Zero Dynamics (HZD) <cit.> for bipedal robots. HZD is a common control approach for bipedal robots since it can handle hybrid and underactuated dynamics associated with this kind of robot.
In this paper, however, our approach leverages the combination of the adaptive control and force control system, which calculates ground reaction forces (GRFs) to achieve highly dynamic locomotion for quadrupeds <cit.>. The use of force control in legged robot systems has several key benefits, including increased robustness in the presence of challenging terrains <cit.> and the ability to accommodate a wide range of dynamic movements <cit.>, such as various types of locomotion gaits. By combining force control with adaptive control strategies that compensate for model uncertainty, achieving an enhanced control system with these advantages is possible.
The overview of our proposed adaptive force-based control system is presented in fig: main adaptive structure. By incorporating a L_1 adaptive controller, we aim to design a combined controller. The force-based controller calculates the optimal GRFs for following the desired trajectory. The adaptive controller calculates the residual parameters for compensating the nonlinear model uncertainty θ in the system dynamic. Therefore, the goal is to adjust adaptive control signal u_a as well as adaptation law to estimate the model uncertainty (θ̂) correctly and make the real model follows the reference model. For the reference model, we employ a similar linear model described in (<ref>), and we will update the reference model in real-time using an ODE solver. Moreover, the vector of uncertainties estimation θ̂ typically has high frequency due to fast estimation in the adaptation law. Thus, we employ a low-pass filter to obtain smooth control signals. We use the same swing leg control to appropriately synchronize the reference and real models. This means that we also use the real model's foot position for the reference model.
In the following sections, we will elaborate on integrating two different force-based control as the baseline controller into the adaptive control. First, in sec: adaptive control, we will describe the proposed method using a QP-based balancing controller, as presented in fig: ControlDiagram_QP. Then, in sec: adaptive MPC, we will show how to incorporate MPC into the adaptive controller in detail, as illustrated in fig: ControlDiagram_MPC.
§ ADAPTIVE FORCE-BASED CONTROL USING THE BALANCE CONTROLLER
In this section, we use the balance controller as the force-based controller, previously demonstrated in <cit.>.
In sec: adaptive MPC, we will present our control framework for integrating the L_1 adaptive control into MPC.
§.§ Closed-loop Dynamics
The L_1 adaptive control is basically designed for trajectory tracking; however, the goal of the balance controller is to compute optimal GRFs. Hence, to integrate the balance controller presented in sec: balance controller into L_1 adaptive control, we should relate the linear model described in (<ref>) into the closed-loop dynamics.
Let us consider the system state error (e) according to equation (<ref>) as the state variable. Therefore, the closed-loop error dynamics in state-space form can be represented as follow:
ė = D_l e + Bu,
where
D_l = [ [ 0_6 1_6; 0_6 0_6 ]]∈ℝ^12 × 12, B = [ [ 0_6; 1_6 ]] ∈ℝ^12 × 6
and u∈ℝ^6 is the control input function. By employing a PD control law, we have
u = [ -K_P -K_D ]e,
where K_P ∈ℝ^6 × 6 and K_D ∈ℝ^6 × 6 are diagonal positive definite matrices. According to definition of matrices D_l and B, from equation (<ref>) it can be obtained that:
ë_p = [[ p̈_c - p̈_c,d; _b - _b,d ]] = u,
where ë_p is the derivative of ė_p presented in (<ref>), p̈_c,d and _b,d are the desired COM linear acceleration and the desired angular acceleration, respectively. Since the desired trajectory is obtained from the velocity command, both desired accelerations p̈_c,d and _b,d are zero vectors. Then from (<ref>) and (<ref>), the desired dynamics can be given by:
b_d = M (u + G),
where M and G are defined in (<ref>).
By substituting (<ref>) into the QP problem (<ref>), we can obtain the optimal GRFs as the input for the low-level leg controller.
The objective of the QP formulation in equation (<ref>) is to find a solution that ensures the actual dynamics AF match the desired dynamics b_d. In general, the QP-based balance controller is capable of achieving the desired control input function outlined in equation (<ref>), thus keeping the error e within a certain range. However, if the desired dynamics vector b_d violates any of the inequality constraints, such as force limits or friction constraints, the controller may yield an optimal solution F^* that may not completely align with the desired dynamics. With this solution, the optimal dynamic b_d^* and u^* can be written as:
b_d^* = AF^*,
u^* = M^-1 b_d^* - G
where in the appendix, we will show that the u^* remains within a bounded range.
Note that the optimal ground reaction force F^* serves as the control input for the robot and the variable u^* acts as an input for the closed-loop dynamic. The closed-loop structure for the robot is depicted in fig: ControlDiagram_QP (the green dashed line).
§.§ Effects of Uncertainty on Dynamic
If we consider uncertainty in the dynamic equation (<ref>) and assume that the matrices D and H are not accurate, then we need to present the dynamic based on the nominal matrices D̅, H̅. The model uncertainty mostly comes from inaccurate values for mass, inertia, and foot position with respect to the center of mass. In addition to that, various terrain (e.g., rough terrain or soft terrain) might have a different impact on the robot, and it is unknown in a practical situation. Therefore, terrain uncertainty should also be considered in the dynamic model. In this section, we solely derive our control equations based on the model uncertainty. In sec: terrain, we will elaborate on how our proposed control system can also consider terrain uncertainty.
There is another parameter involved in the dynamic equation, namely the yaw angle. This angle is obtained through the state estimation, and we assumed that the state estimation has minimal uncertainty.
According to the definition of matrices D and H in (<ref>), the inaccurate value of the dynamic parameter mentioned above reflects on the H matrix. Therefore, the dynamic equation in the presence of uncertainty can be represented as:
Ẋ = DX+ (H̅+H̃) F + [[ 0_6 × 1; G ]]
where H̃ represent the uncertainty in matrix H.
It is worth noting that according to the definition of H in equation (<ref>), the first six rows of H consist of zeros. Thus, we can rephrase the dynamic equation (<ref>) as follows:
Ẋ = DX + H̅F +BG + Bθ
where θ∈ℝ^6
is the vector of uncertainty for six corresponding equations and is defined as follows:
θB^T H̃F
With reference to the state representation given by equation (<ref>), the vector θ can be interpreted as a time-varying disturbance affecting the body and orientation accelerations.
The uncertainty vector θ depends on both time t and F. Since F is obtained through the QP problem (<ref>), it is a function of b_d. Furthermore, b_d is a function of u according to (<ref>). Considering that u is determined by the PD control (<ref>), we can conclude that θ is a function of both the tracking error e and time t. As a result, for any given time t, it is always possible to find α(t)∈ℝ^6 and β(t)∈ℝ^6 satisfying <cit.>:
θ(e,t)=α(t)||e||+β(t).
§.§ Designing Adaptive Controller for Compensating the Uncertainty
By incorporating L_1 adaptive controller, we want to design a combined controller u=u_1+u_2, where u_1 is the control input to follow the desired trajectory for the nominal model as presented in (<ref>) and u_2 is to compensate the nonlinear model uncertainties θ. Therefore, the goal is to adjust the control signal u_2 so that the real model can follow the reference model. For the reference model, we employ a similar linear model described in (<ref>) which, instead of M, the nominal matrix M̅ is being used. The diagram of our proposed force-based adaptive control based on a balance controller is presented in fig: ControlDiagram_QP.
The duplicate version of equation (<ref>) for state space representation presented in (<ref>) by considering combined controller u=u_1+u_2 is as follows:
ė=D_l e+Bu_1 + B (u_2+θ).
Note that the vector of uncertainty θ in equations (<ref>) and (<ref>) are not the same since the state vector of equation (<ref>) is X while the state vector of equation (<ref>) is system error e.
The state representation for the reference model can be expressed as follows:
ê̇=D_l ê+Bû_1+B (u_2+θ̂),
where,
θ̂=α̂||e||+β̂,
and û_1 is defined as:
û_1 = [ -K_P -K_D ]ê.
To compensate the estimated uncertainty θ̂, we can just simply choose u_2=-θ̂ to obtain
ê̇=D_l ê+Bû_1.
However, θ̂ typically has high frequency due to fast estimation in the adaptation law. Therefore, we employ a low-pass filter to obtain smooth control signals as follows:
u_2=-C(s)θ̂,
where C(s) is a second-order low-pass filter with a magnitude of 1:
C(s) = ω_n^2/s^2 + 2 ζω_n s+ ω_n^2 .
According to (<ref>), the b_d for the real model in the presence of uncertainty get the following form:
b_d = M̅ (u_1 + u_2 + G).
Respectively, b̂_d for reference model is as follows:
b̂_d = M̅ (û_1 + u_2 + θ̂ + G).
The QP solver outlined in equation (<ref>) allows us to obtain the optimal GRFs for the real model. Similarly, the optimal GRFs F̂ for the reference model can be obtained as follows:
F̂^* = F̂∈ℝ^12argmin (ÂF̂ - b̂_d)^T S (ÂF̂ - b̂_d)
+ γ_1 F̂^2 + γ_2 F̂ - F̂_prev^* ^2
d≤CF̂≤d̅
F̂_swing^z=0 .
Define the difference between the real model and the reference model ẽ=ê-e, we then have,
ẽ̇=D_l ẽ+Bũ_1+B (α̃||e||+β̃),
where
ũ_1=û_1-u_1, α̃=α̂-α, β̃=β̂-β.
As a result, we will estimate θ indirectly through α and β, or the values of α̂ and β̂ computed by the following adaptation laws based on the projection operators <cit.>,
α̇̂̇=ΓProj(α̂,y_α), β̇̂̇=ΓProj(β̂,y_β)
where Γ∈ℝ^6 × 6 is a symmetric positive definite matrix. The projection functions y_α∈ℝ^6 and y_β∈ℝ^6 are:
y_α =-B^T Pẽ||e||,
y_β =-B^T Pẽ,
where P∈ℝ^12 × 12 is a positive definite matrix that is defined according to the stability criteria using the Lyapunov equation. Moreover, the stability proof of the system is provided in the appendix.
§ ADAPTIVE FORCE-BASED CONTROL USING MPC
Model predictive control (MPC) has been widely used across various fields, from finance to robotics. One of MPC's main advantages is its ability to handle complex systems with multiple inputs and outputs while considering hard control constraints <cit.>.
MPC has also been applied to quadruped robots, providing stable locomotion <cit.>. Thanks to dynamic prediction in MPC, by using the same control framework, it can achieve different dynamic locomotion gaits.
However, MPC's limitations become evident when dealing with significant uncertainty in the dynamic model. For instance, in the case of a quadruped robot carrying an unknown heavy load, MPC fails to track the desired state trajectory, resulting in unstable behavior and deviation from the desired trajectory, especially with dynamic gaits like bounding.
Furthermore, the ability of a robot to traverse soft terrain, where the impact model is unknown, can present a significant challenge. Our proposed approach can tackle this challenge effectively, and we will discuss the details of how it handles the terrain unknown impact model in sec: terrain.
In the previous section sec: adaptive control, we presented an adaptive force-based control framework based on the balance controller.
The balance controller relies on a quadratic program (QP) solver, which is simple to put into practice and well-suited for motions that are slow and safe, like standing and quasi-static walking.
Additionally, the balance controller is an instantaneous control technique, meaning it does not predict the robot's future movement. As a result, the balance controller proves to be ineffective in fast-paced, highly dynamic scenarios. On the other hand, MPC has shown great potential in handling agile motions, even when it comes to underactuated gaits such as bounding.
In this section, we will present a novel control architecture to integrate adaptive control into the MPC framework.
By this proposed framework, we can achieve fast and robust locomotion in the presence of uncertainties. This framework can also be extended to accommodate various dynamic gaits, such as trotting and bounding, in legged robots. As we discussed in a previous section, our approach is not restricted to a specific type of adaptive control, but we have chosen to utilize L_1 adaptive control, which has demonstrated advantages over other adaptive control techniques. The first step in integrating L_1 adaptive control and MPC is to understand the importance of a reference model and the challenges in synchronizing the real model and reference model. We then present our proposed adaptive MPC, which combines conventional MPC <cit.> with adaptive control.
Finally, we address the challenge of real-time computation while having two MPCs in our control system. We will elaborate on how to adjust the frequency of each control component in an optimized manner to allocate enough computation resources for critical control parts and achieve real-time computation.
§.§ Reference Model
Our method aims to design a combined controller based on MPC and L_1 adaptive control that the real model follows the reference model.
In accordance with our previous discussion in sec: L1_adaptive, the combined controller incorporates a control signal u_2 to account for model uncertainty, as indicated in equation (<ref>).
In this section, the auxiliary control signal for this purpose is u_a ∈ℝ^6, thus, the uncertain dynamic equation (<ref>) can be rewritten as follow:
Ẋ = DX + H̅F + BG + B (u_a + θ).
The reference model is similar to the quasi-linear model described in (<ref>) which, instead of H, the nominal matrix H̅ is being used. The proposed adaptive MPC diagram is presented in fig: ControlDiagram_MPC.
We consider a reference model for L_1 adaptive control that arises from MPC. The MPC method is computationally expensive, but replacing it with other simpler control methods, such as the balance controller while simulating the robot's performance using dynamic gaits such as bounding is impossible. The reason is that in bounding gait, the robot's two feet on either the front or rear side touch the ground at each time step, making it challenging to accurately control the height and pitch angle. The MPC approach balances the error in the height and pitch angle and, based on the predicted dynamics of the system in the future, computes the optimal ground reaction forces. As seen in fig: bounding snapshot, the center of mass (COM) height oscillates around the desired value. Thus, the underactuated nature of certain gaits like bounding necessitates the use of MPC as the control system for the reference model.
When implementing MPC for a reference model, one challenge is ensuring that the reference model is synchronized with the real model. This is particularly important when the robot performs a gait with a periodic behavior, such as bounding (see fig: bounding snapshot). In order to correctly compare the real model with the reference model, both should have the same gait schedule. Additionally, the adaptive MPC proposed for legs in the stance phase is independent of the swing leg control. However, the foot position is crucial in calculating the moment of ground reaction force around the center of mass. Therefore, to maintain consistency between the real and reference models, it is important to ensure that the real robot's foot position is fed into the reference model as shown in fig: ControlDiagram_MPC.
The reference model can be expressed as follows:
Ẋ̂̇ = DX̂ + H̅F̂ + BG + B(u_a+θ̂),
where
θ̂=α̂||e||+β̂.
In this case, similar to sec: adaptive control, we use a second-order low-pass filter, same as (<ref>). Therefore, the auxiliary control signal would be:
u_a=-C(s)θ̂.
By defining the difference between the real model and the reference model X̃=X̂-X, we then have:
Ẋ̃̇=DX̃+H̅F̃+B(α̃||e||+β̃),
where
F̃=F̂-F, α̃=α̂-α, β̃=β̂-β.
Since the desired trajectory for both the real model and the reference model is the same (X_d = X̂_d), the difference between the real model and reference model can be defined as:
X̃ = (X̂ - X̂_d) - (X - X_d) = ê - e = ẽ.
Therefore, equation (<ref>) is equal to the following equation:
ẽ̇=Dẽ+H̅F̃+B(α̃||e||+β̃).
The adaption laws and projection functions for computing the value of α and β are the same as equations (<ref>) and (<ref>), respectively. Moreover, the stability of the control system can be proven using the same logic provided in the appendix.
§.§ Adaptive MPC
After computing the auxiliary control signal u_a using the adaptive controller presented in the previous subsection, we will integrate the u_a with the conventional MPC for legged locomotion <cit.> and propose our adaptive MPC framework. We treat the auxiliary control signal u_a as a residual vector in the system's equation to compensate for dynamic uncertainty. Therefore, the u_a should be combined into the state vector and the equation (<ref>) can be written as follow:
η̇ = D^eη + H̅^eF + B^eθ
with the following extended matrices:
η = [[ X^c; u_a ]] ∈ℝ^19
D^e = [[ D^c_13 × 13 0_6 × 6
1_6 × 6
0_1 × 6; 0_6 × 13 0_6 × 6 ]] ∈ℝ^19 × 19
H̅^e = [[ H̅^c; 0_6 × 12 ]] ∈ℝ^19 × 12
B^e = [[ B; 0_7 × 6 ]] ∈ℝ^19 × 6
where H̅^c is the nominal value of H^c. The definition of X^c, D^c, and H^c can be found in (<ref>). Although u_a is considered a part of the state vector in (<ref>), it is just a residual vector for compensating dynamic uncertainty. Therefore, u_a is constant in the state space equation and over the horizons. To this end, the components associated with u_a in matrices D^e and H̅^e are assigned zero, which means u̇_a = 0. Note that the value of u_a will be updated according to the adaptive law, but it is constant during the prediction horizons.
The state representation in (<ref>) is also convenient for discretization methods such as zero-order hold <cit.> for MPC.
Therefore, our adaptive MPC can be designed according to (<ref>) and based on the following discrete-time dynamic:
η_i+1 = D^e_t,iη_t,i + H̅^e_t,iF_i
§.§ Real-time Computation
The main challenge in executing our proposed adaptive MPC framework is ensuring that the computation required is fast enough to be performed in real-time for hardware experiments. If the controller is unable to perform updates at a high frequency, it could result in the robot collapsing during dynamic motion. The control system comprises two MPCs, each with 13 to 19 states predicted over ten horizons. To ensure the robot's balance and allocate sufficient computation resources to each control component, we have devised a scheme, as depicted in fig: ControlDiagram_MPC, to update each control component in an optimized manner.
The robot's sensory data updates in real-time with a frequency of 1 kHz. Thus, the reference model should update with the same frequency to compare the reference model states (X̂) and real model states (X) correctly. The yellow dashed line in fig: ControlDiagram_MPC indicates the update frequency for the reference model. We use the odeint package from Boost software in C++ <cit.> to solve the ODE problem associated with the dynamic equation for the reference model.
One of the critical components in our proposed framework is the adaptive MPC, which is responsible for computing the ground reaction force for the robot, as shown in fig: ControlDiagram_MPC). Through our experimentation, we have determined that for robust locomotion with dynamic gaits, the optimal update frequency for the adaptive MPC should be 300 Hz. In contrast, the reference MPC, which plays a supporting role in the control system, is less sensitive and runs at a slower rate of 30 Hz. In addition, there is a two-millisecond delay between the running of the adaptive MPC and reference MPC to ensure sufficient computational resources are allocated to each component. This means that the two MPC frameworks do not run simultaneously in our control system.
§ ADAPTATION TO UNKNOWN IMPACT MODEL
The dynamic formulation presented in sec: adaptive control and sec: adaptive MPC considers the presence of model uncertainty in real-world situations. It is assumed that the terrain is hard enough to allow the robot receives the desired force as ground reaction forces on its feet. However, this assumption may not hold if the robot walks on soft or elastic terrain with an unknown impact model, which may not generate the desired force needed for stable locomotion.
Some previous studies have included terrain knowledge and contact models in their balancing controllers to address the soft terrain challenge, mainly using a spring-damper model to characterize the soft terrain <cit.>. Some control frameworks for adapting to soft terrain in real-time have also been developed using iterative learning <cit.> and whole-body control <cit.>, without prior knowledge about the terrain. This section demonstrates that the proposed method in sections sec: adaptive control and sec: adaptive MPC can also handle unknown impact models from terrain, allowing the robot to maintain stability while walking on soft terrains.
Assume the computed force F by MPC in (<ref>) cannot be achieved perfectly due to walking on soft terrain. Therefore, equation (<ref>) can be rewritten as follow:
Ẋ = DX + H̅ (F_a + F̃_a) + BG + Bθ
which F_a is the actual ground reaction force exerted on the robot and F̃_a is the difference between the desired ground reaction force and actual reaction force.
Given that F̃_a depends on the tracking error e and time, the uncertainty vector arising from the ground reaction force can be incorporated with θ. Therefore, we can reformulate equation (<ref>) as follows:
Ẋ = DX + H̅F_a + BG + B (θ + θ_F).
where the uncertainty vector θ_F is defined as follow:
θ_F B^T H̅F̃_a
The equation (<ref>) is in the form of equation (<ref>), which uses actual ground reaction force instead of desired ground reaction force. Therefore, all formulations for implementing adaptive controllers are also valid for a situation with an unknown impact model.
§ RESULTS
In this section, we validate our control approach in simulation and hardware experiments on a Unitree A1 robot.
All the hardware experiment's computation runs on a single PC (Intel i7-6500U, 2.5 GHz, 64-bit). For simulation, the control system is implemented in ROS Noetic with the Gazebo 11 simulator, which provides a high-fidelity simulation of the A1 robot. A video showcasing the results accompanies this paper[<https://youtu.be/QmwyysdTk1k>].
We set the control parameters for MPC, the adaption law, and the low-pass filter as presented in Table <ref>. We use one set of parameters for all the experiments with different locomotion gaits, indicating that our approach is easily generalizable.
The following subsections will introduce different experiment results in terms of model and environment uncertainty (see fig: terrain experiment). In each experiment, the robot starts by using a balance controller to stand up and then switches to the MPC framework for walking or running.
§.§ Comparative Analysis
In order to evaluate the performance of our proposed adaptive MPC method, we conduct a comparative experiment with the conventional MPC method presented in <cit.>. The objective is to understand the advantages of integrating the adaptive controller into MPC for quadrupedal locomotion.
§.§.§ Walking with significant model uncertainty
The experiment involves the robot walking and rotating in different directions, using both adaptive and non-adaptive controllers while carrying an unknown load. The results of the experiment show that the adaptive controller provides robust locomotion, with excellent tracking error, even when carrying an unknown 5 kg load. On the other hand, the non-adaptive controller results in a considerable error in the COM height and eventually collapses under the weight of just a 3 kg load. The comparative results for the adaptive and non-adaptive controllers are shown in fig: comparison exp.
§.§.§ Walking on soft terrain
To evaluate the capability of our proposed control method in handling unknown impact models, we conducted an experiment where the robot was made for walking on a double foam, which symbolizes a soft terrain. The performance of both the adaptive and non-adaptive controllers was evaluated and compared. The results are depicted in fig: soft terrain exp, which represents the robot's roll angle. The figure clearly illustrates that the adaptive controller was able to maintain the robot's balance on the soft terrain, while the non-adaptive controller was unable to do so, leading to the collapse of the robot.
§.§ Running with Multiple Gaits
To demonstrate the superiority of our proposed approach for dynamic gaits, we conducted experiments with the robot running while carrying an unknown load. These experiments were carried out for both the trotting and bounding gaits, with an unknown load of 5 kg and 3 kg, respectively. The results of these experiments are shown in <ref>. It can be seen from the figure that the tracking of the center of mass height during the bounding gait is more unstable compared to the trotting gait, which is due to the inherent underactuated nature of the bounding gait.
§.§ Time-varying Load
To demonstrate the effectiveness of our proposed adaptive force control in adapting to model uncertainty, we conducted simulations where the robot carries a time-varying load of up to 92% of its weight during walking. As shown in fig: time_varying result, our approach can enable the robot to adapt to time-varying uncertainty. In the simulation, the robot starts with an unknown 5 kg load. While increasing the robot's velocity, the robot is subjected to a varying external force in the z-direction that rises to 60 N, resulting in an additional unknown 11 kg load. These results indicate that our proposed approach effectively handles high levels of model uncertainty.
§.§ Terrain Uncertainty
To demonstrate the capability of our proposed method to handle terrain uncertainty, we tested the robot navigating various terrain while carrying an unknown 5 kg load. To this end, we tried walking experiments on multiple rough terrains as well as high-sloped terrain, and we got impressive results.
§.§.§ Rough terrain
We tested the robot navigating various rough terrains such as grass and gravel. The robot walks and rotates in multiple directions while carrying an unknown 5 kg load. Some snapshots of the robot walking on diverse rough terrain are presented in fig: terrain experiment. Our approach is based on a force controller and retains the robustness features of the baseline framework, allowing the robot to handle the rough terrain effectively.
§.§.§ Sloped terrain
To enable the robot to climb the sloped terrain perfectly without vision, we need to adjust its orientation to make its body parallel to the walking surface. This is done by using the footstep location to estimate the slope of the ground. For each i-th leg, we can measure the foot position p_i = (p_x,i, p_y,i, p_z,i) and build the vector of feet x-position (p_x), y-position (p_y), and z-position (p_z). Thus, we can model the walking surface as a plane:
z(x,y) = a_0 + a_1 x + a_2 y
and the coefficients (a_0, a_1, and a_2) will be obtained through the solution of the least square problem using p_x, p_x, and p_x data (see <cit.> for more details).
Note that the desired roll and pitch angles for the robot will be modified on the slope according to the following:
roll = arctan (a_2) , pitch = arctan(a_1).
As a result, the reference model's desired pitch and roll angles must be adjusted to the non-zero values determined as described above. It's important to note that the reference model utilizes the actual foot position of the robot, so there is no need to make any changes to the reference model's footstep planning when the robot is attempting to climb a slope.
§ CONCLUSION
In conclusion, a novel control system has been presented that incorporates adaptive control into force control for legged robots walking under significant uncertainties. We have demonstrated our proposed approach's effectiveness using numerical and experimental validations. The experiments show the success of the implementation of the proposed adaptive force control on quadruped robots, allowing them to walk and run while carrying an unknown heavy load on their trunk. The results are remarkable, with the robot being able to carry a load of up to 5 kg (50% of its weight) while still keeping the tracking error within a small range and maintaining stability even in all directions. The experiment demonstrates that the proposed adaptive force control system cannot only adapt to model uncertainty but also leverage the benefits of force control in navigating rough terrains and soft terrain. On the other hand, the baseline non-adaptive controller fails to track the desired trajectory and causes the robot to collapse under uncertainty.
§ ACKNOWLEDGMENTS
The authors would like to thank Yiyu Chen at Dynamic Robotics and Control lab (DRCL) for his help in conducting the hardware experiments.
§.§ Linear Quadratic Lyapunov Theory
According to Lyapunov theory <cit.>, the PD control described in (<ref>) will asymptotically stabilize the system if
A_m = [ 0_6 1_6; -K_P -K_D ]∈ℝ^12 × 12
is Hurwitz. This means that by choosing a control Lyapunov function candidate as follows:
V(e) = e^TPe,
where P∈ℝ^12 × 12 is the solution of the Lyapunov equation
A_m^T P + PA_m = -Q_L,
and Q_L∈ℝ^12 × 12 is any symmetric positive-definite matrix. We then have:
V̇(e,u) + λ V(e) = e^T (D_l^T P + PD_l) e
+ λ V(e) +2 e^T PBu ≤ 0,
where,
λ = λ_min(Q_L)/λ_max(P) > 0.
As a result, the state variable e and the control input u always remain bounded:
e≤δ_η, u≤δ_u.
However, the control signal u^* (<ref>) we construct by solving QP problem (<ref>), is not always the same as u. Based on the friction constraints present in equation (<ref>), the value of F^* is always bounded. Besides, according to the definition of A, M, and G, these matrices also have bounded values. Thus, it implies that:
u^* ≤δ_u^*.
Therefore, the vector of difference between u and u^* can be defined as:
Δ = u^* - u
which is also bounded according to (<ref>) and (<ref>):
Δ≤δ_Δ.
By substituting u^* in (<ref>), we have:
V̇(e,u^*) + λ V(e) ≤ 2 e^T PBΔ≤ϵ_V,
where
ϵ_V = 2 Pδ_ηδ_Δ.
§.§ Stability Analysis
Theorem: Consider the system dynamics with uncertainty described by (<ref>), and a reference model described by (<ref>). Assume the use of an L_1 adaptive controller with the optimal closed-loop control signal given by (<ref>), the adaptive control signal given by (<ref>), and the adaptation laws given by (<ref>). Then, under the aforementioned L_1 adaptive controller, the tracking error between the real model and reference model denoted as ẽ, as well as the errors between the real and estimated uncertainty, denoted as α̃ and β̃, respectively, are bounded.
Proof: Let us consider the following control Lyapunov candidate function:
Ṽ=ẽ^TPẽ+α̃^TΓ^-1α̃+β̃^TΓ^-1β̃.
Therefore, its time derivative will be
Ṽ̇=ẽ̇^TPẽ+ẽ^TPẽ̇ +
α̇̃̇^TΓ^-1α̃+α̃^TΓ^-1α̇̃̇
+
β̇̃̇^TΓ^-1β̃+β̃^TΓ^-1β̇̃̇,
in which we have
ẽ̇^TPẽ+ẽ^TPẽ̇ =
(D_lẽ+BF̃)^TPẽ
+
ẽ^TP(D_lẽ+BF̃)
+
α̃^TB^T||e||Pẽ+ẽ^TPBα̃||e||
+β̃^TB^TPẽ+ẽ^TPBβ̃.
Because ẽ=ê-e satisfies the condition imposed by (<ref>), it implies that:
(D_lẽ+BF̃)^TPẽ +
ẽ^TP(D_lẽ+BF̃)
≤
-λẽ^TPẽ + ϵ_Ṽ,
where
ϵ_Ṽ = 2 Pδ_ẽδ_Δ̃.
Furthermore, with the property of the projection operator <cit.>, we have the following:
(α̂-α)^T(Proj(α̂,y_α)-y_α)≤ 0,
(β̂-β)^T(Proj(β̂,y_β)-y_β)≤ 0.
From (<ref>) and (<ref>), we can imply that
α̃^TΓ^-1α̇̃̇≤α̃^Ty_α-α̃^TΓ^-1α̇,
β̃^TΓ^-1β̇̃̇≤β̃^Ty_β-β̃^TΓ^-1β̇.
We now replace (<ref>), (<ref>) and (<ref>) to (<ref>), which results in
Ṽ̇ ≤
-λẽ^TPẽ + ϵ_Ṽ
+
α̃^T(y_α+B^TPẽ||e||)-α̃^TΓ^-1α̇
+
(y_α^T+ẽ^TPB||e||)α̃-α̇^TΓ^-1α
+
β̃^T(y_β+B^TPẽ)-β̃^TΓ^-1β̇
+
(y_β^T+ẽ^TPB)β̃-β̇^TΓ^-1β̃
So, by using the chosen projection functions (<ref>), then we conclude that:
Ṽ̇+λṼ≤ϵ_Ṽ +
λα̃^TΓ^-1α̃+
λβ̃^TΓ^-1β̃
-α̃^TΓ^-1α̇ -α̇^TΓ^-1α̃
-β̃^TΓ^-1β̇ -β̇^TΓ^-1β̃.
We assume that the uncertainties α, β, and their time derivatives are bounded. Furthermore, the projection operators (<ref>) will also keep α̃ and β̃ bounded (see <cit.> for a detailed proof about these properties.) We define these bounds as follows:
||α̃|| ≤ α̃_b ,
||β̃|| ≤β̃_b ,
||α̇|| ≤ α̇_b ,
||β̇|| ≤β̇_b.
Combining this with (<ref>), we have,
Ṽ̇+λṼ≤λδ_Ṽ,
where
δ_Ṽ=2||Γ||^-1(α̃_b^2+β̃_b^2+1/λα̃_bα̇_b+1/λβ̃_bβ̇_b) + 1/λϵ_Ṽ.
Thus, if Ṽ≥δ_Ṽ then Ṽ̇≤ 0. As a result, we always have Ṽ≤δ_Ṽ.
In other words, by choosing the adaptation gain Γ sufficiently large and P relatively small, we can limit the Control Lyapunov Function (<ref>) in an arbitrarily small neighborhood δ_Ṽ of the origin.
According to (<ref>) and (<ref>), achieving a small value for P depends on choosing a proper value for K_P, K_D, and Q_L. Therefore, the value of PD gains affects the stability of the whole system.
Finally, the tracking errors between the dynamics model (<ref>) and the reference model (<ref>), ẽ, and the error between the real and estimated uncertainty, α̃, β̃ are bounded as follows:
||ẽ|| ≤√(δ_Ṽ/||P||) ,
||α̃|| ≤√(||Γ||δ_Ṽ) ,||β̃|| ≤√(||Γ||δ_Ṽ).
[
< g r a p h i c s >
]Mohsen Sombolestan
received his B.Sc. degree in mechanical engineering in 2017 from Sharif University of Technology, Tehran, Iran, and his M.Sc. degree in mechanical engineering in 2020 from Isfahan University of Technology, Isfahan, Iran. He is working toward a Ph.D. in mechanical engineering from University of Southern California, Los Angeles, CA, USA.
His research interests include control system design in robotic applications, especially legged robots, focusing on adaptive control and reinforcement learning.
[
< g r a p h i c s >
]Quan Nguyen
is an assistant professor of Aerospace and Mechanical Engineering at the University of Southern California (USC). Before joining USC, he was a Postdoctoral Associate in the Biomimetic Robotics Lab at the Massachusetts Institute of Technology (MIT). He received his Ph.D. from Carnegie Mellon University (CMU) in 2017 with the Best Dissertation Award.
His research interests span different control and optimization approaches for highly dynamic robotics, including nonlinear control, trajectory optimization, real-time optimization-based control, and robust and adaptive control. His work on the MIT Cheetah 3 robot leaping on a desk was featured widely in many major media channels, including CNN, BBC, NBC, ABC, etc. Nguyen won the Best Presentation of the Session at the 2016 American Control Conference (ACC) and the Best System Paper Finalist at the 2017 Robotics: Science & Systems Conference (RSS).
|
http://arxiv.org/abs/2307.04298v1 | 20230710013021 | Edge Storage Management Recipe with Zero-Shot Data Compression for Road Anomaly Detection | [
"YeongHyeon Park",
"Uju Gim",
"Myung Jin Kim"
] | cs.SD | [
"cs.SD",
"cs.LG",
"eess.AS"
] |
Edge Storage Management Recipe with Zero-Shot Data Compression for Road Anomaly Detection
YeongHyeon Park
SK Planet Co., Ltd.
Seongnam, Rep. of Korea
[email protected]
Uju Gim
SK Planet Co., Ltd.
Seongnam, Rep. of Korea
[email protected]
Myung Jin Kim
SK Planet Co., Ltd.
Seongnam, Rep. of Korea
[email protected]
August 12, 2023
===============================================================================================================================================================================================================================================
Recent studies show edge computing-based road anomaly detection systems which may also conduct data collection simultaneously.
However, the edge computers will have small data storage but we need to store the collected audio samples for a long time in order to update existing models or develop a novel method.
Therefore, we should consider an approach for efficient storage management methods while preserving high-fidelity audio.
A hardware-perspective approach, such as using a low-resolution microphone, is an intuitive way to reduce file size but is not recommended because it fundamentally cuts off high-frequency components.
On the other hand, a computational file compression approach that encodes collected high-resolution audio into a compact code should be recommended because it also provides a corresponding decoding method.
Motivated by this, we propose a way of simple yet effective pre-trained autoencoder-based data compression method.
The pre-trained autoencoder is trained for the purpose of audio super-resolution so it can be utilized to encode or decode any arbitrary sampling rate.
Moreover, it will reduce the communication cost for data transmission from the edge to the central server.
Via the comparative experiments, we confirm that the zero-shot audio compression and decompression highly preserve anomaly detection performance while enhancing storage and transmission efficiency.
anomaly detection, data compression, edge computing, storage management, transmission efficiency
§ INTRODUCTION
Knowing road conditions is an effective way to prevent traffic accidents <cit.>.
Most of the road hazards are highly related to icy or wet roads which reduce the friction between the road and tires.
When considering a vision sensor-based road anomaly detection system <cit.>, inclement weather will make occlusion on the camera which makes it difficult for understanding road conditions <cit.>.
Moreover, intensity-changing situations such as at night also make it difficult to determine road conditions.
As an approach to solving these problems, an audio-based anomaly detection approach has been developed.
The audio-based system receives information from the medium wave in the air, so it shows a better response-ability than the occlusion situation of the vision sensor.
In addition, sound can be properly transmitted even at night time, an audio-based system is recommended for this situation.
Even in the case of successful anomaly detection as above, continuous updating of the anomaly detection model is required considering that target environments are continuously aging <cit.>.
For this, we need high-quality large data to update the neural network-based anomaly detection model.
We have installed edge computers on the road for anomaly detection, which has limited resources such as storage.
Considering the above, we have a limitation to keep audio data for the long term.
The above limitation can be partially mitigated by transmitting the audio to large central storage in time, but in this case, enormous data transmission costs will be incurred <cit.>.
Also, we can lower the quality of the audio collected from high fidelity (Hi-Fi) to low fidelity (Lo-Fi) to reduce the file size for each audio, but this is not recommended as it leads to fundamental information loss.
Motivated by this, we propose a storage management method that can keep as much data as possible in the edge computer for a long time in limited storage space while minimizing the cost of the data transmission into a central server.
Our method is based on zero-shot encoding and decoding using a pre-trained audio super-resolution (ASR) model <cit.>.
Referring ASR models can convert input audio to high-resolution, so they can handle arbitrary resolution inputs that are lower than the target resolution they are trained on.
The overall scheme of our proposal is shown in Fig <ref>.
The edge computer continuously collects data and determines whether the situation is abnormal or not through a microphone facing the road.
Basically, collected audio is stored in the original resolution, but it is saved after being converted into a latent vector by an encoder of pre-trained ASR.
The latent vectors, stored in edge storage, are sent to the central server at regular intervals or the storage space fills up to a certain level.
Then, at the central computer, they are restored to audio form by a decoder paired with the above encoder <cit.>.
At this time, even if the resolution of the collected sound sources is different, it is characterized by being restored to a similar Hi-Fi quality by the decoder of the central server.
The restored audio data is used for the purpose of developing a novel anomaly detection model or updating existing models.
To verify the proposed method, we collected data from three roads with different characteristics.
The audio is basically collected at 44,100 Hz.
For the purpose of saving storage space experiments in which downsampling is applied to assume a situation in which a microphone such as 11,025 Hz, and our zero-shot audio encoding method are also covered.
Overall, our contributions are summarized below:
* When downsampling is applied, high-frequency information loss occurs, confirming that hinders precise anomaly detection. Our approach, zero-shot encoding and decoding minimize information loss and preserves anomaly detection performance at an appropriate level while maximizing storage efficiency.
* We show that there is no need to train new models to encode and decode for our domain, road noise. Our example deals with road anomaly detection as a target, but our method can also be extended to another edge computer-based data collection approaches in other domains.
§ RELATED WORKS
§.§ Road condition identification
To identify abnormal situations on the road such as road bumps, cracks, or potholes, methods based on vision sensors have been proposed <cit.>.
However, since the abnormal situation on the road is highly related to bad weather, and bad weather can obscure the view of the camera <cit.>, a vision sensor-based approach will show constrained detection performance.
Some approaches using motion or gyroscope sensors rather than a vision sensor can partially ease the above problem but it shows unstable performance that highly depends on pre-defined settings <cit.>.
An audio-based anomaly detection method has been proposed as a way to overcome the problem of invisible situations to make decisions in bad weather or night situations <cit.>.
To construct a reliable road anomaly detection system in outdoor environments, we inherit the above approach from prior research to take advantage of audio-based road anomaly detection.
§.§ Data compression
The most intuitive way to maximize data storage efficiency is to collect data at a lower resolution.
However, when we need to create a high-quality anomaly detection model, we also need high-resolution data <cit.>.
The computational approach which collects high-resolution data and encodes it into lower dimensions can be considered other than the above <cit.>.
When the encoding method is provided with a paired decoding method, we can easily compress the data into small sizes and decompress them into the original resolution.
In this case, some information loss may occur in the process of encoding and decoding, but it is recommended as an alternative to hardware-based capacity-saving methods that block information fundamentally.
An artificial neural network-based encoder-decoder (ED) shows better reconstruction performance than traditional methods <cit.>.
In particular, a model trained for super-resolution (SR) purposes is useful as a method to help roughly estimate the high-frequency region of data collected at lower resolution <cit.>.
We propose a storage space management method based on zero-shot encoding decoding using the pre-trained ED for SR purposes, considering that the resolution of audio sensors installed in each region may be different.
§ APPROACH
§.§ Overview
The overall of our proposal is shown in Fig. <ref>. which is an encoding and decoding system for storing as much audio data as possible in an edge computer.
We separate pre-trained ED into each component encoder and decoder.
Then, we locate each of the above on the edge and central computers respectively.
Our approach only uses a single encoder and decoder pair rather than having each model for each different road environment as shown in Fig. <ref>.
This is dubbed as a zero-shot inference that can eliminate the hassle of preparing models to reflect the surrounding environment of countless sensors installed at outdoor points.
§.§ Audio super-resolution
For data compression, we utilize a pre-trained ASR model, EnCodec <cit.>, which is constructed with an encoder and decoder.
The sensors installed at each site to perform the road anomaly detection that we will cover may include expensive high-resolution microphones or low-resolution microphones, depending on the management budget of the local government.
The advantage of using the ASR model is that the input data can be converted into high-resolution data regardless of the input resolution.
This allows audio data collected from microphones of arbitrary resolutions as aforementioned can be integrated into Hi-Fi audio.
Any ASR model can be employed for the encoding and decoding process, but a structure in which the encoder and decoder can be used separately is recommended.
A method of performing information augmentation in a feature map or latent vector stage may rather increase the capacity of encoded data <cit.>, so it should be avoided.
§.§ Zero-shot encoding and decoding
It is difficult to obtain a pre-trained ASR model on the road noise domain because the audio corresponding to the friction noise between the tire and the road surface, which we deal with, is not commonly used data.
In addition, the number of sensors installed on the roads we deal with is numerous and their characteristics are highly diverse, so it takes a huge amount of time and cost to build a model while guaranteeing the generalization ability.
As a way to easily overcome these methods, we adopt a zero-shot inference that utilizes a highly generalized ASR model which pre-learned with a wide range of audio data including general audio, speech, and music.
Following the above, we propose a method to separate the encoder of the pre-trained ASR model, place it on the edge computer and encode all the collected data.
The encoded data will be transmitted o the central computer and decoded.
§ EXPERIMENTS
§.§ Dataset
To validate the zero-shot inference-based method, we should deal with varied data from different road environments.
Among the collectible road points, we selectively use three points with significantly different environmental characteristics as shown in Fig. <ref>.
We have collected the audio samples for four weather conditions at three locations as summarized in Table <ref>.
§.§ Zero-shot compression
Referring to our purpose, maximizing data compression, it is important to minimize the restoration error as well as the compression capacity of the data.
Note that, we abbreviate 44,100 Hz, 22,050 Hz, and 11,025 Hz as
f_44, f_22, and f_11 respectively.
The f_22 and f_11 in Fig. <ref> show the fundamental loss of high-frequency components compared to f_44 which the sampling rate is set as low to reduce the file size.
Therefore, it should be avoided to set the low sampling rate with a hardware-based approach, and a method of collecting and compressing Hi-Fi data needs to be used in a computational approach.
When applying a pre-trained ASR model for audio compression and decompression, it shows not only preserving high-frequency components but also less information loss than the hardware-based approach as shown in f̂_44 in Fig. <ref> and Table <ref>.
Through this experiment, we confirm that the data compression method based on the ASR model can increase the total amount of samples saved in an edge computer by 34.6× while minimizing information loss.
This means that the cost of data transmission can also be 34.6× reduced.
§.§ Anomaly detection
We simulate the compressed data collecting situation by the zero-shot encoding method in the central computer to check whether an anomaly detection model can perform at the appropriate level when it is trained with the decompressed dataset.
It is clear that zero-shot encoding is a method that can minimize information loss while maximizing compression rate compared to others, but considering that there is a slight difference from the original, we can estimate that anomaly detection performance may also be decreased.
Considering this, the method with the least performance degradation can be considered as the optimal method.
For the experiments, we downsample each audio by 2 and 4 scales to simulate the same Hi-Fi data as collected at Lo-Fi conditions.
Note that, the resolution of the original audio is 44,100 Hz, resolution for each downsampled audio is 22,050 Hz and 11,025 Hz respectively.
Also, the compressed and decompressed audio data are used to verify the ASR case.
The measured anomaly detection performance with the area under the receiver operating characteristic curve (AUROC) <cit.> is summarized in Table <ref>.
In the case of using ASR, the average performance decreased to 92%-level compared to the original Hi-Fi audio case.
However, we confirm that the anomaly detection performance of our method is more compliant than Lo-Fi.
§.§ Resolution integration
We verify that it can be integrated into equal-level of high-resolution audio through the encoding and decoding process when the resolution (sampling rate) of the collected audio is different.
If this premise is satisfied, it can guarantee that the proposed data compression and restoration framework via the ASR model works properly no matter which audio resolution.
If the data collected at low-resolution can be converted into high-resolution, it can be helpful to build a high-performance anomaly detection model considering the case where low-cost and low-resolution microphones are inevitably installed according to the budget.
When the three samples, assuming original high-resolution data and low-resolution data, are upsampled through the ASR model, they show almost the same difference from the original as summarized in Table <ref>.
Thus, we confirm that data of arbitrary resolution can be integrated into high-quality audio at the central server.
§ CONCLUSION
We propose a method based on the pre-trained ASR model for storing as many audio samples as possible in an edge computer with limited storage capacity installed for the purpose of road anomaly detection.
Our method shows that adequate performance can be obtained by using only one generalized encoder-decoder pair instead of each encoder-decoder corresponding to each post or type of road with high environmental diversity.
Each audio is highly compressed from its original size of 173 KiB per second to 5 KiB, showing that it can store up to 34.6× as many audio samples.
In addition, even if an anomaly detection model is trained by collecting compressed audio samples at the central computer, an appropriate level can be achieved.
Some degradation of the anomaly detection performance is caused by a slight information loss during the encoding and decoding process but there is room for improvement via proper encoder-decoder pairs.
In future work, we plan to explore the encoder-decoder model with better generalization ability or trained on the road noise domain as a way to construct more stable systems.
§ ACKNOWLEDGEMENTS
We are grateful to all the members of SK Planet Co., Ltd., who have supported this research, providing equipment for the experiment.
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|
http://arxiv.org/abs/2307.03949v1 | 20230708103948 | Ergodic observables in non-ergodic systems: the example of the harmonic chain | [
"Marco Baldovin",
"Raffaele Marino",
"Angelo Vulpiani"
] | cond-mat.stat-mech | [
"cond-mat.stat-mech"
] |
Institute for Complex Systems - CNR, P.le Aldo Moro 2, 00185, Rome, Italy
Université Paris-Saclay, CNRS, LPTMS,530 Rue André Rivière, 91405, Orsay, France
Dipartimento di Fisica e Astronomia,
Universitá degli Studi di Firenze, Via Giovanni Sansone 1, 50019, Sesto Fiorentino, Italy
Dipartimento di Fisica, Sapienza Universitá di
Roma, P.le Aldo Moro 5, 00185, Rome, Italy
In the framework of statistical mechanics the properties of
macroscopic systems are deduced starting from the laws of their microscopic
dynamics. One of the key assumptions in this procedure is the ergodic property,
namely the equivalence between time averages and ensemble averages.
This property can be proved only for a limited number of systems; however,
as proved by Khinchin <cit.>, weak forms of it hold even in systems that
are not ergodic at the microscopic scale, provided that extensive observables are considered.
Here we show in a pedagogical way the
validity of the ergodic hypothesis, at a practical level,
in the paradigmatic case of a chain of harmonic oscillators. By
using analytical results and numerical computations, we provide evidence that
this non-chaotic integrable system shows ergodic behavior in the
limit of many degrees of freedom. In particular, the
Maxwell-Boltzmann distribution turns out to fairly describe the
statistics of the single particle velocity. A study of the
typical time-scales for relaxation is also provided.
Ergodic observables in non-ergodic systems: the example of the harmonic
chain
Angelo Vulpiani
August 12, 2023
==============================================================================
§ INTRODUCTION
Since the seminal works by Maxwell, Boltzmann and Gibbs, statistical mechanics
has been conceived as a link between the microscopic world of atoms and
molecules and the macroscopic one where everyday phenomena are
observed <cit.>. The same physical system can be
described, in the former, by an enormous number of degrees of freedom N (of
the same order of the Avogadro number) or, in the latter, in terms of just a few
thermodynamics quantities. Statistical mechanics is able to describe in a
precise way the behavior of these macroscopic observables, by exploiting the
knowledge of the laws for the microscopic dynamics and classical results from
probability theory. Paradigmatic examples of this success are, for instance, the
possibility to describe the probability distribution of the single-particle velocity
in an ideal gas <cit.>, as well as the detailed behavior
of phase transitions <cit.> and
critical phenomena <cit.>.
In some cases (Bose-Einstein condensation <cit.>, absolute negative temperature
systems <cit.>) the results
of statistical mechanics were able to predict states of the matter that were never been observed before.
In spite of the above achievements, a complete consensus about the actual reasons for such a success
has not been yet reached within the statistical mechanics community. The main
source of disagreement is the so-called “ergodic hypothesis”, stating that time
averages (the ones actually measured in physics experiments) can be computed as
ensemble averages (the ones appearing in statistical mechanics calculations).
Specifically, a system is called ergodic when the value of the time average of
any observable is the same as the one obtained by taking the average over the
energy surface, using the microcanonical distribution <cit.>. It is worth mentioning that, from a mathematical
point of view, ergodicity holds only for a small amount of physical systems: the
KAM theorem <cit.>
establishes that, strictly speaking, non-trivial dynamics cannot be ergodic.
Nonetheless, the ergodic hypothesis happens to work extremely well also for
non-ergodic systems. It provides results in perfect agreement with the numerical and experimental observations, as seen in a wealth of physical situations <cit.>.
Different explanations for this behavior have been provided. Without going into
the details of the controversy, three main points of view can be identified: (i)
the “classical” school based on the seminal works by Boltzmann and the important
contribution of Khinchin, where the main role is played by the presence of many
degrees of freedom in the considered systems
<cit.>; (ii) those, like the Prigogine school, who recognize in the chaotic
nature of the microscopic evolution the dominant ingredient <cit.>; (iii) the maximum entropy point of view, which does
not consider statistical mechanics as a physical theory but as an inference
methodology based on incomplete information <cit.>.
The main aim of the present contribution is to clarify, at a pedagogical level,
how ergodicity manifests itself for some relevant degrees of freedom, in
non-ergodic systems. We say that ergodicity occurs “at a practical level”.
To this end, a classical chain of N coupled harmonic oscillators turns out to
be an excellent case study: being an integrable system, it cannot be suspected
of being chaotic; still, “practical” ergodicity is recovered for relevant
observables, in the limit of N≫1. We believe that this kind of analysis
supports the traditional point of view of Boltzmann, which identifies the large
number of degrees of freedom as the reason for the occurrence of ergodic
behavior for physically relevant observables. Of course, these conclusions are
not new. In the works of Khinchin (and then Mazur and van der
Lynden) <cit.> it is rigorously shown that the ergodic
hypothesis holds for observables that are computed as an average over a finite
fraction of the degrees of freedom, in the limit of N ≫ 1. Specifically, if
we limit our interest to this particular (but non-trivial) class of observables,
the ergodic hypothesis holds for almost all initial conditions (but for a set
whose probability goes to zero for N →∞), within arbitrary accuracy.
In addition, several numerical results for weakly non-linear systems
<cit.>, as well
as integrable systems <cit.>,
present strong indications of the poor role of chaotic behaviour, implying the
dominant relevance of the many degrees of freedom. Still, we think it may be
useful, at least from a pedagogical point of view, to analyze an explicit
example where analytical calculations can be made (to some extent), without
losing physical intuition about the model.
The rest of this paper is organized as follows. In Section <ref> we
briefly recall basic facts about the chosen model, to fix the notation and
introduce some formulae that will be useful in the following.
Section <ref> contains the main result of the paper. We present an explicit calculation of the empirical distribution of the single-particle momentum, given a system starting from out-of-equilibrium initial conditions. We show that in this case the Maxwell-Boltzmann distribution is an excellent approximation in the N→∞ limit. Section <ref> is devoted to an analysis
of the typical times at which the described ergodic behavior is expected to be
observed; a comparison with a noisy version of the model (which is ergodic by
definition) is also provided. In Section <ref> we draw our final
considerations.
§ MODEL
We are interested in the dynamics of a one-dimensional chain of N classical
harmonic oscillators of mass m.
The state of the system is described by the canonical coordinates {q_j(t),
p_j(t)} with j=1,..,N; here p_j(t) identifies the momentum of the j-th
oscillator at time t, while q_j(t) represents its position. The j-th and
the (j+1)-th particles of the chain interact through a linear force of
intensity κ|q_j+1-q_j|, where κ is the elastic constant. We will
assume that the first and the last oscillator of the chain are coupled to
virtual particles at rest, with infinite inertia (the walls), i.e. q_0≡ q_N+1≡ 0.
The Hamiltonian of the model reads therefore
ℋ(𝐪,𝐩)=∑_j=0^N p_j^2/2 m +
∑_j=0^Nm ω_0^2 /2(q_j+1 - q_j)^2,
where ω_0=√(κ/m).
Such a system is integrable and, therefore, trivially non-ergodic. This can
be easily seen by considering the normal modes of the chain, i.e. the set of
canonical coordinates
Q_k=√(2/N+1)∑_j=1^N q_j sinj k π/N+1
P_k=√(2/N+1)∑_j=1^N p_j sinj k π/N+1 ,
with k=1, ..., N. Indeed, by rewriting the Hamiltonian in terms of these new
canonical coordinates one gets
ℋ(𝐐,𝐏)=1/2∑_k=1^N
P_k^2/m + ω_k^2 Q_k^2 ,
where the frequencies of the normal modes are given by
ω_k=2 ω_0 sinπ k/2N +2 .
In other words, the system can be mapped into a collection of independent
harmonic oscillators with characteristic frequencies {ω_k}. This system
is clearly non-ergodic, as it admits N integrals of motion, namely the
energies
E_k=1/2P_k^2/m + ω_k^2 Q_k^2
associated to the normal modes.
In spite of its apparent simplicity, the above system allows the investigation
of some nontrivial aspects of the ergodic hypothesis, and helps clarifying the
physical meaning of this assumption.
§ ERGODIC BEHAVIOR OF THE MOMENTA
In this section we analyze the statistics of the single-particle momenta of the
chain. We aim to show that they approximately follow a Maxwell-Boltzmann
distribution
𝒫_MB(p)=√(β/2π m)e^-β p^2/2m
in the limit of large N, where β is the inverse temperature of the
system. With the chosen initial conditions, β=N/E_tot. Firstly, extending
some classical results by Kac <cit.>, we focus on the empirical
distribution of the momentum of one particle, computed from a unique long
trajectory, namely
𝒫_e^(j)p=1 T∫_0^T dt δp -p_j(t) .
Then we consider the marginal probability distribution
𝒫_ep,t computed from the momenta {p_j} of all the
particles at a specific time t, i.e.
𝒫_ep,t=1 N∑_j=1^N δp -p_j(t) .
In both cases we assume that the system is prepared in an atypical initial
condition. More precisely, we consider the case in which Q_j(0)=0, for all
j, and the total energy E_tot, at time t=0, is equally distributed
among the momenta of the first N^⋆ normal modes, with 1 ≪ N^⋆≪ N:
P_j(0)=
√(2m E_tot/N^⋆) for 1 ≤ j ≤ N^⋆
0 for N^⋆< j ≤ N .
In this case, the dynamics of the first N^⋆ normal modes is given by
Q(t) =√(2 E_tot/ω_k^2N^⋆)sinω_k t
P(t) =√(2 m E_tot/N^⋆)cosω_k t .
§.§ Empirical distribution of single-particle momentum
Our aim is to compute the empirical distribution of the momentum of a given
particle p_j, i.e., the distribution of its values measured in time. This
analytical calculation was carried out rigorously by Mazur and Montroll in
Ref. <cit.>. Here, we provide an alternative argument that has the advantage of being more concise and intuitive, in contrast to the mathematical rigour of <cit.>. Our approach exploits the computation of
the moments of the distribution; by showing that they are the same, in the limit
of infinite measurement time, as those of a Gaussian, it is possible to conclude
that the considered momentum follows the equilibrium Maxwell-Boltzmann
distribution. The assumption N≫1 will enter explicitly the calculation.
The momentum of the j-th particle can be written as a linear combination of
the momenta of the normal modes by inverting Eq. (<ref>):
p_j(t) =√(2/N+1)∑_k=1^N sinj k π/N+1
P_k(t)
=2√(m
E_tot/(N+1)N^⋆)∑_k=1^N^⋆sinkjπ/N+1cosω_k t
where the ω_k's are defined by Eq. (<ref>), and the
dynamics (<ref>) has been taken into account. The n-th empirical moment
of the distribution is defined as the average p_j^n of the n-th
powerof p_j over a measurement time T:
p_j^n =1/T∫_0^Tdt p_j^n(t)
=1/T∫_0^Tdt (C_N^⋆)^n
∏_l=1^n∑_k_l=1^N^⋆sink_l
jπ/N+1cosω_k_l t
=(C_N^⋆)^n
∑_k_1=1^N^⋆…∑_k_n=1^N^⋆sink_1jπ/N+1
…sink_njπ/N+1 1/T∫_0^Tdt cosω_k_1 t…cosω_k_n t
with
C_N^⋆=2√(m E_tot/(N+1)N^⋆) .
We want to study the integral appearing in the last term of the above equation.
To this end it is useful to recall that
1/2 π∫_0^2πd θcos^n(θ)=
(n-1)!!/n!! for n even
0 for n odd .
As a consequence, one has
1/T∫_0^Td t cos^n(ω t)≃(n-1)!!/n!! for n even
0 for n odd .
Indeed, we are just averaging over ≃ω T/2 π periods of the
integrated function, obtaining the same result we get for a single period,
with a correction of the order O(ω T)^-1. This correction comes from the fact that T
is not, in general, an exact multiple of 2 π/ω.
If ω_1, ω_2, ..., ω_q are incommensurable (i.e., their
ratios cannot be expressed as rational numbers), provided that T is much
larger than (ω_j-ω_k)^-1 for each choice of 1 ≤ k < j ≤ q, a
well known result <cit.> assures that
1/T∫_0^Td t cos^n_1(ω_1 t)·...·cos^n_q(ω_q t) ≃ 1/T∫_0^Td t
cos^n_1(ω_1 t)·...·1/T∫_0^Td t
cos^n_q(ω_1 t)
≃ (n_1-1)!!/n_1!!· ...·(n_q-1)!!/n_q!! if all n's are even ,
where the last step is a consequence of Eq. (<ref>). Instead, if at
least one of the n's is odd, the above quantity vanishes, again with
corrections due to the finite time T. Since the smallest sfrequency is
ω_1, one has that the error is at most of the order Oq(ω_1
T)^-1≃ O(qN /ω_0 T).
Let us consider again the integral in the last term of Eq. (<ref>). The
ω_k's are, in general, incommensurable. Therefore, the integral vanishes
when n is odd, since in that case at least one of the {n_l}, l=1,...,q,
will be odd. When n is even, the considered quantity is different from zero
as soon as the k's are pairwise equal, so that n_1=...=n_q=2. In the following we
will neglect the contribution of terms containing groups of four or more equal
k's: if n≪ N^⋆, the number of these terms is indeed ∼
O(N^⋆) times less numerous than the pairings, and it can be neglected if
N^⋆≫1 (which is one of our assumptions on the initial condition).
Calling Ω_n the set of possible pairings for the vector
𝐤=(k_1,...,k_l), we have then
p_j^n≃C_N^⋆/√(2)^n ∑_𝐤∈Ω_n∏_l=1^n sink_ljπ/N+1 ,
with an error of O(1/N^⋆) due to neglecting groups of 4, 6 and so on,
and an error O(nN/ω_0 T) due to the finite averaging time T, as
discussed before. Factor 2^-n/2 comes from the explicit evaluation of
Eq. (<ref>) .
At fixed j, we need now to estimate the sums appearing in the above equation,
recalling that the k's are pairwise equal. If j> N/N^⋆, the
arguments of the periodic functions can be thought as if independently extracted
from a uniform distribution 𝒫(k)=1/N^⋆. One has:
sin^2 kj π/N+1≃∑_k=1^N^⋆1/N^⋆sin^2 kj π/N+1≃1/2 π∫_-π^πd θ sin^2(θ)=1/2 ,
and
∏_l=1^n sink_ljπ/N+1≃ 2^-n/2 ,
if 𝐤∈Ω_n.
As a consequence
p_j^n ≃C_N^⋆/2^n (N^⋆)^n/2 𝒩(Ω_n)≃m
E_tot/N+1^n/2𝒩(Ω_n) ,
where 𝒩(Ω_n) is the number of ways in which we can choose the
pairings. These are the moments of a Gaussian distribution with zero average and
m E_tot/N+1 variance.
Summarising, it is possible to show that, if n ≪ N^⋆≪ N, the first
n moments of the distribution are those of a Maxwell-Boltzmann distribution.
In the limit of N≫1 with N^⋆/N fixed, the Gaussian distribution is
thus recovered up to an arbitrary number of moments. Let us note that the
assumption Q_j(0)=0, while allowing to make the calculations clearer, is not
really relevant. Indeed, if Q_j(0)≠ 0 we can repeat the above computation
while replacing ω_k t by ω_k t + ϕ_k, where the phases ϕ_k
take into account the initial conditions.
Fig. <ref> shows the standardized histogram of the relative
frequencies of single-particle velocities of the considered system, in the N
≫ 1 limit, with the initial conditions discussed before. As expected, the
shape of the distribution tends to a Gaussian in the large-time limit.
§.§ Distribution of momenta at a given time
A similar strategy can be used to show that, at any given time t large enough,
the histogram of the momenta is well approximated by a Gaussian distribution.
Again, the large number of degrees of freedom plays an important role.
We want to compute the empirical moments
p^n(t)=1/N∑_j=1^N p_j^n(t) ,
defined according to the distribution 𝒫_e^(j)p introduced
by Eq. (<ref>).
Using again Eq. (<ref>) we get
p^n(t)= 1/N∑_j=1^N(C_N^⋆)^n∑_k=1^N^⋆sinkjπ/N+1cosω_k t^n
= 1/N(C_N^⋆)^n∑_k_1^N^⋆…∑_k_n=1^N^⋆∏_l=1^Ncosω_k_lt∑_j=1^Nsink_1 j
π/N+1…sink_n j π/N+1 .
Reasoning as before, we see that the sum over j vanishes in the large N
limit unless the k's are pairwise equal. Again, we neglect the terms including
groups of 4 or more equal k's, assuming that n≪ N^⋆, so that their
relative contribution is O(1/N^⋆). That sum selects paired values of k
for the product inside the square brackets, and we end with
p^n(t)≃1/N(C_N^⋆)^n∑_𝐤∈Ω_n∏_l=1^Ncosω_k_lt .
If t is “large enough” (we will come back to this point in the following
section), different values of ω_k_l lead to completely uncorrelated
values of cos(ω_k_l t). Hence, as before, we can consider the
arguments of the cosines as extracted from a uniform distribution, obtaining
p^n(t)≃C_N^⋆/2^n (N^⋆)^n/2 𝒩(Ω_n)≃m
E_tot/N+1^n/2𝒩(Ω_n) .
These are again the moments of the equilibrium Maxwell-Boltzmann distribution.
We had to assume n ≪ N^⋆, meaning that a Gaussian distribution is recovered
only in the limit of large number of degrees of freedom.
The
empirical distribution can be compared with the Maxwell-Boltzmann by looking at
the Kullback-Leibler divergence K(𝒫_e(p,t), 𝒫_MB(p))
which provides a sort of distance between the empirical 𝒫_e(p,t) and
the Maxwell-Boltzmann:
K[𝒫_e(p,t), 𝒫_MB(p)]= - ∫𝒫_e(p,t) ln𝒫_MB(p)/𝒫_e(p,t) dp.
Figure <ref> shows how the Kullback-Leibler divergences
approach their equilibrium limit, for different values of N. As expected, the transition happens on a time scale that depends linearly on N.
A comment is in order: even if this behaviour
may look similar to the H-Theorem for diluited gases, such a resemblance is
only superficial. Indeed, while in the cases of diluited gases the approach to
the Maxwell-Boltzmann is due to the collisions among different particles that actually exchange
energy and momentum, in the considered case the “thermalization” is due to a dephasing
mechanism.
§ ANALYSIS OF THE TIME SCALES
In the previous section, when considering the distribution of the momenta at a
given time, we had to assume that t was “large enough” in order for our
approximations to hold. In particular we required cos(ω_k_1t) and
cos(ω_k_2t) to be uncorrelated as soon as k_1 k_2. Such a dephasing
hypothesis amounts to asking that
|ω_k_1t-ω_k_2t|> 2π c ,
where c is the number of phases by which the two oscillator have to differ before
they can be considered uncorrelated. The constant c may be much larger than 1,
but it is not expected to depend strongly on the size N of the system. In other words,
we require
t> c/|ω_k_1-ω_k_2|
for each choice of k_1 and k_2. To estimate this typical relaxation time, we
need to pick the minimum value of |ω_k_1-ω_k_2| among the
possible pairs (k_1,k_2). This term is minimized when k_1=k̃ and
k_2=k̃-1 (or vice-versa), with k̃ chosen such that
ω_k̃-ω_k̃-1
is minimum. In the large-N limit this quantity is approximated by
ω_k̃-ω_k̃-1=ω_0sink̃π/2N+2-ω_0sink̃π- π/2N+2≃ω_0cosk̃π/2N+2π/2N+2 ,
which is minimum when k̃ is maximum, i.e. for k̃=N^⋆.
Dephasing is thus expected to occur at
t> 4cN/ω_0cosN^⋆π/2N ,
i.e. t>4cN/ω_0 in the N^⋆/N ≪ 1 limit.
It is instructive to compare this characteristic time with the typical
relaxation time of the “damped” version of the considered system. For doing so, we assume that our chain of oscillators is now in contact with a viscous medium which acts at the same time as a thermal bath and as a source of viscous friction. By
considering the (stochastic) effect of the medium, one gets the Klein-Kramers
stochastic process <cit.>
∂ q_j/∂ t=p_j/m
∂ p_j/∂ t=ω_0^2(q_j+1 - 2 q_j + q_j-1)
-γ p_j + √(2 γ T)ξ_j
where γ is the damping coefficient and T is the temperature of the
thermal bath (we are taking the Boltzmann constant k_B equal to 1). Here the
{ξ_j} are time-dependent, delta-correlated Gaussian noises such that
ξ_j(t)ξ_k(t')=δ_jkδ(t-t').
Such a system is surely ergodic and the stationary probability distribution is
the familiar equilibrium one
𝒫_s(𝐪,𝐩) ∝
e^-H(𝐪,𝐩)/T.
Also in this case we can consider the evolution of the normal modes. By taking
into account Eqs. (<ref>) and (<ref>) one gets
Q̇_̇k̇ =1/m P_k
Ṗ_̇k̇ =- ω_k^2 Q_k - γ/m P + √(2 γ T)ζ_k
where the {ζ_k} are again delta-correlated Gaussian noises. It is
important to notice that also in this case the motion of the modes is
independent (i.e. the friction does not couple normal modes with different
k); nonetheless, the system is ergodic, because the presence of the noise
allows it to explore, in principle, any point of the phase-space.
The Fokker-Planck equation for the evolution of the probability density function
𝒫Q_k,P_k,t of the k-th normal mode can be derived using
standard methods <cit.>:
∂_t𝒫=-∂_Q_kP_k𝒫+∂_P_kω_k^ 2Q_k𝒫+γ/mP_k𝒫+γ
T∂_P_k^2 𝒫 .
The above equation allows to compute also the time dependence of the correlation
functions of the system in the stationary state. In particular one gets
d/dtQ_k(t) Q_k(0)=1/mP_k(t)Q_k(0)
and
d/dtP_k(t) Q_k(0)-ω_k^2 m Q_k(t) Q_k(0)
-γ/mP_k(t) Q_k(0) ,
which, once combined together, lead to
d^2/d t^2Q_k(t) Q_k(0)+γ/md/dtQ_k(t)
Q_k(0)+ ω_k^2Q_k(t) Q_k(0)=0 .
For ω_k <γ/m the solution of this equation admits two characteristic
frequencies ω̃_±, namely
ω̃_±=γ/2m1 ±√(1-m^2
ω_k^2/γ^2).
In the limit ω_k ≪γ/m one has therefore
ω̃_- ≃m/4 γω_k^2 ≃m ω_0^2
π^2 k^2/γ N^2 .
Therefore, as a matter of fact, even in the damped case the system needs a time
that scales as N^2 in order to get complete relaxation for the modes. As we
discussed before, the dephasing mechanism that guarantees for “practical”
ergodicity in the deterministic version is instead expected to occur on time scales of order O(N).
§ CONCLUSIONS
The main aim of this paper was to expose, at a pedagogical level, some aspects
of the foundation of statistical mechanics, namely the role of
ergodicity for the validity of the statistical approach to the study of complex
systems.
We analyzed a chain of classical harmonic oscillators (i.e. a paradigmatic
example of integrable system, which cannot be suspected to show chaotic
behaviour). By extending some well-known results by Kac <cit.>, we showed that the
Maxwell-Bolzmann distribution approximates with arbitrary precision (in the
limit of large number of degrees of freedom) the empirical distribution of the
momenta of the system, after a dephasing time which scales with the size of the
chain. This is true also for quite pathological initial conditions, where only a
small fraction of the normal modes is excited at time t=0. The scaling of the
typical dephasing time with the number of oscillators N may appear as a limit
of our argument, since this time will diverge in the thermodynamic limit; on the
other hand one should consider, as explicitely shown before, that the damped
version of this model (which is ergodic by definition) needs times of the order
O(N^2) to reach thermalization for each normal mode.
This comparison clearly shows that the effective thermalization observed in
large systems has little to do with the mathematical concept of ergodicity, and
it is instead related to the large number of components concurring to define the
global observales that are usually taken into account (in our case, the large
number of normal modes that define the momentum of a single particle). When
these components cease to be in phase, the predictions of statistical mechanics
start to be effective; this can be observed even in integrable systems, without
need for the mathematical notion of ergodicity to hold.
In other words, we believe that the present work give further evidence of the
idea (which had been substantiated mathematically by Khinchin, Mazur and van
der Linden) that the most relevant ingredient of statistical mechanics is the
large number of degrees of freedom, and the global nature of the observables
that are typically taken into account.
§ ACKNOWLEDGEMENTS
RM is supported by #NEXTGENERATIONEU (NGEU) and funded by the Ministry of University and Research (MUR), National Recovery and Resilience Plan (NRRP), project MNESYS (PE0000006) "A Multiscale integrated approach to the study of the nervous system in health and disease" (DN. 1553 11.10.2022).
|
http://arxiv.org/abs/2307.06208v1 | 20230712145318 | Decoherence effects on lepton number violation from heavy neutrino-antineutrino oscillations | [
"Stefan Antusch",
"Jan Hajer",
"Johannes Rosskopp"
] | hep-ph | [
"hep-ph"
] |
[
[
Received May 01, 2023 / Accepted May 31, 2023
=================================================
We study decoherence effects and phase corrections in , based on with external wave packets.
Decoherence damps the oscillation pattern, making it harder to resolve experimentally.
Additionally, it enhances for processes in symmetry-protected low-scale seesaw models by reducing the destructive interference between mass eigenstates.
We discuss a novel time-independent shift in the phase and derive formulae for calculating decoherence effects and the phase shift in the relevant regimes, which are the and .
We find that the phase shift can be neglected in the parameter region under consideration since it is small apart from parameter regions with large damping.
In the oscillation formulae, decoherence can be included by an effective damping parameter.
We discuss this parameter and present averaged results, which apply to simulations of in the dilepton-dijet channel at the .
We show that including decoherence effects can dramatically change the theoretical prediction for the ratio of over events.
§ INTRODUCTION
The origin of the observed neutrino masses is one of the great open questions in current particle physics.
When the new particles involved in the neutrino mass generation have masses close to the scale, it is possible to investigate this question at the and future accelerators.
One possible extension of the of elementary particles that explains the observed light neutrino masses is based on the introduction of sterile neutrinos, fermions which are uncharged under the gauge symmetry of the <cit.>, see also <cit.>.
When they form Yukawa interaction terms with the lepton and Higgs doublets and, in addition, have Majorana mass terms <cit.>, light neutrino masses can be generated, which are then of Majorana-type.
However, when the sterile neutrino masses are around the scale, care has to be taken not to exceed the bounds on the light neutrino masses <cit.>.
When the Yukawa couplings are not tiny, the smallness of the light neutrino masses is realised by an approximate .
The sterile neutrinos then form pseudo-Dirac pairs of nearly mass-degenerate heavy neutrinos.
Although is significantly suppressed for prompt heavy neutrino decays, <cit.>, it can lead to observable effects via the phenomenon of <cit.>, see also <cit.>.
Since the light neutrino masses become zero in the limit of exact , observing processes is crucial for probing the origin of neutrino masses.
Due to the , the number of and events in a given process depends on the time difference between the production and decay of the heavy neutrinos.
Recently it has been shown for a selected point consistent with present constraints, featuring a long-lived pseudo-Dirac heavy neutrino pair, that could be resolved during the <cit.>, see also <cit.>.
However, even when the oscillations are not resolvable, they can induce .
The total ratio of over events, R_ll, can be used to quantify the effect.
Decoherence and phase correction effects on are so far unexplored at the quantitative level.
Previous studies have used estimates to verify that decoherence effects can be neglected for the considered parameters, <cit.>, or have assumed this to be the case.
While decoherence can, in principle, depend on various parameters, it has to be a function of the mass splitting of the heavy neutrinos.
This can be argued from the fact that for experimentally resolvable mass splittings, the pseudo-Dirac pair must reproduce the phenomenology of two separate Majorana neutrinos.
In such cases, are expected to vanish.
Thus, in regions where decoherence effects are relevant, the simple formulae have to be modified.
Phase corrections for have not yet been discussed.
One can calculate the possible decoherence and phase correction effects in using with external wave packets.
The formalism is discussed in <cit.> and has been adapted to the case of in <cit.>.
In <cit.>, the effective damping parameter λ is introduced, which contains the collective effects of decoherence onto the oscillation formulae.
In the present work, we explore how decoherence and a time-independent phase shift affect as well as the quantitative prospects for observing .
The remainder of this publication is organised as follows:
In <ref>, we introduce the external wave packet formalism.
Afterwards, in <ref>, we describe the derivation of the damped oscillation probability for the general case and subsequently apply the results to the case of in the .
We show that the effects of decoherence can be summarised by a damping parameter λ, leading to a simple extension of the oscillation formulae.
Results for λ, including its impact on R_ll and searches for , are discussed in <ref>.
Finally, we conclude in <ref>.
Details of the analytical derivations of the oscillation probabilities are presented in the appendices.
The detailed steps necessary to integrate the transition amplitude over the intermediate particles' momentum and travelled distance are presented in <ref>, respectively.
The constant phase shift is discussed in <ref>.
The algorithm to compute the damping parameter λ numerically is discussed in <ref>, where the kinematics of the considered process is simulated using the introduced in <cit.>.
§ EXTERNAL WAVE PACKET FORMALISM
In this section, we derive the transition amplitude between two external states that are prepared as wave packets.
This essential quantity is the main ingredient to derive an oscillation probability following the arguments made in <cit.>.
It is defined as a function of a distance (t,x⃗) in spacetime
[Quantities with a suppressed vectorial index are indicated by boldface.]
𝒜(t,x⃗) = *Φ(t^'',x⃗^'')𝒯exp[- ı∫[ṭ^‵] ∫[^̣3 x⃗^‵] ℋ(t^‵,x⃗^‵)] - 1Φ(t^',x⃗^') ,
where ℋ(t^‵,x⃗^‵) is the interaction Hamiltonian and 𝒯 is the time ordering operator.
In comparison to the usual approach, in which plane wave states |Φ(p⃗)⟩ with momentum p⃗ are used, the initial |Φ(t^',x⃗^')⟩ and final ⟨Φ(t^'',x⃗^'')| states are wave packets centred at the indicated points in spacetime and can be written as a function of a plane wave state using
|*⟩Φ(t,x⃗) = ∫[^̣3 p⃗/(2 π)^3 √(2 E(p⃗))] ψ(t,x⃗,p⃗,p⃗_0) |*⟩Φ(p⃗) ,
where E(p⃗) is the energy of the particle, and ψ(t,x⃗,p⃗,p⃗_0) is the wave packet envelope which describes the shape of the wave packet and is centred around the momentum p⃗_0.
Assuming that the external wave packets are Gaussian and approximating the matrix element at the peak of those Gaussian functions, it is possible to evaluate the momentum integration over the external wave packets, yielding the transition amplitude for a mass eigenstate i <cit.>
𝒜_i(t,x⃗) = 𝒩∫[Ẹ] ∫[^̣3 p⃗] M_i(E,p⃗) G_i(s) exp[-f(E,p⃗) - ıϕ(t,x⃗,E,p⃗)] .
Here G_i(s) is the denominator of the renormalised propagator with s = E^2 - p⃗^2, M_i(E, p⃗) denotes the interaction amplitude, defined as the matrix element without the denominator of the propagator, and 𝒩 is a normalisation constant.
[
The precise form of the normalisation constant 𝒩 changes throughout the paper.
However, the normalisation constant can always be evaluated using an appropriate normalisation condition, as discussed in <ref>.
]
The imaginary part of the exponent contains the phase
ϕ(t,x⃗,E,p⃗) = E t - p⃗⃗̇x .
Here x⃗ is the distance, and t is the time difference between the production and detection point.
The real part of the exponent contains the .
This name is derived from the fact that it describes the shape of the intermediate particle's wave packet as a function of its energy and momentum.
Its shape is defined by the shapes of the external particles' wave packets at the production P and detection D vertices V and is given by <cit.>
f(E,p⃗) = *p⃗ - p⃗_0/2 σ_p⃗P^^2 + [e^_P(E, p⃗)/2 σ_EP^]^2 + (P→ D) ,
where
e^_V(E, p⃗) = E - E_0 - (p⃗ - p⃗_0) ⃗̇v_V ,
V ∈{P, D} ,
Here E_0 and p⃗_0 are the energy and momentum of the intermediate particle obtained from the peaks of the external particles' wave packets using energy-momentum conservation either at the detection or production vertex.
They are thus called reconstructed energy and momentum.
The (P→ D) is a shorthand notation where quantities at production are replaced by similar quantities at detection.
If the is approximated as a Gaussian, its width can be interpreted as the effective width σ_eff of the intermediate particles' wave packet.
The total energy and momentum widths are given by the reciprocal sum of the respective widths at the production and detection vertices
1/σ_E^2 = 1/σ_EP^2 + 1/σ_ED^2 , 1/σ_p⃗^2 = 1/σ_p⃗P^2 + 1/σ_p⃗D^2 .
Each of these widths can be expressed in terms of the widths of the external particles in position space.
The widths of the external particles in position space are parameters of the theory and are determined by the experimental situation under consideration.
In the following, we only explicitly write definitions for quantities at production, while analogous definitions hold for quantities at detection.
The energy and momentum widths at this vertex are given by
σ_EP^2/σ_p⃗P^2 = Σ_P - v⃗_P^^2 , σ_p⃗P^σ_x⃗P^ = 1/2 , 1/σ_x⃗P^2 = ∑_n 1/σ_x⃗P_n^2 ,
where σ_x⃗P_n^ is the width of the external particle n in position space and
Σ_P = σ_x⃗P^2 ∑_n v⃗_P_n^^2/σ_x⃗P_n^2 .
The velocity of the production region is defined by
v⃗_P^ = σ_x⃗P^2 ∑_n v⃗_P_n^/σ_x⃗P_n^2 , v⃗_P_n^ = p⃗_P_n^/E_P_n .
The particle with the smallest width dominates these terms unless its velocity is much smaller than the velocities of the other particles.
Since it holds that <cit.>
0 ≤*v⃗_P^^2 ≤Σ_P ≤ 1 ,
0 ≤Σ_P - v⃗_P^^2 ≤ 1 ,
one can calculate that the energy and momentum widths obey the inequality
σ_E^≤σ_p⃗ .
From the (<ref>), it follows that energies E and momenta p⃗ far from the reconstructed energy E_0 and momentum p⃗_0 are exponentially suppressed, where far is defined according to the energy or momentum width, respectively.
Additionally, damping from the propagator is expected when the reconstructed energy and momentum are such that the reconstructed mass
m_0^2 = E_0^2 - p⃗_0^2
is far from the intermediate particles' masses m_i.
This damping defines the shape of the resonance, which in plane wave would be given by the Breit–Wigner distribution.
§ DERIVATION OF THE DAMPED OSCILLATION PROBABILITY
The transition amplitude (<ref>) needs to be integrated over the energy and momentum of the intermediate particle.
The strategy to perform these integration steps is depicted in <ref>.
While the energy integral is performed using the theorem in the following section, the subsequent momentum integral is evaluated in <ref>.
The final step is a distance average performed in <ref>, contrasting the time average used in <cit.>.
In this publication, we express the oscillation probability as a function of elapsed time instead of distance since the relevant observables naturally depend on the proper time of the oscillating particles rather than their travelled distance.
For example, since the heavy neutrinos, once they are detected at a collider experiment, will exhibit a range of Lorentz boosts, the oscillation pattern has to be translated into the proper time frame of the heavy neutrino in order to be reconstructable, see <cit.>.
Therefore, an oscillation probability as a function of time, averaged over the distance, is more suited for this purpose.
The same applies to measurements of the R_ll ratio, which is sensitive to the interplay between and the decay of heavy neutrinos and, hence, naturally defined in the proper time frame of the neutrinos.
Furthermore, a distance average is more straightforward from a technical point of view since there are fewer distance-dependent terms than time-dependent terms in the relevant exponential, as can be seen in <ref>.
§.§ Energy integration via theorem
JS
To further evaluate the transition amplitude (<ref>), the energy integral is evaluated using the theorem <cit.>.
This theorem states that for times larger than a threshold time t_, which is estimated in <ref>, and for functions Ψ(E, p⃗) that are non-zero only for a finite range of s = E^2 - p⃗^2, the energy integral can be approximately evaluated using
∫[Ẹ] Ψ(E,p⃗) G_i(s) exp[- ı E t] ≈𝒩Ψ(E_i^'(p⃗),p⃗) exp[- ı E_i^'(p⃗) t] ,
where the complex pole energy is defined in terms of the complex pole of the propagator as
E_i^'2(p⃗) = p⃗^2 + z_i ,
z_i = m_i^2 - ı m_i Γ_i .
while m_i and Γ_i are the mass and decay width of the mass eigenstate i, respectively.
After this approximate integration, the transition amplitude reads
𝒜_i(t,x⃗) = 𝒩∫[^̣3 p⃗] M_i(E^'(p⃗),p⃗) exp[- f(E_i^'(p⃗),p⃗) - ıϕ(t,x⃗,E_i^'(p⃗),p⃗)] .
The pole energy can be rewritten as
E_i^'2(p⃗) =
[1 - 2 ıϵ_i(p⃗)] E_i^2(p⃗) ,
where the decay width expansion parameter, ϵ_i(p⃗), and the mass eigenstate energy, E_i(p⃗), are defined as
ϵ_i(p⃗) := γ_i(p⃗)/E_i(p⃗) , γ_i(p⃗) := m_i Γ_i/2 E_i(p⃗) ,
E_i^2(p⃗) = p⃗^2 + m_i^2 .
Under the assumption that the decay width is small compared to the mass eigenstate energy, the phase, (<ref>), can be expanded in the decay width expansion parameter, which yields
ϕ(t,x⃗,E_i^'(p⃗),p⃗) = [1 - ıϵ_i(p⃗) + *ϵ_i^2(p⃗)] E_i(p⃗) t - p⃗⃗̇x , ϵ_i(p⃗) ≪ 1 .
The real part results in the phase of the mass eigenstate i, while the imaginary part generates an exponential decay term
ϕ_i(t,x⃗,p⃗) := E_i(p⃗) t - p⃗⃗̇x , γ_i(t,p⃗) := γ_i(p⃗) t .
After the energy integration, the (<ref>) of the mass eigenstate i can be approximated to be
f(E_i^'(p⃗),p⃗) = f(E_i(p⃗),p⃗) + *ϵ_i(p⃗) ,
such that the term reads
f_i(p⃗) := f(E_i(p⃗),p⃗) = *p⃗ - p⃗_0/2 σ_p⃗P^^2 + [e^_iP(p⃗)/2 σ_EP^]^2 + (P→ D) ,
where
e^_iV(p⃗) := e^_V(E_i(p⃗), p⃗) = E_i(p⃗) - E_0 - (p⃗ - p⃗_0) ⃗̇v_V .
The derivation from which follows that higher orders in the decay width expansion parameter can generically be neglected is presented in <ref>.
However, for very short times, the *ϵ_i(p⃗) terms can lead to
a time-independent phase shift, discussed in <ref>.
In the numerical calculation presented in <ref>, these corrections are explicitly taken into account by identifying the imaginary part as a correction to the phase.
Finally, the transition amplitude (<ref>) after the energy integration (<ref>) takes the form
𝒜_i(t,x⃗) = 𝒩∫[^̣3 p⃗] M_i(p⃗) exp[-f_i(p⃗) - γ_i(t,p⃗) - ıϕ_i(t,x⃗,p⃗)] ,
where terms in the decay width expansion of the interaction amplitude M_i(p⃗) = M_i(E_i,p⃗) are neglected.
The remaining integrals are the three-momentum integral and an integral that averages over distance or time.
§.§ Applicability of the formalism
JS
The theorem used in the previous section is only valid for times larger than the threshold time t_.
It is defined via the diameter of the support of the intermediate particle's wave packet
t ≥ t_ := 1/*suppexp[-f(E,p⃗)] .
Since the interaction amplitude, and therefore the wave packet envelope, must vanish for values of √(s) - m_i larger than the uncertainties, this support is estimated by the experimental uncertainty in reconstructing the mass m_i in <cit.>.
In the case of , where the mass of the heavy neutrino has to be reconstructed from semi-leptonic decay products, assuming this uncertainty to be of the order of one per cent of the heavy neutrino mass yields a time threshold of
t_≈100/m = [1]GeV/m[6.58-23]s .
In order to have a fraction f of particles decaying beyond that time requires decay widths of
Γ≤Γ_ := γ/t_ln1/f ,
where γ denotes the Lorentz boost factor.
Demanding [99]% of all particles decaying later than that time results in a width of
Γ_≈m/[1]GeV[100]MeV ,
when assuming a Lorentz boost factor of γ≈ 10, which is a reasonable estimate for the parameter region considered in this work.
In contrast, the width of the wave packets of the external particles is estimated in <ref> using the size of the silicon atom radius and the proton-proton distance in a beam bunch for final and initial states, respectively.
This line of argument suggests that the neutrino wave packet should be zero outside a range defined by the width of the wave packet in the squared four-momentum s = p^2
1/2σ_s^2 = s/2 E_0^2 σ_E^2 + *p⃗ - p⃗_0
Using the approximations derived in <ref>, the numerical values given in <ref> and further approximating √(s) = m, this estimate leads to a time threshold of
t_≈γ/n σ_E^≈2 γσ_p/n≈γ/n[200]nm≈γ/n[6.66-16]s
where n is the number of standard deviations which is taken to define the support of the Gaussian distribution.
Requiring a fraction f = [99]% of particles decaying later than that time leads to a decay width of
Γ_≈ n [0.0199]eV .
Taking the [5]σ range leads to a decay width of about [0.1]eV.
In order to be conservative, we use this more restrictive value in the following.
§.§ Dispersion regimes of the momentum integration
The integration over the three-momentum is carried out differently in three separate regimes depending on how fast the phase varies over the effective width.
For slowly varying phases, the integral is evaluated using Laplace's method.
This regime is called the since time-dependent dispersion effects can be neglected.
In this regime, the argument of the exponential in the transition amplitude (<ref>) is expanded up to second order in the momentum p⃗ around the position of the minimum of the at p⃗_i.
Therefore, the momentum p⃗_i maximises the exponential of the
p⃗_i = _p⃗exp[-f_i(p⃗)] ,
The Hessian of the (<ref>) with respect to the momentum is given at , neglecting the mass splitting, by (<ref>)
[Quantities with suppressed matrix indices are indicated by sans-serif font.]
Σ_0 = /2 σ_p⃗P^2 + u⃗_P^⊗u⃗_P^/2 σ_EP^2 + (P→ D) , u⃗_V^ := v⃗_V^ - v⃗_0 ,
and defines the inverse of the effective width of the intermediate particle.
2 σ_eff^2 = Σ_0^-1 .
The matrix structure of Σ_0 is defined by the two vectors u⃗_P^ and u⃗_D^.
Therefore, there exists a vector which is orthogonal to both, and the corresponding eigenvalue is
*Σ_0_smallest = 1/2 σ_p⃗^2 .
Due to the inequality (<ref>), this is the smallest eigenvalue leading to the largest effective width.
The other two eigenvalues, which are dominated by the energy width, are therefore larger and approximately given by
*Σ_0_larger = u⃗^2/2σ_E^2 + *σ_E^/σ_p⃗, *v⃗_P^ - v⃗_D^ .
This approximation is justified when the velocity vectors are almost aligned u⃗ :≈u⃗_P^≈u⃗_D^ and the inequality (<ref>) is large σ_E^≪σ_p⃗.
The applies to times shorter than the short-time threshold (<ref>)
t ⪅ t^short ,
t^short = Σ_0_smallest E_0 = E_0/2 σ_p⃗^2 .
Since, in its derivation, the phase is required to vary slowly over the effective width of the in all directions, the short-time threshold depends on the largest effective width and, therefore, the smallest eigenvalue of the Hessian.
The detailed computation is described in <ref>.
When wave packets travel longer, the phase oscillates more rapidly as a function of p⃗, such that Laplace's method, used in the short time regime, becomes unsuitable.
Since the wave packets are broader in directions transversal to the reconstructed momentum p⃗_0, an intermediate regime exists in which Laplace's method can only be used for the longitudinal direction.
In contrast, transversal directions are integrated using the method of stationary phase.
This intermediate regime is called the .
The method of stationary phase yields p_x = p_y = 0, assuming that the longitudinal component, indicated by hatted variables, is p⃗_0 = p_0 e⃗_z.
Laplace's method in the longitudinal direction results in
p⃗_i_z = p_i = _ pexp[-f_i( p)] , p⃗_x = p⃗_y = 0 .
The Hessian at is given by (<ref>)
Σ_0 = 1/2σ_p⃗P^2 + u_P^2/2 σ_EP^2 + (P→ D) = u^2/2 σ_E^2 + *σ_E^/σ_p⃗, v_P^ - v_D^ , u_V^ := v_V^ - v_0 ,
where the last approximation holds for u :≈ u_D^≈ u_P^ and when the inequality (<ref>) is large, σ_E^≪σ_p⃗.
Similar to the definition in the (<ref>), the effective width in the is defined as
2 σ_eff^2 = Σ_0^-1 .
and the long-time threshold, which forms the upper bound for this regime (<ref>), is defined by
t^short⪅ t ⪅ t^long ,
t^long = Σ_0 E_0^3/m_0^2 = u^2/2 σ_E^2E_0^3/m_0^2 + *σ_E^/σ_p⃗, v_P^ - v_D^ .
The computation leading to this result is presented in detail in <ref>.
Longer times t ⪆ t^long are not relevant for the discussion of heavy neutrinos in the parameter space of interest for this paper.
However, for the respective regime, called the , the distance-dependent formulae derived in <cit.> can be used.
The short- and long-time thresholds (<ref>) are given in the lab frame.
Using τ = t m_0E_0 they can be reexpressed in the proper time frame as
τ_short = Σ_0_smallest m_0 = m_0/2 σ_p⃗^2 , τ_long = Σ_0 E_0^2/m_0 = u^2/2 σ_E^2E_0^2/m_0 + *σ_E^/σ_p⃗, v_P^ - v_D^ .
For heavy neutrinos appearing in the process presented in <ref>, these regimes are depicted in <ref> after averaging over 100 events.
The partition based on distances is presented in <ref>.
It shows that for experimental length scales smaller than about [100]km, only the and are relevant, and the short-time threshold is of dm.
The regimes in the proper time frame are shown in <ref>.
For decay widths leading to lifetimes comparable to the short-time threshold, it becomes relevant to quantify the fraction of events that fall into the and the , respectively.
Decay widths Γ≈τ_short^-1 result in a fraction of 1-e^-1 events decaying before the threshold, and therefore inside the .
For decay widths Γ⪆ 10 τ_short^-1 practically all events decay before the threshold.
Contrary, for decay widths Γ⪅ 10^-1τ_short^-1, practically all events decay beyond the threshold, and therefore in the .
For decay widths Γ⪅[10]peV, the , becomes relevant.
§.§ Time dependent oscillation probability
The probability for a superposition of mass eigenstates i and j to yield the transition between the given initial and final states, defined in the amplitude (<ref>), is given by
𝒫(t) = 𝒩∫[̣⃗x]_x⃗_0 - Δx⃗^x⃗_0 + Δx⃗∑_ij𝒜_i(t,x⃗) 𝒜_j^*(t,x⃗) .
The normalisation constant 𝒩 can be evaluated using the condition
∑_outgoing𝒫(t) = 1 ,
where the sum is understood to include all possible processes, decay channels of the intermediate particle.
Since mass eigenstates acquire a complex phase while propagating, the superposition of distinct eigenstates depends on a phase difference that varies with time and distance, leading to a periodic fluctuation of the probability.
Therefore, we refer to the probability (<ref>) as an oscillation probability.
The position space integral in this probability is performed in <ref> for the and in <ref> for the .
After this integration, the oscillation probability reads, according to results (<ref>),
𝒫(t) = 𝒩∑_ij M_ijexp[- λ^'_ij(t) - ıϕ_ij(t)] , λ^'_ij(t) = f_ij + Λ_ij + γ_ij(t) + F_ij(t) ,
where the product of the interaction amplitudes with their momenta evaluated at the peak of the intermediate particle's wave packet is
M_ij = M_i M_j^* ,
M_i = M_i(p⃗_i) .
From the definition of the oscillation probability (<ref>), it can be seen that it depends on the sum over the two mass eigenstates of the absolute value squared transition amplitudes.
Since the (<ref>) and the decay term (<ref>) are real-valued, the probability depends on their sum
f_ij = f_i + f_j , γ_ij(t) = γ_i(t) + γ_j(t) ,
where their values at the minimum of the are given by
f_i = f(E_i, p⃗_i) , γ_i(t) := γ_i t , γ_i := γ_i(p⃗_i) = m_i Γ_i/2 E_i ,
E_i^2 = p⃗_i^2 + m_i^2 .
Contrary, the exponential term describing the phase (<ref>) is imaginary, such that the probability depends on the phase difference calculated in (<ref>),
ϕ_ij(t) = m_ijτ(t) ,
m_ij = m_i - m_j , τ(t) = m_0/E_0 t ,
where E_0m_0 is the Lorentz factor of the intermediate particle and τ(t) denotes the proper time the intermediate particle travels between production and decay.
It vanishes when the mass splitting becomes zero, if i = j.
This expression is modified by subdominant terms <cit.> and augmented by a time-independent shift, see <ref>.
From the derivation of the localisation term Λ_ij in (<ref>) as well as the dispersion term F_ij(t) in (<ref>), it can be seen that they inherit this dependence on the mass splitting and that they are given by
Λ_ij = 1/4p⃗_ij^Σ_0 p⃗_ij ,
Σ_0 p_ij^2 ,
F_ij(t) = 1/4
0 ,
Σ_0^-1 v_ij^2 t^2 ,
with
p⃗_ij = p⃗_i - p⃗_j , p_ij = p_i - p_j , v_ij = v_i - v_j ,
where the inverse of the effective width Σ_0 is defined in (<ref>).
Both the localisation and the dispersion term are decoherence terms and thus lead to a damping of the oscillations.
While the time-dependent dispersion term is absent in the , it becomes relevant in the .
§.§ probability
SPSS
NNO
From here on, we restrict to the phenomenology of symmetry-protected low-scale seesaw models and appearing in processes such as the one presented in <ref>.
The discussion is based on the , recently introduced with its minimal phenomenological version, the , in <cit.>.
The generation of light neutrino masses in seesaw models is directly related to the presence of .
The process in <ref> is if the two charged leptons have opposite charges and if they have equal charge.
In this scenario the oscillation probability (<ref>) for these two possible processes takes the form
𝒫^/_αβ(t) = 𝒩_α(t) ∑_i,j V^/_αβ ijexp[- λ^'_ij(t) - ıϕ_ij(t)] .
In comparison to reference <cit.>, the exponential has been replaced by the one of the oscillation probability (<ref>) containing, apart from the phase, additional terms due to the wave packet nature of the involved particles.
An additional term in <cit.>, which summarises the effects of the mass splitting in the interaction amplitudes, is neglected here since we treat the oscillations at .
The factors of the leptonic mixing matrix at production α and decay β are collected in the terms
V^_αβ ij := V^_β i V^*_α i V^*_β j V^_α j ,
V^_αβ ij := V^*_β i V^*_α i V^_β j V^_α j .
For the this results at in <cit.>
V^/_αβ ij =
±θ_α^2θ_β^2/4 for and for with i = j ,
for with i≠ j .
where the active-sterile mixing angle is defined by
θ⃗= y⃗v/m_M^ ,
with the Higgs v ≈[174]GeV and the Yukawa coupling of one sterile neutrino labelled y⃗, see <cit.>.
The normalisation condition (<ref>) for this scenario is evaluated for each flavour at production and yields
1
= ∑_β∑_
𝒫^/_αβ(t)
= ∑_β∑_
𝒩_α(t) ∑_ij V^/_αβ ijexp[- λ^'_ij(t) - ıϕ_ij(t)]
= ∑_β𝒩_α(t) θ_α^2θ_β^2/2∑_i=jexp[- f_ij - γ_ij(t) ] .
In the last step, it has been used that the sum of leptonic mixing matrix factors over and processes vanishes for i≠ j.
Since the dispersion term, the localisation term, and the phase difference vanish for i = j, they are absent in the last line.
The oscillation probability (<ref>) between the two mass eigenstates N_4 and N_5 is then given by
𝒫^/_αβ(t) = θ_β^2/2 ∑_γθ_γ^2(1 ±exp[- λ_45(t)] cos[ϕ_45(t)]) ∀ α ,
where the damping parameter takes the form
exp[-λ_45(t)] =
2 exp[- λ^'_45(t)]/exp[- f_44 - γ_44(t) ] +exp[- f_55 - γ_55(t) ] ,
and can be expressed as
λ_45(t) := Λ_45 + F_45(t) - ln[f_4 - f_5 + γ_4(t) - γ_5(t)] .
Here (x) denotes the hyperbolic secant function, which is equal to one at the origin, and decays exponentially for values x≫1.
For the parameter region and time scales considered in this work, it is justified to
assume that the two decay parameters are approximately equal such that
λ_45(t) = Λ_45 + F_45(t) - ln[f_4 - f_5] + *ε , ε = γ_4(t) - γ_5(t) ,
From the (<ref>), it can be seen that its minimum goes to zero if m_i = m_0.
However, heavy neutrinos with distinct masses cannot have m_4 = m_0 and m_5 = m_0 simultaneously.
Therefore, we consider two more limiting cases:
On the one hand, in cases where the reconstructed mass is near the mean of the heavy neutrino masses, with respect to the energy and momentum widths, the values of the are approximately equal f_4 ≈ f_5.
The normalisation then cancels these contributions, such that the damping factor becomes
λ_45(t) = Λ_45 + F_45(t) + *ε^2 , ε = []*γ_4(t) - γ_5(t) = []*f_4 - f_5 .
On the other hand, configurations in which either the or the decay terms are significantly different between the two mass eigenstates can lead to a damping of the oscillations.
For example, for mass splittings much larger than the energy and momentum widths, the minima f_4 and f_5 are very different.
The result is that one of the mass eigenstates is favoured by the available energy and momentum of the process, such that each event is dominated by one of the two Majorana particles, and the phenomenology is that of a pair of Majorana neutrinos without .
If, , ε = γ_4(t) - γ_5(t)≪ 1 but f_4 ≪ f_5 the damping parameter is given by
λ_45(t) = Λ_45 + F_45(t) - ln f_5 + *ε
= Λ_45 + F_45(t) + f_5 + ln1 + e^-2 f_5/2 + *ε .
This leads to significant damping if f_5 ≫ 1.
A similar argument holds for γ_4(t) ≪γ_5(t).
The interpretation, in this case, is that if one of the mass eigenstates decays much faster than the other, oscillations are significantly suppressed, and damping is large.
The reconstructed mass m_0 has to be near to one of the pole masses m_4 or m_5 since otherwise, the whole process is suppressed.
This effect is similar to the resonant scattering, described by an s-channel process with an intermediate particle of mass m.
For cases in which s - m≫Γ, the process is suppressed compared to s - m≪Γ.
In the present case, s is labelled m_0, and the width of the resonance is dominated by the energy and momentum widths.
While the definition of the damping parameter is derived in the context of in the , the presented strategies for its evaluation also apply to more general processes.
§ DAMPED
For the simulation of the damped oscillations discussed in the previous section, the parameters that can impact the damping are
* The masses of the heavy neutrinos.
* The decay widths of the heavy neutrinos, correlated with the time the heavy neutrinos propagate between production and decay.
* The momentum configuration of the external particles.
* The wave packet widths of the external particles.
The masses of the heavy neutrinos can be described in terms of their mean mass and their mass splitting
m = m_4 + m_5/2 , Δ m = m_45 = m_5 - m_4 .
The considered process and the heavy neutrinos' mean mass restrict the external particles' momentum configuration.
Since the exact momentum configuration changes on an event-per-event basis, a general result is obtained by averaging the computed damping parameter over several events.
Realistic momentum configurations are generated using the general purpose generator MadGraph5_aMC@NLO <cit.> together with the FeynRules <cit.> implementation of the defined in <cit.>.
The numerical computation, obtained using the algorithm presented in <ref>, takes the decay time of the heavy neutrino into account and is accordingly performed either in the or in the .
In order to simulate the process shown in <ref>, the widths of the external particles' wave packets in position space need to be estimated.
When the heavy neutrino is lighter than the W boson, the first W boson can be on-shell such that its width can be directly estimated.
In contrast, the second W boson is off-shell, and the external widths of its decay products must be estimated.
The situation is reversed if the heavy neutrino is heavier than the W boson.
The wave packet widths in configuration space of the incoming particles σ_p are assumed to be defined by the average distance between two protons in a beam bunch <cit.>.
The wave packet width of the outgoing leptons σ_l^ is assumed to be defined by the atom radius of silicon present in the detector material.
Final quarks and the final W boson are expected to have a larger uncertainty than final leptons, such that their width σ_j is given by 10 σ_l^.
See <ref> for more details.
§.§ Decay width dependence of the damping parameter
The mean decay width of the heavy neutrinos determines the time range the neutrino can propagate before it decays.
It is thus possible to examine the time dependence of the damping parameter λ = λ_45(t) by studying its dependence on the decay width.
A numerical computation of the damping parameters for fixed mean masses of heavy neutrinos is presented in <ref>.
The shape of the contours depicting constant damping consists of three regions:
* To the right is a plateau stretching over several orders of magnitude.
The plateau demonstrates that the effects due to varying decay widths, and with it, the time dependence of λ, are not significant in this region.
The plateau can be understood from the result (<ref>), noting that neither the momentum differences p⃗_ij nor the matrix Λ_ij contains any terms in Γ or t at .
* For smaller decay widths Γ⪅[0.1]μ eV the damping increases with decreasing decay width.
This effect is due to non-identical group velocities of wave packets of different mass eigenstates, which causes the wave packets to separate over time and, in turn, causes decoherence.
The effect becomes larger for heavy neutrinos that live longer.
However, if only the first 100 oscillation cycles are considered, the effect vanishes, and the plateau in the central section continues for small decay widths.
Alternatively, the effect also vanishes if decays inside a sphere of radius [50]cm are considered.
[
Note that the [50]cm represents a somewhat randomly chosen value for which we have checked that the effects can be neglected.
It does not represent a boundary at which those effects become relevant.
]
Since these two restrictions cover most phenomenologically interesting cases, the effects of decoherence due to the separation of wave packets can be neglected in the parameter region under consideration.
In the following discussions, we assume these restrictions.
They imply that the damping parameter depends, in addition to the width of the external wave packet in position space, only on the mean mass and the mass splitting of the heavy neutrinos, λ = λ(m, Δ m).
In <ref> the plateau extends beyond Γ = [0.1]eV, until where our numerical calculations are applicable, as we discussed in <ref>.
Since the physics leading to the damping as a function of Δ m is independent of Γ at according to our analytical derivations in <ref>, we conjecture that we can extrapolate the plateau also to larger Γ.
We make use of this conjecture when we analyse the consequence of damping on R_ll in <ref>.
The analytical damping formula (<ref>), together with the approximated expressions for decoherence (<ref>), reproduces the plateau found in the numerical evaluation for both regimes, as shown in <ref>.
Since the time-dependent dispersion is disregarded in the , the respective formulae do not feature the increased damping for small decay widths.
In the region where they are applicable, the analytical formulae are in good agreement with the numerical results for m=[10]GeV and in almost perfect agreement for m=[500]GeV.
§.§ Mass dependence of the damping parameter
With the restrictions established in the last section, the damping parameter is time-independent and can be studied as a function of the mean mass m and the mass splitting Δ m of the heavy neutrino.
This dependency is presented in <ref>.
The numerical results are shown in <ref>, the results for the time-dependent analytic formulae (<ref>) are presented in <ref>, and the results for a distance-dependent oscillation probability, as obtained in <cit.>, are shown in <ref>.
While the numerical results in the and the are approximately equal, the results for the approximated analytic formulae derived in <ref> differ between those regimes.
Therefore, the results obtained from the analytical computation are presented for each regime individually, while the presented numerical results are valid in both regimes.
Since the time-dependent formulae for the damping factor are similar in the and , the effects of the momentum dependence can be explained by studying the time-independent part of the damping factor (<ref>)
λ_45 = Λ_45 + *ε, t/t^short = 2(p⃗_45%̇ṡ/̇%̇ṡu⃗_P^2 σ_EP^)^2 + *ε, t/t^short, σ_p⃗/σ_E^ ,
where (<ref>) are used, and the last approximation is obtained by observing that, in both regimes, the energy width at production dominates the reciprocal sum in the localisation term for the baseline estimate of external widths, defined in <ref>.
Although the exact dependence of the damping factor on the mass is complicated since the process-dependent orientation of momenta and velocities change with varying mass, the sudden decrease of damping around the W boson mass can be explained by a change in the energy width.
The energy width at production is given by a sum of all external particles at the production vertex.
For heavy neutrinos lighter than the W mass, this includes the initial W boson and the initial charged lepton.
For heavy neutrinos above the W mass, the initial W boson is off-shell, and thus, the relevant particles are given by the two incoming quarks and the initial charged lepton.
An increase in the number of external particles participating in the production process results in a sudden increase in the energy width, which results in a sudden decrease in damping.
The shape of the contours describing constant damping is very similar for all computation methods in all regimes, except for the distance-dependent formulae in the .
This difference can be traced back to the localisation term in the , which takes the form
Λ_ij = 1/2p⃗_ij^ (∂_x⃗,x⃗ F_ij(t, x⃗))^-1p⃗_ij
E_ij (∂_t,t F_ij(t, x⃗))^-1 E_ij
= 1/4p⃗_ij ^Σ_0^p⃗_ij time dependent ,
E_ij^2(v⃗_0 ^Σ_0^-1v⃗_0)^-1 distance dependent .
where E_ij := E_i - E_j and F_ij(t, x⃗) is the in the (<ref>).
The time-dependent formula is given in (<ref>), and the distance-dependent one can be found in <cit.>.
While the upper formula is proportional to the eigenvalues of Σ_0, the lower formula is proportional to the inverse of the eigenvalues of Σ_0^-1.
Therefore, as long as the reconstructed neutrino velocity v⃗_0 is not parallel to one of the eigenvectors of the effective width, this lead to significantly different damping behaviour.
The absence of the dispersion term in the can yield different damping for each regime.
However, for the restrictions discussed in <ref>, the effects of this term are expected to be negligible.
Together with the fact that all other additional effects considered in the numerical derivation compared to the analytic one are small, the numerical and time-dependent analytical results in both regimes are expected to be approximately identical.
This is confirmed by the results shown in <ref>.
Therefore, the time-dependent results feature a smooth transition of the damping between the and the .
In contrast, for the distance-dependent results, the damping in the differs significantly from the results in the .
This smooth transition can be seen as a further advantage of the time-dependent formulae, as derived in this work, over the distance-dependent ones.
§.§ Wave packet widths dependencies
In order to understand the impact of the estimates for the widths of the external wave packets, we rederive the previous results after rescaling individual widths by a factor of one hundred and present their dependence on this scaling in <ref>.
[This scaling factor is chosen to generate large deviations from the baseline estimates to illustrate the parameter dependence and does not represent uncertainty in the baseline estimates.]
The effects on the relevant time thresholds are presented in <ref>.
They can be understood by considering their dependence on the external width given in (<ref>).
While the momentum width σ_p⃗ depends on the smallest width in configuration space at production and detection, which is given by the widths of the charged leptons σ_l^, the energy width σ_E^ depends, according to the following argument, mainly on the proton width σ_p.
The energy width depends on the two smallest widths at production and detection.
While the smallest width is cancelled by the global factor of σ_p⃗ P or σ_p⃗ D, respectively, the second smallest width dominates the energy width (<ref>).
For the baseline of external widths, see <ref>, the energy width at production is much smaller than the corresponding energy width at detection.
The reciprocal sum of those energy widths, precisely σ_E^, is thus dominated by the proton width.
In <ref>, we show that the main impact on the damping parameter is due to the proton width by individually varying the external widths.
This relation can be traced back to the fact that the damping is dominated by the energy width, see approximation (<ref>).
The effect of the non-monotonous behaviour of the contour around the W boson mass in <ref> is reduced when varying the jet width.
The non-monotonous behaviour can be explained by a change in the number of particles participating in production and detection, as described in <ref>.
The change in the number of particles is the opposite for production and detection, such that it weakens if the energy widths at production and detection are similar.
Multiplying the jet width by a factor of 100 has precisely the effect of equalising those energy widths.
From the above results, it becomes clear that the value of the external widths plays an essential role in the prediction of decoherence and merits further dedicated studies.
§.§ Decoherence effects on R_ll
The ratio between the number of and decays is called R_ll.
Since it is calculated by integrating over the , it is affected by decoherence.
Therefore, it is necessary to know the amount of damping to predict its value as a function of, , the mean mass and the active-sterile mixing parameter.
The probability of obtaining an or event, between proper times τ_min and τ_max, is given by the integral <cit.>
P_ll^/(τ_min, τ_max) = ∫_τ_min^τ_max
P_decay(τ)
P^/_osc(τ)
τ̣ ,
where τ = m_0/E_0 t is the proper time.
Here,
the decay probability density is given by
P_decay(τ) = - τexp(- Γτ) = Γexp(- Γτ) ,
and the oscillation probability is given by (<ref>)
P^/_osc(τ) = 𝒩(1 ±exp[- λ - μ^2 τ^2/4] cos[Δ m τ]) ,
where the time dependence of the damping parameter (<ref>) has been made explicit by defining the parameter μ using the dispersion term in the (<ref>).
In the limit τ_min→ 0 and τ_max→∞ the integral and the ratio of over events is given by
P_ll^/(λ,μ) ∝1 ± f(λ,μ)/2 ,
R_ll(λ,μ) = 2/1 + f(λ, μ) - 1 .
where the function that appears in both quantities is
f(λ,μ) = erfcx[Γ^'_-(μ)] + erfcx[Γ^'_+(μ)]/2Γ^'_λ(μ) ,
which is defined in terms of
erfcx(x) = exp(x^2) [1 - erf(x)] , Γ^'_±(μ) = Γ±ıΔ m/μ , Γ^'_λ(μ) = Γ/μ√(π)/expλ .
Here erfcx(x) is the scaled complementary error function, which decays exponentially for negative x approaches one for small x and is inversely proportional to x for large x.
For a subleading time dependence in the damping parameter, the function can be approximated using
f(λ,μ) = f(λ,0) (1 - Γ^2 - 3 Δ m^2/Γ^2 + Δ m^2ε/2 + *ε^2) , ε = 1/Γ_-^'(μ) Γ_+^'(μ) = μ^2/Γ^2 + Δ m^2 .
The term corresponds to the limit μ→ 0, which captures the .
In this limit the equations simplify to
f(λ,0) = f(0,0)/expλ ,
R_ll(λ) := R_ll(λ,0) = 1 - 2/1 + (1 + Δ m^2/Γ^2) expλ ,
where the damping independent term f(0,0) corresponds to the term that appears when taking furthermore the limit that also the constant localisation term can be neglected, λ→0, which recovers for the ratio of over events the known result <cit.>
f(0,0) = Γ^2/Γ^2 + Δ m^2 ,
R_ll(0,0) = Δ m^2/Δ m^2 + 2 Γ^2 .
In cases where the damping is large λ(τ) ≫ 1, the coherence between the propagating mass eigenstates is lost, and the phenomenology is that of two independent Majorana neutrinos.
Increasing λ, therefore, increases the observed R_ll compared to the naive case that does not take damping into account.
This behaviour is depicted for time-independent damping in <ref>.
Therefore, when considering R_ll(λ) as a function of Δ m and Γ, parameter regions with large damping must have a large R_ll and lines representing a smaller R_ll cannot penetrate those regions.
As depicted in <ref>, the contours representing the naive R_ll are given by constant ratios between Γ and Δ m.
However, damped R_ll contours are bound from above by regions of large damping.
Therefore, once damping becomes relevant, the R_ll bands follow the contour lines of the damping parameter shown in <ref>.
§.§ Decoherence effects on prompt searches for
BM
For the part of the parameter space where heavy neutrinos decay promptly, direct discovery of oscillations by resolving them as proposed in <cit.> is not possible.
However, the integrated R_ll ratio introduced in the last section can still be measured.
Hence, it is relevant to predict this ratio for the realistic of the linear and inverse seesaw, introduced and motivated in <cit.> and summarised in <ref>.
While the heavy neutrino mass splitting in the inverse seesaw scenarios depends on the active-sterile mixing parameter, defined in (<ref>),
Δ m = m_νθ⃗^-2 ,
the mass splitting in the linear seesaw scenarios is fixed for each point.
Therefore, in the inverse seesaw, it is always possible to restore coherence and the naive value of R_ll by considering larger values of the active-sterile mixing parameter and, accordingly, smaller mass splittings.
However, in the linear seesaw, the damping solely depends on the mass of the heavy neutrinos.
When it becomes relevant, coherence is lost independently of the value of the active-sterile mixing parameter.
The effects of decoherence onto the R_ll bands of the linear and inverse seesaw scenarios are depicted in <ref>.
For our baseline estimates of external particle widths, the damping becomes relevant for mass splittings Δ m ⪆[1]eV as shown in <ref>.
The presented points for the inverse seesaw feature such mass splittings for values of the active-sterile coupling in the range of 10^-4 < θ⃗^2 < 10^-1.
Therefore, the R_ll bands deviate from the naive ones in that region.
As discussed above, the bands then follow the contour lines of the damping parameter, such that large damping results in an R_ll of one.
The mass splittings for the points for the linear seesaw are such that decoherence effects can be neglected for our baseline estimate of external particle widths, defined in <ref>.
Consequently, the R_ll bands do not deviate from the naive ones in this case.
However, the situation changes if different widths of the external wave packets are assumed.
As shown in <ref>, the most significant effect on the damping is given by varying the proton width σ_p.
Hence the effect of this variation on the R_ll bands are depicted in <ref>.
If the proton width is divided by 100, the damping becomes relevant only above mass splittings Δ m ⪆[120]eV.
This results in the deviation from the naive R_ll only at smaller values of the active-sterile mixing parameter in the inverse seesaw, while the linear seesaw is still not affected as depicted in <ref>.
On the contrary, if the proton width is multiplied by 100, the damping already becomes relevant for mass splittings Δ m ⪆[10]meV.
The R_ll bands of the inverse seesaw points now deviate from the naive results already for larger values of the active-sterile mixing parameter, and the effects on the linear seesaw points are also relevant, as shown in <ref>.
The mass splitting of the linear seesaw with inverted ordering of the light neutrino masses is still too small for decoherence to become relevant.
However, for the normal ordered linear seesaw, damping must be considered.
From <ref>, it can be seen that the mass splitting of the normal ordered linear seesaw results in a small damping for masses m ⪅[10]GeV and [120]GeV⪅ m.
However, for masses in the range [10]GeV⪅ m ⪅[120]GeV, corresponding to the local minimum observable in <ref>, decoherence becomes relevant.
The largest damping is around m ≈[50]GeV, and thus the observed R_ll deviates from the naive one in this region.
Since, as discussed above, coherence cannot be restored by varying the active-sterile mixing parameter as in the case of the inverse seesaw, the observed R_ll is close to one in this mass range for all values of the active-sterile mixing angle.
Therefore, the R_ll bands form a pole around m ≈[50]GeV as shown in <ref>.
The effects of the damping parameter are crucial when reinterpreting prompt searches for signals.
Regions that result in an R_ll close to zero for a considered model do not yield any events and, thus, are not restricted by these searches.
However, if damping is significant, the true value of R_ll might significantly deviate from the naive one, as discussed above.
Therefore, decoherence effects can result in prompt searches for becoming applicable in regions of the parameter space that seem unconstrained when neglecting decoherence.
§ CONCLUSION
NNO
Low-scale symmetry-protected seesaw models generically predict the appearance of pseudo-Dirac pairs of heavy neutrinos.
It is usually expected that for such models, is severely suppressed <cit.>.
However, these considerations omit the crucial possibility of which can lead to a sizable number of events if the oscillation period is shorter or of the same order as the lifetime of the heavy neutrinos <cit.>.
In particular, for long-lived heavy neutrinos, the oscillation pattern between and events may be reconstructible in collider experiments <cit.>.
Apart from , decoherence can yield observable amounts of by reducing the destructive interference between propagating mass eigenstates and, with it, the suppression of .
While this obstructs the reconstruction of the oscillation pattern, it also enhances the chances of observing when the heavy neutrino's lifetime is smaller than its oscillation period.
In this work, we have quantitatively studied this effect for the first time.
To this end, we have derived oscillation probabilities for in the framework of with external wave packets, extending previous results, see <cit.>, with a reformulation that depends on the time difference between production and decay of the heavy neutrinos.
The derivations presented here are not only technically simpler than results that depend on the distance between production and decay of the heavy neutrinos but also more readily applicable to cases where an interplay between the oscillation period and lifetime is relevant, such as the over event ratio R_ll and the reconstruction of the pattern, which requires a translation of the oscillations into the proper time frame of the heavy neutrinos <cit.>.
The analytical calculation relies on expansions in small parameters and differentiates between the and the that apply for short and longer-lived particles.
The numerical comparison shows that the time-dependent results for the and are almost identical, such that a smooth transition connects the two regimes.
On the contrary, in the distance-dependent results, the differences between the two regimes are significant such that no smooth transition is possible.
We have compared the time-dependent analytical results with a numeric calculation using data and confirmed a broad range of applicability.
In <cit.>, we proposed using a single damping parameter λ that encodes all decoherence effects in .
Here, we provide the formulae necessary to calculate this damping parameter from first principles as a function of the heavy neutrinos' mean mass and mass splitting.
To that end, we provide the conditions under which the dependence on the decay width Γ and, therefore, the dependence on the propagation time between the production and detection of the heavy neutrinos can be neglected.
Based on our analytical results, we conjecture that the dependence of λ on m and Δ m also extends to Γ larger than the ones we calculated numerically.
Furthermore, we discussed a novel time-independent contribution to the phase and derived analytical formulae for this phase shift in the and .
However, in the parameter region under consideration, the phase shift can be neglected since it is either small or damping is large, which results in a suppression of the phase shift effects.
The employed framework depends on the widths of the external particles' wave packets in position space.
These input parameters need to be adjusted to the described experimental situation.
We have picked well-motivated values in order to present our results.
Additionally, we have discussed the dependence of relevant quantities on changes in these parameters.
We have also discussed possible limitations of applicability of the formalism from the time threshold.
In our computations, we followed a conservative approach restricted to Γ < Γ_ = [0.1]eV.
We illustrate the impact of decoherence by presenting bands of R_ll in the (m, θ⃗^2)-parameter plane and show significant deviations from the predictions when decoherence is neglected.
Hence, we quantify for the first time how the transition of a coherently oscillating pseudo-Dirac pair to two independently acting Majorana particles affects the phenomenology and, therefore, the discovery prospects of symmetry-protected low-scale seesaw models.
From the results of this work, it is clear that the possibility of decoherence has to be considered when studying signatures.
Large decoherence not only suppresses the oscillation pattern but also disables the mechanism that suppresses for a pseudo-Dirac pair.
Therefore, the phenomenology of a pseudo-Dirac pair changes significantly in regions of parameter space that exhibit sizable decoherence.
§.§ Acknowledgements
The work of Jan Hajer was partially supported by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) through the projects CFTP-FCT unit UIDB/00777/2020 and UIDP/00777/2020.
§ MOMENTUM INTEGRATION
STE
NDR
TDR
LDR
The integration of the transition amplitude (<ref>) over the three-momentum of the intermediate particle is performed using different techniques depending on how fast the complex phase varies over the width of the intermediate particles' wave packet.
In the , the phase varies slowly such that the integral is approximated around the maximum of the intermediate wave packet up to the second order in the momentum.
This method is referred to as Laplace's method.
If the phase varies rapidly over the intermediate wave packet, the method of stationary phase is used, where the largest contribution to the integral is obtained near the point for which the phase has an extremum.
In the , the phase varies slowly in directions transversal to the reconstructed momentum p⃗_0, while in the longitudinal direction, Laplace's method can still be used.
The third regime, called the , in which the method of stationary phase has to be used for all directions of the momentum p⃗, is not relevant to the phenomenology considered in this work.
In the following, we treat the and the separately and derive an oscillation probability in each of them.
§.§
For short times the phase (<ref>) varies slowly, as a function of the momentum, over the size of the intermediate particles' wave packet given by the (<ref>).
Thus the momentum integration can be performed using Laplace's method, where the integral is approximated around the momentum p⃗_i for which the together with the decay term (<ref>) is minimal.
Hence a necessary condition that needs to be fulfilled in order to apply Laplace's method is
*p⃗(f_i(p⃗) + γ_i(t,p⃗))_p⃗ = p⃗_i = 0 .
While the yields a Gauss-like shape for the amplitude with a maximum near the reconstructed momentum, the decay term favours large momenta.
Therefore we are interested in the minima of the .
For the analytic computation, it is assumed that the impact of the decay term on the position of the maximum is negligible, which can be justified if the decay term varies slowly over the width of the intermediate particles' wave packet.
The condition for this assumption to be valid is derived in (<ref>).
However, in the numerical computation in <ref>, the effects of the decay term are taken into account.
With the decay term neglected, the position of the maximum is expanded around the maximum of the reconstructed momentum
p⃗_i = p⃗_0 + p⃗_1 δ_i + *δ_i^2 .
where the mass splitting expansion parameter and the term are
δ_i = m_i^2 - m_0^2/2 E_0^2 , p⃗_1 = E_0 Σ_0^-1(u⃗_P^/2 σ_EP^2 + u⃗_D^/2 σ_ED^2) , u⃗_V = v⃗_V^ - v⃗_0 ,
V ∈{P,D} ,
The matrix Σ_0^-1 appearing in the term of the expansion is the inverse of the Hessian of the at in the momentum expansion, which is given in (<ref>).
Its eigenvalues can be interpreted as effective widths of the wave packet of the intermediate particle mass eigenstates in different directions, see <ref>.
The corresponding expansion of the energy of the mass eigenstate at the minimum of the around the reconstructed energy yields
E_i = E_0 + E_1 δ_i + *δ_i^2 ,
E_1 = p⃗_1 ⃗̇v_0 + E_0 ,
E_i = E_i(p⃗_i) .
Finally, the expansion of the velocity of the mass eigenstate at the minimum reads
v⃗_i = v⃗_0 + v⃗_1 δ_i + *δ_i^2 , v⃗_1 = p⃗_1 - v⃗_0 E_1/E_0 , v⃗_i = v⃗_i(p⃗_i) .
§.§.§ Expansion
In order to evaluate the momentum integral in the amplitude (<ref>), each exponential term is expanded up to second order in the momentum p⃗ around the minimum of the at p⃗_i resulting in a function that can be evaluated using Gaussian integration as demonstrated in <ref>.
The expansion of the (<ref>) around its minimum at p⃗_i results in
f_i(p⃗) = f_i + 1/2 (p⃗ - p⃗_i)^Σ_i (p⃗ - p⃗_i) + *p⃗ - p⃗_i^3 ,
where the linear term vanishes since the evaluation is performed at the minimum.
The constant term and the Hessian of the at the minimum read
f_i = f_i(p⃗_i) , Σ_i = Σ_0 + *δ_i ,
where the constant term to in the mass splitting expansion defines the mass width
f_i = f_1 δ_i^2 + *δ_i^4 ,
f_1 = *p⃗_1/2 σ_p⃗P^^2 + [e^_1P/2 σ_EP^]^2 + (P→ D) , σ_m := E_0/2 √(f_1)
where
e^_iV := e^_iV(p⃗_i) = e^_1Vδ_i ,
e^_1V := E_1 - p⃗_1 ⃗̇v_V .
and the term in the mass splitting expansion of the Hessian is
Σ_0 = /2 σ_p⃗P^2 + u⃗_P^⊗u⃗_P^/2 σ_EP^2 + (P→ D) .
From the inequality (<ref>) follows that the Hessian can be estimated to take values within
1/2σ_p⃗^2⪅Σ_i⪅1/2σ_E^2 ,
where Σ_i denotes that the considered inequality has to hold for all eigenvalues of Σ_i.
integration
Solving the energy integral via the theorem as demonstrated in <ref> introduces the complex pole energy (<ref>).
The dependence on the mass eigenstate energy (<ref>) can then be used to estimate the effects of the decay width expansion parameter (<ref>) on the
f_i(E_i^'(p⃗),p⃗) = [e^_P(E_i^'(p⃗), p⃗)/2 σ^_EP]^2 + … ,
where
e^_P(E_i^'(p⃗), p⃗) = √(1 - 2 ıϵ_i(p⃗))√(p⃗^2 + m_i^2) - E_0 - (p⃗ - p⃗_0) ⃗̇v_P .
and the ellipses denote terms that do not depend on the decay width expansion parameter, and for simplicity, we only consider the term at production, keeping in mind that the same arguments hold for the equivalent term at detection.
Furthermore, assuming that the energy and mass splitting expansion parameters are of the same order, the constant term and the Hessian of the momentum expanded (<ref>) are to
f_i = (δ_i - ıϵ_i(p⃗)/2 σ_EP^ E_0)^2 + … , Σ_i = (u⃗_P^/2 σ_EP^)^2 + *δ_i + *ϵ_i^2(p⃗) + … .
The effects of the decay width expansion parameter on the Hessian are subleading and can therefore be neglected.
While the effects on the are relevant, the term itself does only contribute to the damping of the amplitude (<ref>) if it differs greatly between different mass eigenstates.
This is due to the normalisation discussed in <ref>.
The effects of the decay width expansion parameter are therefore neglected in the analytical derivation by using (<ref>) while they are taken into account in the numerical calculations in <ref>.
Phase
The expansion of the phase (<ref>) around the minimum of the results in
ϕ_i(t,x⃗,p⃗) = ϕ_i(t,x⃗) + Δ⃗_i(t, x⃗) (̇p⃗ - p⃗_i) + 1/2 (p⃗ - p⃗_i)^ R_i(t) (p⃗ - p⃗_i) + *p⃗ - p⃗_i^3 ,
where the constant term, the linear coefficient, the Hessian, and the velocity of the i-th mass eigenstate are
ϕ_i(t,x⃗) = ϕ_i(t,x⃗,p⃗_i) , Δ⃗_i(t, x⃗) = v⃗_i t - x⃗ , R_i(t) = - v⃗_i ⊗v⃗_i/E_i t , v⃗_i = p⃗_i/E_i .
Since the inverse of Σ_i can be interpreted as an effective width of the intermediate wave packet (<ref>), it can be used to quantify the condition that the phase varies slowly over the width of the intermediate wave packet.
Replacing p⃗ - p⃗_i^2 in the expansion series of the phase (<ref>) with 2 Σ_i^-1 and requiring that the linear and quadratic terms are small results in
2 Δ⃗_i(t, x⃗)^2 ≪Σ_i , R_i(t) ≪Σ_i ,
These conditions can be approximated to read
2v⃗_i t - x⃗^2 ≪Σ_0 , - v⃗_i ⊗v⃗_i/E_i t ≪Σ_0 .
Since the averaging over the distance (or time) in a later step ensures that v⃗_i t ≈x⃗, the linear condition is automatically satisfied.
The quadratic condition can be approximated as
t/E_i≪1/2 σ_p⃗^2 ,
where the eigenvalue of Σ_0 containing σ_p⃗^2 is used since it imposes the most restrictive condition, see (<ref>).
For the same reason, the velocity-dependent parts in the contribution from the phase can be neglected.
This condition defines the short-time threshold up to which this integration method is valid.
It is given by
t_i^short = t^short + *δ_i = E_i Σ_i_smallest ,
t^short = E_0/2 σ_p⃗^2 ,
t ⪅ t^short ,
and it is usually sufficient to work with the approximation.
Decay term
The decay term (<ref>) is also expanded up to second order in its momenta around the minimum of the yielding
γ_i(t,p⃗) = γ_i(t) - ϝ⃗_i (̇p⃗ - p⃗_i) + 1/2 (p⃗ - p⃗_i)^ W_i(t) (p⃗ - p⃗_i) + *p⃗ - p⃗_i^3 ,
where the constant term, the linear coefficient, the Hessian and the parameter appearing in (<ref>) evaluated at the minimum of the are given by
γ_i(t) = γ_i t , ϝ⃗_i(t) = v⃗_i ϵ_i t , W_i(t) = 3 v⃗_i ⊗v⃗_i - /E_iϵ_i t , γ_i = m_i Γ_i/2 E_i , ϵ_i = γ_i/E_i .
Similar to the conditions appearing in the evaluation of the phase (<ref>), two conditions can be derived, ensuring that the decay term varies slowly over the width of the wave packet
2 ϝ⃗_i(t)^2 ≪Σ_i , W_i(t) ≪Σ_i .
For times earlier than the short-time threshold (<ref>), these conditions become
p⃗_iϵ_i ≪σ_p⃗ , ϵ_i ≪1/*3 v⃗_i ⊗v⃗_i - .
The linear condition holds as long as the decay width is of the same order or smaller as the momentum width, while the quadratic condition is satisfied for the assumptions used in the decay width expansion during the integration in <ref>.
§.§.§ Integration
The momentum integral can then be evaluated using a standard technique for multidimensional Gaussian integrals over the coordinates x⃗ with a symmetric positive definite matrix A and a linear term b⃗
∫[^̣3 x⃗] exp[b⃗⃗̇x - 1/2x⃗^ A x⃗] = exp[1/2b⃗^ A^-1b⃗] .
The vector b⃗ contains the linear order coefficients appearing in the momentum expansion of the phase (<ref>) and the decay term (<ref>) and reads
b⃗ = ıΔ⃗_i(t, x⃗) + ϝ⃗_i(t) = ıΔ⃗_i(t, x⃗) + *ϵ_i ,
For the analytical derivation, we neglect the correction from the decay term and proceed solely with the linear coefficient from the expansion of the phase around the minimum of the energy-momentum integral.
In the numerical calculation presented in <ref> the minimum p⃗_i is computed for the sum of the and the decay term, such that the linear contribution ϝ⃗_i(t) is absent all together.
The matrix A collects the Hessian matrices resulting from the momentum expansion of the (<ref>), the phase (<ref>), and the decay term (<ref>) around the minimum of the and reads
A = - Σ_i - W_i(t) - ı R_i(t) = - Σ_i + *t/t^short .
For times earlier than the short-time threshold (<ref>), it is justified to approximate this sum with just the contribution from the , see the conditions (<ref>).
Hence the Hessian from the expansion (<ref>) together with the linear term of the phase expansion (<ref>) integrated over the momentum using the Gaussian integral (<ref>) yield the
F_i(t,x⃗) := 1/2Δ⃗_i^(t, x⃗) Σ_i^-1Δ⃗_i(t, x⃗) .
For a given time and velocity, this term leads to an exponential damping of the transition amplitude for values of x⃗ far from v⃗_i t.
Here, far is defined by the eigenvalues of Σ_i.
We have explicitly discussed the effects of the decay width expansion parameter after (<ref>) and found the to agree with <cit.>.
The oscillation amplitude in the after momentum integration is therefore given by
𝒜_i(t,x⃗) ∝exp[- f_i - γ_i(t) - F_i(t,x⃗) - ıϕ_i(t,x⃗)] .
where the terms in the exponential are the constant (<ref>), the time dependent decay term (<ref>), the spacetime dependent (<ref>), and the spacetime dependent phase (<ref>).
§.§
For times later than the short-time threshold (<ref>), derived in the previous section, the oscillations are fast compared to the width of the wave packet such that Laplace's method is no longer the preferred method for evaluating the momentum integral.
However, this argument depends on the direction of the heavy neutrino momentum.
The second order term of the phase (<ref>) yields a contribution p⃗^2 tE for momenta orthogonal to v⃗_i, while momenta in the direction of v⃗_i obtain an additional Lorentz contraction factor that leads to (1 - v⃗_i^2) p⃗^2 tE.
Since at v⃗_i = v⃗_0, this factor slows down the oscillations in the direction of the reconstructed momentum.
Laplace's method is therefore still preferred in the direction along p⃗_0, while in directions orthogonal to it, the method of stationary phase is used.
§.§.§ Stationary phase
The linear term of the expanded phase (<ref>) averages oscillations to zero for momenta transversal to x⃗.
Therefore it can be assumed that p⃗ is parallel to x⃗.
Additionally, the requires that p⃗ is parallel to p⃗_0, which yields that x⃗ has to be parallel to p⃗_0.
Assuming that p⃗_0 is in the z direction the method of stationary phase can be used for x⃗_x and x⃗_y which yields
p_xϕ(t,x⃗,p⃗) = p_yϕ(t,x⃗,p⃗) = 0 ,
resulting in p_x = p_y = 0.
Using those, the argument of the exponential of the amplitude (<ref>) does not depend on p_x and p_y anymore such that only the integration over p = p⃗_z is left.
This integration is done using Laplace's method, as in <ref>.
The argument of the (<ref>) can then be expressed as
f_i(E_i( p), p) = ( p - p_0/2 σ_p⃗P^)^2 + (e^_iP( p)/2 σ_EP^)^2 + (P→ D) ,
where
e^_iV( p) = E_i( p) - E_0 - ( p - p_0) v_V .
Similar to the , the phase, the , and the decay term are all expanded around the momentum p_i for which the is minimal.
The effects of the decay term onto the position of the maximum are neglected in the analytical derivation, and the exact conditions for this approximation to hold are given in (<ref>).
The position of the maximum up to linear order in the mass splitting expansion parameter (<ref>) is
p_i = p_0 + p_1 δ_i + *δ_i^2 , p_1 = E_0/Σ_0( u_P/2σ_EP^2 + u_D/2σ_ED^2) , u_V := v_V^ - v_0 ,
the mass splitting expansion of the mass eigenstate energy reads
E_i = E_0 + E_1 δ_i + *δ_i^2 ,
E_1 = E_0 + p_1 v_0 ,
E_i = E_i( p_i)
,
and the mass splitting expansion of the mass eigenstate velocity is
v_i = v_0 + v_1 δ_i + *δ_i^2 , v_1 = p_1 - E_1 v_0/E_0 , v_i = v_i( p_i) .
The effects of the decay width expansion parameter are neglected here for the same reasons as in <ref>.
§.§.§ Expansion
Just as in the , the terms in the exponent of the amplitude (<ref>) need to be expanded up to second order in the momentum in order to perform a Gaussian integration in <ref>.
The expansion of the (<ref>) around the momentum of the minimum is given by
f_i(E_i( p), p) = f_i + 1/2Σ_i ( p - p_i)^2 + []* p - p_i^3 ,
where the linear term vanishes since the expansion is evaluated at the minimum, while the constant term and the Hessian are given by
f_i = f_1 δ_i^2 + *δ_i^4 = f_i(E_i, p_i) , Σ_i = Σ_0 + *δ_i ,
and their contributions in the mass splitting expansion are given by
f_1 = ( p_1/2 σ_ pP^)^2 + (E_1 - p_1 v_P/2 σ_EP^)^2 + (P→ D) , Σ_0 = 1/2σ_p⃗P^2 + u_P^2/2σ_EP^2 + (P→ D) .
Phase
The expansion of the phase (<ref>) around the momentum of the minimum of the yields
ϕ_i(t, x, p) = ϕ_i(t, x) + Δ_i(t, x) ( p - p_i) + 1/2 R_i(t) ( p - p_i)^2 + []* p - p_i^3 ,
where the constant term, the linear coefficient, the Hessian, and the velocity of the mass eigenstate are given by
ϕ_i(t, x) = ϕ_i(t, x, p_i) , Δ_i(t, x) = v_i t - x , R_i(t) = m_i^2/E_i^3 t , v_i = p_i/E_i .
Similar to the case of the , a time threshold can be obtained by requiring that the phase varies slowly over the width of the wave packet.
The wave packet width is approximated by Σ_i^-1/2 and used to reexpress the momentum deviations.
The conditions resulting from the requirement that the linear and quadratic terms are small are
2 Δ_i^2(t, x) ≪Σ_i , R_i(t) ≪Σ_i .
The linear condition is ensured by the distance average performed in the next section, while the quadratic condition defines the long-time threshold.
A time threshold independent of the mass eigenstates can be defined by approximating it at in the mass splitting expansion (<ref>), leading to the validity range of the
t_i^long = t^long + *δ_i =
Σ_i E_i^3/m_i^2 ,
t^long := Σ_0 E_0^3/m_0^2 ,
t^short ⪅ t ⪅ t^long .
Decay term
The expansion of the decay term (<ref>) is the same as in the (<ref>) where all vector quantities are replaced by their corresponding longitudinal component.
γ_i(t, p) = γ_i(t) - ϝ_i (̇ p - p_i) + 1/2 ( p - p_i)^ W_i(t) ( p - p_i) + * p - p_i^3 .
The constant term, the linear coefficient, the Hessian and the terms appearing in (<ref>) are given by
γ_i(t) = γ_i t , γ_i = m_i Γ_i/2 E_i , ϝ_i(t) = v_i ϵ_i t , W_i(t) = 3 v_i^2 - 1/E_iϵ_i t , ϵ_i = γ_i/E_i .
The conditions for the decay term to vary slowly over the width of the neutrino wave packet read
2 ϝ_i^2(t) ≪Σ_i , W_i(t) ≪Σ_i .
assuming t ⪅ t_i^long, these conditions reduce to
ϵ_i p_i E_i^2/m_i^2 ≪σ_E^ , (3 v_i^2 - 1) ϵ_i ≪m_i^2/E_i^2 .
Similar to the situation in the (<ref>), the quadratic condition is automatically satisfied.
The linear condition requires the decay width to be of the same order or smaller than the effective momentum width of the wave packet, which is typically of the order of the energy width σ_E^ and thus much smaller compared to the width in the .
Additionally, the factor E_im_i can result in a violation of the condition for ultra-relativistic particles.
This reflects the fact that for such particles, the is valid up to arbitrarily large times, such that eventually, the decay term becomes important.
In the numerical estimation of the damping, the effect of the decay term is taken into account.
§.§.§ Integration
The integral in the transition amplitude (<ref>) can be solved using the general result (<ref>) where the vector b⃗, now scalar, contains the first order terms from the expansion of the phase (<ref>) and the decay term (<ref>) and reads
b = ıΔ_i(t, x) + ϝ_i(t) = ıΔ_i(t, x) + *ϵ_i .
In the following, the decay width expansion parameter correction is neglected for the analytical derivation but taken into account for the numerical calculation.
The Hessians resulting from the momentum expansion of the (<ref>), the phase (<ref>), and the decay term (<ref>) are collected in A which is now a scalar and reads
A = - Σ_i - W_i(t) - ı R_i(t) = - Σ_i + *t/t^long .
For times earlier than the long-time threshold (<ref>), the contribution is given by the Hessian of the , see conditions (<ref>).
For the numerical results, the contribution from the Hessian of the phase and the decay term are taken into account.
The integration (<ref>) over the momentum yields the
F_i(t, x) := Δ_i^2(t, x)/2 Σ_i(t) ,
and the amplitude (<ref>) after integration is given by
𝒜_i(t, x) ∝exp[- f_i - γ_i(t) - F_i(t, x) - ıϕ_i(t, x)] ,
where the terms in the exponent are the (<ref>), the decay term (<ref>), the (<ref>), and the phase (<ref>).
§ DISTANCE INTEGRATION
In order to obtain an oscillation probability, the amplitudes for different mass eigenstates are coherently summed over
𝒫(t,x⃗) ∝∑_ij𝒜_i(t,x⃗) 𝒜_j^*(t,x⃗) .
An average over the oscillation distance is performed to obtain a formula dependent on the time.
For the as well as for the the only distance-dependent components in the amplitudes (<ref>) are the phases (<ref>) and the (<ref>).
Since the restricts the values of t and x⃗ to a range, similar to how the (<ref>) restricts the values of p and E, the probability is expanded up to second order around x⃗_ij, the position of the minimum of the , before the resulting Gauss-like integral is evaluated.
Typically, neither the oscillation distance x⃗ nor the oscillation time t are measured with perfect precision.
Therefore, the oscillation probability (<ref>) is either integrated over a span of time, as in <cit.>, or over a region of space in order to address the experimental uncertainty.
In the following, the second possibility is employed, which yields an oscillation probability as a function of time.
The distance-integrated oscillation probability reads
𝒫(t) ∝∫[̣⃗x]_x⃗_0 - Δx⃗^x⃗_0 + Δx⃗𝒫(t,x⃗) ,
where Δx⃗ labels the experimental uncertainty in determining the distance travelled by the intermediate particle between its production and decay.
This distance has to be larger than the width of the intermediate particle in spacetime, which describes the minimal uncertainty due to the wave packet nature of the intermediate particle.
Due to the exponential damping stemming from the (<ref>) for values x⃗ - x⃗_0⪆Σ_0, the distance integration can be taken to infinity Δx⃗→∞.
§.§
Since the oscillation probability (<ref>) depends on the absolute value square of the transition amplitude, the (<ref>) and (<ref>) need to be summed
f_ij = f_i + f_j = f_1 (δ_i^2 + δ_j^2) + *δ_i^4 , γ_ij(t) = γ_i(t) + γ_j(t) = (γ_i + γ_j) t .
The same holds true for the (<ref>)
F_ij(t,x⃗) := F_i(t,x⃗) + F_j(t,x⃗) ,
Since dispersion effects are neglected in the , the can be approximated by its term in the mass splitting expansion
F_ij(t, x⃗) = F_0(t, x⃗) + *δ_i ,
F_0(t, x⃗) = Δ⃗_0^(t, x⃗) Σ_0^-1Δ⃗_0(t, x⃗) , Δ⃗_0(t, x⃗) = v⃗_0 t - x⃗ .
Since it is quadratic in the spacial coordinates, there are no constant or linear terms.
The position of the minimum of the is thus at given by
x⃗_ij(t) = x⃗_0(t) + *δ_i , x⃗_0(t) = v⃗_0 t .
Expansion
The expansion of the in x⃗ around x⃗_ij(t) reads at in the mass splitting expansion
F_0(t,x⃗) = (x⃗ - x⃗_0(t))^Σ_0^-1 (x⃗ - x⃗_0(t)) + *δ_i .
Since dispersion is neglected, there is no constant term.
The phase (<ref>) is rewritten in terms of the deviation from the maximum in position space x⃗ - x⃗_ij(t) yielding
ϕ_i(t,x⃗) = ϕ_i(t) - p⃗_i (̇x⃗ - x⃗_ij(t)) , ϕ_i(t) = E_i t - p⃗_i ⃗̇x_ij(t) .
Since the oscillation probability (<ref>) is proportional to the absolute value square of the transition amplitude, the phase difference
ϕ_ij(t,x⃗) = ϕ_i(t,x⃗) - ϕ_j(t,x⃗) = ϕ_ij(t) - p⃗_ij(̇x⃗ - x⃗_ij(t)) ,
needs to be considered.
The constant term encodes the phase difference in time, while the linear coefficient consists of a momentum difference
ϕ_ij(t) := ϕ_i(t) - ϕ_j(t) = E_ij t - p⃗_ij⃗̇x_ij(t) ,
E_ij = E_i - E_j , p⃗_ij = p⃗_i - p⃗_j .
Using the mass splitting expansion of the energy (<ref>) and momentum (<ref>) yields
E_ij = (E_0 - v⃗_0 ⃗̇p_1) δ_ij + *δ_i^2 , p⃗_ij = p⃗_1 δ_ij + *δ_i^2 , δ_ij := δ_i - δ_j ,
and the appearing difference in the mass splitting expansion parameter (<ref>) can be approximated to be
δ_ij = m_i^2 - m_j^2/2 E_0^2 = m_ij m_0/E_0^2 + *m_ij^2/E_0^2 ,
m_ij = m_i - m_j .
It is used that the reconstructed mass m_0 cannot be far from the mean of the mass eigenstate masses m, since otherwise, it is not possible to have p⃗_i ≈p⃗_0 and E_i ≈ E_0 for both mass eigenstates at the same time, and thus one of the two f_i terms in the amplitude (<ref>) would lead to large damping of oscillations.
Therefore, the phase difference is at in terms of the proper time
ϕ_ij(t) = m_ijτ(t) + *δ_i^2, m_ij^2/E_0^2 , τ(t) = m_0/E_0 t .
Integration
The integral (<ref>) can be evaluated as a Gaussian integral, using relation (<ref>) with the linear term from the phase expansion (<ref>) and the Hessian from the expansion (<ref>)
b⃗ = p⃗_ij , A = 2 Σ_0^-1 ,
resulting in the time-independent localisation term
Λ_ij = 1/4p⃗_ij^Σ_0 p⃗_ij + *δ_i^3 .
The transition amplitude after this distance integration is, therefore,
𝒫(t) ∝∑_ijexp[- λ^'_ij(t) - ıϕ_ij(t)] , λ^'_ij(t) = f_ij + Λ_ij + γ_ij(t) ,
where the and the decay term are given by (<ref>), the phase is given by (<ref>), and the localisation term is given by (<ref>).
§.§
The calculations for the are very similar to the ones described in <ref>.
However, compared to the dispersion is not neglected, and therefore, the approximation v_i ≈ v_j does not apply, which leads to a separation of wave packets of different mass eigenstates over time.
The sum of the (<ref>) and the decay terms (<ref>) are
f_ij = f_i + f_j = f_1 (δ_i^2 + δ_j^2) + *δ_i^4 , γ_ij(t) = γ_i(t) + γ_j(t) = (γ_i + γ_j) t ,
The distance average is performed by extending the terms in the exponent of the transition amplitude (<ref>) around the minimum of the sum of the (<ref>) of the two mass eigenstates appearing in the oscillation probability
F_ij(t, x) = F_i(t, x) + F_j(t, x) .
The minimum is given at the position
x_ij(t)
= Σ_i(t) v_j + Σ_j(t) v_i/Σ_i(t) + Σ_j(t) t
= v_i + v_j/2 t + δ_i ,
Expansion
The expansion of the yields
F_ij(t, x) = F_ij(t) + 1/2 Z_ij( x - x_ij(t))^2 + []* x - x_ij(t)^3 .
where the constant term and the Hessian are
F_ij(t) = v_ij^2/2t^2/Σ_i(t) + Σ_j(t)
= 1/4Σ_0^-1 v_ij^2 t^2 + *δ_i^3 ,
v_ij = v_i - v_j ,
Z_ij(t) = 1/Σ_i(t) + 1/Σ_j(t) = 2/Σ_0(t) + *δ_i ,
using the expansion (<ref>) the velocity difference can be written as
v_ij = v_1 δ_ij + *δ_i^2 = p_1 - E_1 v_0/E_0δ_ij + *δ_i^2 , δ_ij := δ_i - δ_j .
Since the expansion is performed around the minimum, the linear term is still absent.
However, since dispersion is not neglected, the constant term does not vanish.
The phase can be rewritten similarly to the in terms of a constant and a linear term
ϕ_i(t, x) = ϕ_i(t) - p_i ( x - x_ij) , ϕ_i(t) = E_i t - p_i x_ij .
Since the phase contains the imaginary contributions to the amplitude, the difference between the two mass eigenstates appears in the oscillation probability
ϕ_ij(t, x) = ϕ_i(t, x) - ϕ_j(t, x) =
ϕ_ij(t) + p_ij (x - x_ij) ,
the constant term and the linear coefficient are
ϕ_ij(t) = E_ij t - p_ij x_ij ,
E_ij = E_i - E_j , p_ij = p_i - p_j .
Using the expansions (<ref>) yielding
E_ij = E_1 δ_ij + *δ_i^2 = (E_0 - v_0 p_1) δ_ij + *δ_i^2 , p_ij(t) = p_1 δ_ij + *δ_i^2 ,
together with the approximation (<ref>), leads for the phase difference to
ϕ_ij(t) = m_ijτ(t) + *δ_i^2, m_ij^2/E_0^2 , τ(t) = m_0/E_0 t .
which is identical to the phase difference in the (<ref>).
Integration of the oscillation probability
The integration
𝒫(t) ∝∫[x]_ x_ij - Δ x^ x_ij + Δ x𝒫(t, x) ,
can be performed using (<ref>) with
b = p_ij ,
A = Z_ij ,
resulting in the constant localisation term
Λ_ij = 1/2 Z_ij^-1 p_ij^2 = 1/2Σ_i(t) Σ_j(t)/Σ_i(t) + Σ_j(t) p_ij^2 = 1/4Σ_0 p_ij^2 + *δ_i^3 .
Finally, the oscillation probability reads
𝒫(t) ∝∑_ijexp[λ^'_ij(t) - ıϕ_ij(t)] , λ^'_ij(t) = f_ij + Λ_ij + F_ij(t) + γ_ij(t) ,
where the and the decay term are given by (<ref>), the phase difference is given by (<ref>), the dispersion term is given by (<ref>), and the localisation term is given by (<ref>).
§ PHASE SHIFT
For short times, corrections to the expression of the phase (<ref>), which scale linearly with the decay width expansion parameter ϵ_i become relevant.
The expansion in the mass splitting expansion parameter δ_i is given in (<ref>) in detail.
Therefore, we derive an analytical correction to the phase for parameter points in which ϵ_i is non-negligible and test how the numerical phase as obtained via the algorithm described in <ref> deviates from the expression for the phase (<ref>).
As presented in <ref>, the contribution in ϵ_i to the phase results in the usual decay term γ_i(t, p⃗) (<ref>).
However, there is also a contribution from the , which yields
ϕ_i(t, x⃗, p⃗) = E_i(p⃗) t - p⃗⃗̇x - γ_i(p⃗) [e^_iP(p⃗)/2 σ_EP^2 - e^_iD(p⃗)/2 σ_ED^2] + *ϵ_i^2 .
While the direct contribution of the correction term to the phase ϕ_i(t, x⃗, p⃗) is negligible, it has a significant effect on the (<ref>) since the corrections appear in the linear term
Δ⃗_0(t, x⃗) = v⃗_0 t - x⃗ + γ_0 (u⃗_P/2σ_EP^2 - u⃗_D/2σ_ED^2) + *ϵ_i^2 . γ_0 = m_0 Γ/2E_0 ,
This correction to the results in a shift of the position of its minimum (<ref>), which is now at
x⃗_ij(t) = v⃗_0 t + γ_0 (u⃗_P/2σ_EP^2 - u⃗_D/2σ_ED^2) + *δ_i, ϵ_i^2 .
The correction becomes relevant for large decay widths since then the particle's lifetime becomes small, while simultaneously, the correction term becomes larger.
The resulting phase is then given by
ϕ_ij(t) = ϕ_ij + m_ijτ(t) ,
where the constant phase shift is
ϕ_ij = - p⃗_ij(u⃗_P/2σ_EP^2 - u⃗_D/2σ_ED^2) γ_0 .
Since Σ_0 is symmetric, this can be simplified to read
ϕ_ij = u⃗_D^/2σ_ED^2Σ_0^-1u⃗_D/2σ_ED^2 m m_ijϵ_0 - (D→ P) + *ϵ_i^2, δ_i^2, m_ij^2/E_0^2 , ϵ_0 = γ_0/E_0 .
Numerical results for the phase shift are shown in <ref>.
As can be seen from <ref>, in the considered parameter region, the total value of the modulus of the phase shift is small for most of the parameter space except for part of the region with large Δ m, where it can get large.
However, we remark that in this parameter region, many oscillations take place before the heavy neutrino decays, and thus the phase shift per oscillation is still small.
This is illustrated in <ref>, which shows the modulus of the relative phase shift
ϕ_ij^rel := ϕ_ij/max(2π, m_ijτ) .
Furthermore, comparing with <ref>, one can see that in the region with a large total phase shift, the damping λ is very large, such that the oscillation term is strongly suppressed, making the phase shift practically unobservable.
In summary, the numerical results show that, for current collider simulations, one can safely neglect the phase shift in the considered parameter region.
However, outside the applicability region the phase shift can become significant, which can be seen from (<ref>).
The numerical results match this behaviour.
Since these results are based on values of the decay width larger than Γ_, and since we do not see a physical reasons for the phase shift to become arbitrarily large, we neglect the phase shift for the main part of this work.
[
float,
captionpos=b,
caption=[Algorithm describing the strategy to calculate decoherence effects.]
Algorithm describing the strategy to calculate decoherence effects.
,label=lst:code
]
define E_0, p⃗_0, m_0||
for V in P, D do define σ_x⃗V, σ_p⃗V, σ_EV, Σ_V, v⃗_V^||
for i in 4, 5 do define m_i, Γ_i||
t = rand.variate[expdistri[mean[Γ_4, Γ_5]]] E_0m_0||
for V in P, D do
e_V[E,p⃗] = E - E_0 - (p⃗ - p⃗_0) ·v⃗_V^
f_V[E, p⃗, γ] = p⃗ - p⃗_0^2(2 σ_p⃗V)^2 + (e_V^2[E, p⃗] - e_V[0, p⃗] γ^2E)/(2 σ_EV)^2||
for i in 4, 5 do
E_i[p⃗] = sqrt[p⃗^2 + m_i^2]||
γ_i[p⃗] = m_i Γ_i2 E_i[p⃗]||
f_i[p⃗] = f_P[E_i[p⃗], p⃗, γ_i[p⃗]] + f_D[E_i[p⃗], p⃗, γ_i[p⃗]] ||
λ_i[p⃗] = f_i[p⃗] + γ_i[p⃗]t
e_i[p⃗] = e_P[E_i[p⃗], p⃗](2σ_EP^2) - e_D[E_i[p⃗], p⃗](2σ_ED^2)
ϕ_i[p⃗, x⃗] = E_i[p⃗] t - p⃗·x⃗ - γ_i[p⃗] e_i[p⃗]||
α_i[p⃗, x⃗] = λ_i[p⃗] + i ϕ_i[p⃗, x⃗]||
for i in 4, 5 do // momentum integral
p⃗_i = argmin[λ_i[p⃗]]||
Σ_i[x⃗] = hessian[α_i[p⃗, x⃗], p⃗] at p⃗ = p⃗_i||
Δ⃗_i[x⃗] = ∂_p⃗ϕ_i[p⃗, x⃗] at p⃗ = p⃗_i||
F_i[x⃗] = Δ⃗_i[x⃗] ·Σ_i^-1[x⃗] ·Δ⃗_i[x⃗]2
α_45[x⃗] = α_4[p⃗_4, x⃗] + F_4[x⃗] + conj[α_5[p⃗_5, x⃗] + F_5[x⃗]]||
// distance integral
x⃗_45 = argmin[Re[α_45[x⃗]]]||
Z_45 = hessian[α_45[x⃗], x⃗] at x⃗ = x⃗_45||
Π⃗_45 = ∂_x⃗Im[α_45[x⃗]] at x⃗ = x⃗_45
Λ_45 = Π⃗_45· Z_45^-1·Π⃗_452||
β_45 = Λ_45 + α_45[x⃗_45]||
𝒩_45 = log[exp[- 2 λ_4[p⃗_4]]/2 + exp[- 2 λ_5[p⃗_5]]/2]||
λ_45 = Re[β_45 + 𝒩_45]||
ϕ_45 = Im[β_45 + 𝒩_45]||
§ NUMERICAL DECOHERENCE DERIVATION
In this section, the algorithm for the numerical computation of the damping parameter λ is presented.
The algorithm expects the momenta of the external particles and the widths of the wave packets of the external particles as input.
Our estimates for the external widths in the process in <ref> are given in <ref>.
Realistic momentum configurations can be generated using a generator such as MadGraph together with the implementation of the introduced in <cit.>.
We present an algorithm for the derivation of the numerical results in <ref>.
In the first lines, the kinematics of the event and the wave packet widths of the external particles are used to define the given quantities as input to the algorithm.
In particular, in <ref>, the kinematics of the external particles are used to define the reconstructed quantities (<ref>), in <ref> the definitions (<ref>) are used to define relevant widths, and in <ref> the masses and decay widths of the heavy neutrino mass eigenstates are defined.
In <ref>, the propagation time between production and decay of the heavy neutrino superposition is drawn from an exponential distribution, defined by the mean decay width of the neutrinos.
In <ref>, the (<ref>) at production and detection are defined, where the corrections in the decay width expansion (<ref>), are taken into account.
In the following lines, quantities for the mass eigenstates are defined.
In <ref>, the decay term is calculated,
<ref> defines the for the heavy neutrino, and in
<ref> the phase, taking into account the imaginary part stemming from the decay width expansion of the , is calculated.
All exponential terms relevant for the transition amplitude (<ref>) are collected in <ref>.
The integration over the three-momentum of the heavy neutrino wave packet is performed by approximating all terms up to second order around the maximum of the wave packet.
The maximum of the wave packet is defined in <ref> by the minimum of the , where the effects of the decay term are taken into account.
Subsequently, the terms of the exponential quadratic in p⃗ - p⃗_i are computed in <ref>.
Since the expansion is around the minimum of f_i(p⃗) + γ_i(p⃗), only the complex phase has to be considered for the linear terms in <ref>.
The integration results in the in <ref>, which is defined in (<ref>).
The following steps are valid for the 𝒜_i 𝒜_j^* terms in the probability since the damping term is relevant for terms i ≠ j, which are responsible for oscillations.
The distance integral is evaluated in the same fashion as the three-momentum integral.
The only terms in the exponential that depend on the distance are the F_ij(x⃗) and the complex phase ϕ_ij(x⃗, t).
The expansion is around the minimum of the computed in <ref>.
Since the phase is linear in x⃗, the only contribution to the quadratic terms in x⃗ - x⃗_ij are given by the Hessian of the computed in <ref>.
After the integration, the final exponent term is defined by
𝒜_i(t) 𝒜_j^*(t) = 𝒩_ij(t) exp(-β_ij(t))
in <ref>.
The normalisation is computed in <ref> based on the condition (<ref>).
For the computation of the damping in the case of in this paper, the decay terms γ_i[p⃗_i] in β and in 𝒩 have been neglected, as we found them to be not significant.
Finally, the damping parameter and the complex phase are computed in <ref>.
While the algorithm is presented with the in mind, it can easily be applied to the by replacing vector quantities denoted with bold font by their projection onto the direction of p⃗_0.
LDR
|
http://arxiv.org/abs/2307.04268v1 | 20230709214743 | Optical Properties of Charged Defects in Monolayer MoS$_2$ | [
"Martik Aghajanian",
"Arash A. Mostofi",
"Johannes Lischner"
] | cond-mat.mtrl-sci | [
"cond-mat.mtrl-sci",
"cond-mat.mes-hall"
] |
[email protected]
Departments of Physics and Materials and the Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, London, SW7 2AZ, UK
We present theoretical calculations of the optical spectrum of monolayer MoS_2 with a charged defect. In particular, we solve the Bethe-Salpeter equation based on an atomistic tight-binding model of the MoS_2 electronic structure which allows calculations for large supercells. The defect is modelled as a point charge whose potential is screened by the MoS_2 electrons. We find that the defect gives rise to new peaks in the optical spectrum approximately 100-200 meV below the first free exciton peak. These peaks arise from transitions involving in-gap bound states induced by the charged defect. Our findings are in good agreement with experimental measurements.
Optical Properties of Charged Defects in Monolayer MoS_2
Johannes Lischner
August 12, 2023
========================================================
§ INTRODUCTION
Monolayer transition-metal dichalcogenides (TMDs) are two-dimensional (2D) materials which have been intensely studied in recent years because of their attractive electronic properties for applications in transport and optoelectronic devices <cit.>. Many materials in this class exhibit a direct band gap in the optical range as well as multiple band extrema <cit.> which give rise to rich valley physics <cit.>. The reduced dimensionality of 2D materials causes a weaker electronic screening of electron-electron interactions compared to 3D systems and the stronger interaction results in large binding energies of excitons, bound electron-hole pairs. In monolayer TMDs, exciton binding energies can be as large as several hundred meV <cit.>.
Charged defects can have a significant impact on the electronic structure and transport properties of TMDs. In particular, doped carriers can increase the conductivity, while scattering from charged defects reduces it. The optical properties of TMDs are also influenced by the presence of charged defects. For example, Greben and coworkers <cit.> demonstrated that irradiating monolayer MoS_2 with an electron beam gives rise to an additional peak (approximately 200 meV below the first neutral exciton peak) in the photoluminescene spectrum, which they interpreted as the signature of neutral excitons that are bound to an ionized donor defect. Similarly, Shang et al. <cit.> studied the optical properties of monolayer WS_2 and MoS_2 and found that the photoluminescence could be tuned from donor-bound to acceptor-bound excitons by changing from n-doping to p-doping.
To gain insight into the microscopic properties of excitons bound to charged defects, Ganchev and coworkers <cit.> solved a three-particle Schroedinger equation based on an effective mass approximation for the electronic structure of monolayer TMDs. Similarly, Wu <cit.> used the effective mass approximation to study transitions between bound defect states. However, such models do not capture the delicate effects associated with bound defect states arising from the multi-valley electronic structure of TMDs. To address this shortcoming of effective mass methods which typically only capture defect states from the K and K' valleys, we previously developed an atomistic approach to describe the electronic structure of a TMD monolayer with a charged defect <cit.>. In particular, we used the tight-binding approach to model the large supercells required to describe the long-ranged electrostatic potential of the charged defect. Our calculations demonstrated that the most strongly bound acceptor states derive from the Γ valley for a wide range of dielectric environments and defect charges. These predictions were verified by scanning tunneling spectroscopy experiments <cit.>. For donor impurities, we predicted that the most strongly bound in-gap states derive from the Q valleys for a range of dielectric environments and defect charges <cit.>.
In this paper, we extend our atomistic modelling approach to calculate the optical properties of monolayer TMDs with charged defects. For this, we solve the Bethe-Salpeter equation using the tight-binding states as input. We calculate optical spectra of both donor and acceptor defects in MoS_2 on a SiO_2 substrate <cit.>. We find that the charged defects induce additional low-energy peaks in the optical spectrum. These arise from electronic transitions which involve bound defect states. The binding energy of these excitations, which can be interpreted as defect-bound excitons, are between 100 and 200 meV in good agreement with experimental findings.
§ METHODS
§.§ Bethe-Salpeter equation
To study the effect of a charged adsorbate on the optical properties of monolayer MoS_2, we solve the Bethe-Salpeter equation (BSE) for an N× N MoS_2 supercell with a single adsorbate, which is modelled as a point charge that creates a screened potential acting on the electrons in the MoS_2.
The BSE is given by
∑_c'v'𝐤' H^BSE_cvc'v'(𝐤,𝐤') A^M_c'v'𝐤' = E_MA^M_cv𝐤,
where E_M denotes the energy of the M-th excited state and A^M_cv𝐤 is the corresponding eigenvector. Here, c and v label conduction and valence states, respectively, and 𝐤 is a crystal momentum in the first Brillouin zone.
The BSE Hamiltonian is given by <cit.>
H^BSE_cvc'v'(𝐤,𝐤') = δ_vv'δ_cc'δ_𝐤𝐤'(E_c𝐤-E_v𝐤)
- [ D^cc'_vv'(𝐤,𝐤')-X^cc'_vv'(𝐤,𝐤') ],
where E_n𝐤 denotes the energy of a quasiparticle state, with corresponding wavefunction ψ_n𝐤(𝐱) where 𝐱=(𝐫,α) comprises both a position 𝐫 and a spin variable α, and D and X are the direct and exchange integrals, respectively, given by
D^cc'_vv'(𝐤,𝐤') =
∫d𝐱∫d𝐱'ψ_v𝐤(𝐱)ψ^*_c𝐤(𝐱')W(𝐫,𝐫')ψ^*_v'𝐤'(𝐱)ψ_c'𝐤'(𝐱'),
X^cc'_vv'(𝐤,𝐤') =
∫d𝐱∫d𝐱'ψ_v𝐤(𝐱)ψ^*_c𝐤(𝐱)v(𝐫,𝐫')ψ^*_v'𝐤'(𝐱')ψ_c'𝐤'(𝐱').
Here, W(𝐫,𝐫') and v(𝐫,𝐫')=e^2/(ε_bg|𝐫-𝐫'|) denote the screened and bare Coulomb interaction, respectively, with ε_bg being the background dielectric constant and e the proton charge. For a MoS_2 layer placed on a substrate material with dielectric constant ε_sub, we use ε_bg=(ε_sub+1)/2. The screened interaction in real space is obtained using a Hankel transform according to
W(r=|𝐫-𝐫'|) = e^2∫_0^∞dq e^-qdJ_0(qr)/ε_bg + ε_2D(q) ,
where J_0(x) is the zeroth order Bessel function of the second kind and ε_2D(q) is the 2D dielectric function of MoS_2, which is calculated from first-principles DFT with the random-phase approximation <cit.>. In the above, the parameter d regularizes the divergence of the screened interaction when the electron and the hole reside on the same atom. We have found that d=1.2 Å reproduces the experimentally measured binding energy of the lowest exciton <cit.>.
To efficiently calculate the quasiparticle energies and wavefunctions of MoS_2 with a charged adsorbate, we use the tight-binding (TB) approach. Following Liu and coworkers <cit.>, we express the wavefunctions as a linear combination of Mo 4d_z^2, 4d_xy and 4d_x^2-y^2 orbitals according to
ψ_n𝐤(𝐱) = 1/√(N_k)∑_ljc_n𝐤lj∑_𝐑e^i𝐤·(𝐑 + τ_j)ϕ_l(𝐫-𝐑-τ_j,α) ,
where ϕ_l denotes an atomic basis function, 𝐑 is a lattice vector and τ_j denotes the position of the j-th atom relative to the origin of the supercell. Also, N_k denotes the number of k-points used to sample the first Brillouin zone of the supercell and c_n𝐤lj are complex coefficients obtained by diagonalizing the TB Hamiltonian.
The TB Hamiltonian of MoS_2 with a charged adsorbate is constructed by starting from the Hamiltonian of pristine MoS_2 of Liu and coworkers, which has been fitted to reproduce the ab initio DFT band structure <cit.>, and create an 18× 18 supercell. Next, the screened potential induced by the charged adsorbate is added as an onsite potential. We assume that the defect has a charge of Ze (with Z=±1) and is located at x=y=0 at a distance D above a Mo atom. The corresponding screened potential is then given by ZW(r) with d in Eq. (<ref>) replaced by D.
Inserting the TB ansatz for the quasiparticle wavefunctions into the exchange and direct integrals and exploiting the localization of the atomic basis functions yields
D^cc'_vv'(𝐤,𝐤') = ∑_ij(T^(i)_c𝐤,c'𝐤')^* W_ij(𝐤-𝐤')T^(j)_v𝐤,v'𝐤' ,
X^cc'_vv'(𝐤,𝐤') = ∑_ij(T^(i)_c𝐤,v𝐤)^* v_ij(0)T^(j)_c'𝐤',v'𝐤',
where we define T^(j)_n𝐤,n'𝐤' =∑_l c_n𝐤ljc^*_n'𝐤'lj, W_ij(𝐤)=∑_𝐑exp(-i𝐤·𝐑)W(𝐑+τ_i - τ_j) and v_ij(𝐤)=∑_𝐑exp(-i𝐤·𝐑)v(𝐑+τ_i - τ_j). We have found that the effect of exchange interactions on the absorption spectrum is small (see Fig. 4 in the Appendix) and have therefore neglected it in our calculations.
As the size of the BSE Hamiltonian increases rapidly with the supercell size, we only include those conduction states at each k-point which fulfill E_c𝐤≤ E_v𝐤^max+E_cut with E_cut being a cutoff parameter and E_v𝐤^max being the highest valence band energy at 𝐤. Similarly, we only include valence states with E_v𝐤≥ E_c𝐤^min-E_cut with E_c𝐤^min denoting the lowest conduction band energy at 𝐤. We then increase E_cut until the energies of the lowest excitons are converged. The resulting cutoffs are shown in Table <ref>. In our calculations, we use Γ point sampling (N_k=1) of the first Brillouin zone associated with the supercell.
The real part of the optical conductivity is obtained from the eigenvectors and eigenvalues of the BSE according to <cit.>
Re σ_xx(ω) =e^2/ħ m^2_0A∑_M |∑_𝐤,c,v A^M_cv𝐤𝐱̂·𝐩_cv𝐤|^2/E_Mδ(ħω-E_M),
where A=N_k A_SC (with A_SC being the area of the supercell), m_0 denotes the bare electron mass and 𝐞=𝐱̂ is the polarization direction of the electric field of the electromagnetic wave with frequency ω. The delta function is approximated by a normalized Lorentzian function with a full width at half maximum of 0.04 eV. The momentum matrix elements are given by <cit.>
𝐩_cv𝐤 = m_0/ħ∑_limjc^*_c𝐤lic_v𝐤mj∇_𝐤H^TB_limj(𝐤),
where H^TB_limj(𝐤) denotes the tight-binding Hamiltonian in the atomic orbital basis, see Appendix for details.
As we do not include a GW correction to our quasiparticle energies, it is necessary to shift the calculated optical spectrum such that the peak associated with the A exciton agrees with the experimental value of 1.93 eV <cit.>. We have used this to align the spectrum both with and without the defect.
§ RESULTS
§.§ Quasiparticle states
To understand the effect of charged adsorbates on the optical properties of MoS_2, we first discuss the quasiparticle states of this system. This discussion follows closely our previous work <cit.>. Charged defects induce localized bound states in the band gap of the MoS_2. Fig. <ref>(a-f) shows the real-space wavefunctions of the most strongly bound acceptor states induced by a negatively charged adsorbate. We have used ε_sub=3.8 corresponding to a SiO_2 substrate. The most strongly bound defect state is highly localized and has 1s symmetry (Fig. <ref>(a)). It is composed of states from the Γ valley of the MoS_2 band structure, even though the valence band maximum of the pristine material is located at the K/K' points, as shown in Fig. <ref> in the Appendix. The Γ valley has a large effective mass which gives rise to a highly localized state with a large binding energy. The next state also has 1s symmetry (Fig. <ref>(b), but is less localized. It is composed of states from the MoS_2 K and K' valleys which have a smaller effective mass than the Γ valley. The state shown in Fig. <ref>(c) has a similar shape as the state in (b). Indeed, this state originates from the lower of the spin-split valence bands at K and K'. The other states in Fig. <ref> correspond to 2s and 2p states derived from the Γ valley.
The wavefunctions of a donor defect are shown in Fig. <ref>. Again, we find that the three most strongly bound defect states are of 1s symmetry and highly localized. The most strongly bound donor states is composed of monolayer states from the Q valleys, as shown in Fig. <ref> of the appendix. Since the conduction band spin-orbit splitting is very small, the defect states from different Q valleys of the Brillouin zone can hybridize and form different linear combinations whose energy splitting is determined by the Fourier component of the defect potential whose wave vector connects the different Q valleys. In contrast, the state in Fig. <ref>(b) is a linear combination of 1s donor states from the K and K' valleys. The less strongly bound defect states, again, correspond to higher energy hydrogenic orbitals (and their linear combinations) from the Q valleys and the K/K' valleys.
§.§ Optical properties
The optical conductivity of monolayer MoS_2 on a SiO_2 substrate in the presence of an acceptor defect (Z=-1) is shown in Fig. <ref>(a) and compared to result for the pristine defect-free material. Without defects, the conductivity is characterized by two large peaks at approximately 1.93 eV and 2.05 eV, corresponding to the well-known A and B excitons from the K and K' valleys. The energy difference between the two peaks reflects the spin splitting of the highest valence bands in these valleys from spin-orbit coupling. In the presence of the defect, the A and B exciton peaks are still present in the optical spectrum, but with significantly reduced intensities. Also, the A peak is now split into two overlapping peaks. In addition, a new smaller peak arises at ∼ 1.8 eV, i.e., at an energy approximately 130 meV lower than the A exciton peak.
To understand these findings, we also plot the squared magnitudes of the projections of the BSE eigenvectors A^M_cv(𝐤=0) onto the quasiparticle states, see Fig. <ref>(a). This reveals that the new low-energy peak originates from several excitons which are predominantly composed of transitions from the most strongly bound defect state (of 1s character composed of Γ valley states) to conduction band states. Transitions from the second most strongly bound defect state (of 1s character composed of K/K' valley states) make a smaller contribution to the peak. Interestingly, the A and B peaks also contain transitions involving low-lying defect states.
For MoS_2 on SiO_2 with a donor impurity, see Fig. <ref>(b), the optical conductivity exhibits more peaks than for an acceptor impurity. In particular, both the A and the B peak of the pristine spectrum break into several smaller peaks. In contrast to the case of the acceptor impurity, we now observe two low-energy peaks: one at approximately 1.75 eV and another one at approximately 1.83 eV. The lowest peak is dominated by transitions from the valence band maximum to the two most strongly bound defects states. The peak at 1.83 eV involves transitions from the valence band maximum to the three most strongly bound defect states. The calculated energies of the low-energy peaks are similar to those reported in the experimental work of Greben and coworkers who also study MoS_2 on an SiO_2 substrate <cit.> find the first neutral exciton peak at 1.96 eV and the defect-bound exciton peak at 1.77 eV.
§ CONCLUSIONS
We have calculated the optical absorption spectrum of monolayer MoS_2 in the presence of a charged defect by solving the Bethe-Salpeter equation. We find that the presence of the defect gives rise to additional peaks in the spectrum approximately 100 - 200 meV below the A exciton peak in good agreement with experimental observations. These peaks arise from transitions involving bound defect states.
§ ACKNOWLEDGEMENTS
This work was supported through a studentship in the Centre for Doctoral Training on Theory and Simulation of Materials at Imperial College London funded by the EPSRC (EP/L015579/1). We acknowledge the Thomas Young Centre under Grant No. TYC-101.
§.§ Appendix
Exchange interactions: It is well known that the exchange term of the BSE kernel does not strongly influence the absorption spectrum of the pristine monolayer <cit.>. To test whether this is still the case in the presence of a charged defect, we have calculated the optical conductivity with and without the exchange term for a 12 × 12 supercell containing a single acceptor defect, see Projections: Fig. <ref>. It is clear that also in the presence of the charged defect, exchange interactions influence the optical spectrum only weakly.
Defect state projections: Figures <ref> and <ref> show the projections of acceptor and donor defect states onto the states of the defect-free system, respectively.
Optical matrix elements: Using the tight-binding basis convention
ψ_n𝐤(𝐱) = 1/√(N_k)∑_lic̃_n𝐤lj∑_𝐑e^i𝐤·𝐑ϕ_li^𝐑(𝐫-𝐑-τ_j,α),
Pedersen et al. <cit.> write the momentum matrix element as
𝐩_cv𝐤 = m_0/ħ∑_limjc̃^*_c𝐤lic̃_v𝐤mj∇_𝐤H̃_limj(𝐤)
+ im_0/ħ(E_c𝐤 - E_v𝐤)∑_limjc̃^*_c𝐤lic̃_v𝐤mj𝐝_limj,
where 𝐝_limj=δ_lmδ_ijτ_i denotes the intra-atomic contribution to the matrix element. In this work, we use a different tight-binding basis convention that includes an additional phase factors exp(i 𝐤·τ_j), see Eq. (<ref>). The Hamiltonians and the eigenvectors of the two different conventions are related through <cit.>
c̃_n𝐤li = c_n𝐤lie^i𝐤·τ_i
H̃_limj(𝐤) = H_limj(𝐤) e^i𝐤·(τ_i-τ_j).
Applying this transformation to the expression of the momentum matrix element, we find that
𝐩_cv𝐤 = m_0/ħ∑_limjc^*_c𝐤lic_v𝐤mj[∇_𝐤H_limj(𝐤) +i(τ_i-τ_j)H_limj(𝐤)]
+ im_0/ħ(E_c𝐤 - E_v𝐤)∑_lic^*_c𝐤lic_v𝐤liτ_i
= m_0/ħ∑_limjc^*_c𝐤lic_v𝐤mj∇_𝐤H_limj(𝐤)
+im_0/ħ∑_lic^*_c𝐤liτ_i(∑_mjH_limj(𝐤)c_v𝐤mj)
- im_0/ħ∑_mjc_v𝐤mjτ_j(∑_li c^*_c𝐤liH_limj(𝐤))
+ im_0/ħ(E_c𝐤 - E_v𝐤)∑_lic^*_c𝐤lic_v𝐤liτ_i
= m_0/ħ∑_limjc^*_c𝐤lic_v𝐤mj∇_𝐤H_limj(𝐤),
i.e. the terms involving τ_j cancel out.
|
http://arxiv.org/abs/2307.05684v1 | 20230711180007 | The Signature of the Northern Galactic Center Region in Low-Velocity UV Absorption | [
"Christian Soto",
"Trisha Ashley",
"Andrew J. Fox",
"Rongmon Bordoloi"
] | astro-ph.GA | [
"astro-ph.GA",
"astro-ph.HE"
] |
0000-0001-7840-2972]Christian Soto
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
[email protected]
0000-0002-6541-869X]Trisha Ashley
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
[email protected]
0000-0003-0724-4115]Andrew J. Fox
AURA for ESA, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA
0000-0002-3120-7173]Rongmon Bordoloi
Department of Physics, North Carolina State University, 421 Riddick Hall, Raleigh, NC 27695-8202
The Galactic Center (GC) is surrounded by plasma lobes that extend up to 14 kpc above and below the plane.
Until now, UV absorption studies of these lobes have only focused on high-velocity components (|v_ LSR|>100 ) because low- and intermediate-velocity (LIV) components (|v_ LSR|<100 ) are blended with
foreground interstellar medium. To overcome this difficulty, we present a differential experiment to compare the LIV absorption between different structures within the GC region, including the Fermi Bubbles (FBs; seen in γ-rays), the eROSITA Bubbles (eBs; seen in X-rays), and the Loop I North Polar Spur (LNPS) association, an X-ray and radio feature within the northern eB.
We use far-UV spectra from Hubble Space Telescope to measure
LIV absorption
in 61 AGN sight lines, of which 21 pass through the FBs, 53 pass through the
eBs, and 18 pass through the LNPS. We also compare our measurements to those in the literature from sight lines covering the disk-halo interface and CGM.
We find that the FBs and eBs have enhancements in measured columns of
0.22-0.29 dex in log. We also remove the contribution of a modeled disk and
CGM component from the measured columns and find that the northern eB still retains a enhancement of
0.62 dex in log. A similar enhancement is not seen in the southern eB. Since
the LNPS model-subtracted residuals show an enhancement compared to the rest of the northern eB of 0.69 dex, the northern eB enhancement may be caused by the LNPS.
§ INTRODUCTION
The Fermi Bubbles (FBs) are two bipolar plasma lobes launched from the center of the Milky Way, reaching 55 in Galactic latitude
above and below the Galactic plane. Given their close proximity, we can use them as a laboratory to study the effect of nuclear feedback on the baryonic ecosystems of galaxies in greater detail than is possible in any other galaxy. Recent evidence suggests that the FBs likely formed ≈3-6 Myr ago through an energetic outburst from Sagittarius A* <cit.>. For example, <cit.> and <cit.> find an excess in Hα, , and in the Magellanic Stream passing directly below the Galactic Center (GC). Models produced by <cit.> show that a Seyfert flare coming from Sagittarius A* 3.5±1 Myr ago is necessary to produce those elevated present-day levels of Hα in the Magellanic Stream. This timescale agrees with models that suggest the FBs formed via a jet or flare emanating from Sagittarius A* in the past 1-6 Myr <cit.>.
The timescale also agrees with the 6-9 Myr age of the UV-observed outflow modeled by <cit.>.
An alternative hypothesis for the growth of the FBs is nuclear star formation; although this would operate on a longer timescale of 30-100 Myr <cit.>.
While the FBs are defined by the gamma-ray emitting regions of the nuclear outflow, counterparts at other wavelengths have also been detected in microwave, optical, and polarized radio emission <cit.>. Recently, an X-ray counterpart has also been discovered, the eROSITA Bubbles <cit.>. The eBs are similar in shape but extend further than the FBs, reaching ±85 in latitude or 14 kpc in height. The northern eB also encompasses the multi-wavelength feature, called the Loop I and North Polar Spur (hereafter, LNPS) association <cit.>. The origin of the LNPS has long been debated. It is believed to trace either nearby supernova remnant projected in front of the GC 100-200 pc away or a structure physically associated with the GC, or an overlapping projection of both a GC feature and nearby supernova remnant <cit.>. Until recently, one strong argument against the LNPS being associated with the GC was that the LNPS had no southern Galactic counterpart. The discovery of a southern X-ray bubble (a.k.a. the southern eB) has reignited that debate <cit.>.
Discrete high-velocity clouds (HVCs) thought to be associated with the FBs have been detected in emission, CO emission, UV H_2 absorption, and UV-metal absorption towards AGN sight lines <cit.>. The velocities of these clouds (|v_ LSR|>100 ) are high enough to distinguish them from Milky Way disk gas at |v_ LSR|<100 . On the other hand, diffuse gas at low and intermediate velocities (LIV; |v_ LSR| <100 ) have largely gone unexplored because they overlap with the Milky Way interstellar medium in velocity space, and are thus contaminated by foreground absorption.
This contamination issue can be overcome with a differential experiment in which absorption in FB sight lines is compared to absorption in nearby sight lines outside the bubbles. Since the plasma inside the FBs has a high ionization level, sight lines passing inside the bubbles are expected to show stronger high-ion absorption than sight lines passing outside. High ions are therefore an ideal tracer for this experiment. For example, <cit.> study a pair foreground-background stellar sightlines, which show enhancement of LIV and absorption towards the southern FB, but they do not explore its significance. This effect was also seen by <cit.> and <cit.>, who measured the amount of LIV absorption towards 58 and 132 AGN sight lines, respectively, through the disk-halo interface and CGM, probing gas at temperatures of 10^4-10^5K <cit.>. and find an excess of 0.1–0.26 dex in LIV column densities towards the GC. suggest that this enhancement could be due to the FBs, but they do not explore it further.
In this paper, we present a differential survey of LIV absorption in sight lines that pass through various structures in the GC region, including the FBs, eBs, and LNPS, using UV spectra from the Hubble Space Telescope Cosmic Origins Spectrograph (HST/COS) and Space Telescope Imaging Spectrograph (HST/STIS). We chose to use doublet for this study because among the high-ion doublets in the COS bandpass, is often undetected and is often saturated, while tends to yield a measurable column density <cit.>. Additionally, using allows us to make direct comparisons of GC pointings to the extensive sample in <cit.>, , and probing the rest of the disk-halo interface and CGM. In Section <ref> we present our data-set and discuss our observations and data reduction. In Section <ref> we present the results of our survey and discuss how our results compare to those in the literature. Our conclusions are then presented in Section <ref>.
§ OBSERVATIONS AND DATA REDUCTION
Our sample consists of 61 AGN sight lines with HST/COS and HST/STIS data drawn from the samples of <cit.>, <cit.>, <cit.>, and <cit.>; 21 pass through the FBs, 53 pass through the eBs, 18 pass through the LNPS association, and 8 pass through none of these GC features.
For the sight lines passing through the FBs, the set also included targets that were listed in the literature as being on the “edge" of the bubble. Seven of the sight lines in our sample are close to the Magellanic System or have HVCs associated with the Magellanic Stream identified in their spectrum <cit.>. As discussed in Section <ref>, the Magellanic Stream does not contribute an excess of LIV absorption to our measurements. Figure <ref> shows the location of the sight lines relative to the FBs, eBs, and LNPS. In Table <ref>, we present basic information on each sight line.
Each sight line has a COS FUV G130M spectrum and one sight line (NGC5548) has STIS E140M spectrum. The COS data were initially calibrated with the calibration pipeline. We then applied customized velocity alignment and co-addition steps described in <cit.>, which align the Galactic absorption with emission.
We measure the apparent optical depths (AODs) and the apparent column densities (ACDs) of the LIV absorption using a custom Python package called [https://github.com/cmagness/spectrAOD] <cit.>. AODs are used to analyze interstellar absorption lines; they represent the true optical depth with the addition of instrumental blurring <cit.>. To make these measurements, first normalizes the spectrum using a straight line between two sections of user-determined continuum. Continuum ranges were chosen for each ion visually, avoiding absorption features such as redshifted intergalactic absorbers and HVCs identified in previous work <cit.>. The velocity-dependent AOD, τ_a(v), is calculated as:
τ_a(v)= -ln[F_c(v)/F(v)],
where F(v) is the flux of the absorption and F_c(v) is the continuum flux. τ_a(v) is then used to find the apparent column density (N_a):
N_a = ∫_v_-^v_+ 3.768×10^14 τ_a(v)fλ dv cm^-2,
where v_- and v_+ are the LSR velocity limits of the absorption, f is oscillator strength of the transition of interest <cit.>, and λ is the transition wavelength in Angstroms. For each sight line, we measure the AOD and ACD between -100 and 100 for λλ1393 and 1402. Figure <ref> shows an example of 's graphical output, including the sight lines' spectra, continuum fits, and ACD measurements.
determines the noise using the standard deviation of the flux around the continuum. If the absorption reaches a minimum flux less than the noise anywhere within the line profile, then it is labeled as saturated. If the minimum flux is greater than the noise and the line is detected at 3σ significance (i.e., the equivalent width is at least three times the equivalent width error), then the absorption is labeled as detected. In all other cases, the absorption is labeled as not detected.
For lines that are unblended and unsaturated, the measured ACD from each member of the doublet should match within their respective measurement errors. However, different measurements may arise due to contamination from intergalactic absorption, continuum-fit errors, and/or unresolved saturation. The column-density measurements from each member of the doublet were considered to match if the following condition was met:
log(N_a2)- log(N_a1)√(σ_2^2+σ_1^2)< 3
where σ is the error on each logarithmic ACD measurement. If the value was >3, then we inspected the spectrum to determine which doublet line was more reliable. For example, if an intergalactic absorber was found in the stronger line, then the weaker lines measurement was considered more reliable and was used.
In cases where both members of the doublet were detected and unsaturated, and the requirement of Equation <ref> was met, we used the average of the two ACD measurements in our analysis. If both lines were saturated and visual inspection of the data did not reveal any issues with either line, then the weaker line was used to derive a lower limit on the ACD. If the stronger line was saturated and the weaker line was detected without apparent saturation, then we adopt the measurement from the weaker line. In this case, the weaker line was labeled as detected and unsaturated if the Equation <ref> requirement was met, otherwise it was labeled as saturated to account for unresolved saturation.
§ RESULTS AND DISCUSSION
Of the 61 sight lines measured in our study, one sight line (SDSS J141038.40+230447.0) was unusable due to intergalactic absorption blending in both lines. Of the remaining 60 sight lines, the detection rate was 100%. For sight lines passing though the FBs, eBs, and LNPS 45%, 39%, and 60% respectively, are labeled as saturated.
§.§ Column-density dependence on Latitude and Longitude
, , and measured the LIV absorption through the MW disk and CGM
with 31, 58, and 132 extragalactic sight lines covering the entire sky. and found that their LIV column density measurements increase slightly by 0.1–0.26 dex in directions towards the GC regions, and suggest this enhancement could be due to the FBs.
To understand how the measurements in our GC sample compare to in the rest of the disk-halo interface and CGM, we combine the , , and results with our own in Figure <ref> where we plot log(N_a sin|b|) vs. Galactic latitude and longitude. log(N_a sin|b|) measures the z-axis-projected column densities, where the z-axis perpendicular to the Galactic plane, and is designed to correct for disk projection effects. In a simple exponential disk model, log(N_a sin|b|) is a constant
and thus becomes independent of Galactic latitude <cit.>.
Of the 31, 58, and 132 sight lines from the , , and samples, 3, 3, and 23 sight lines overlap with our sample, respectively.
We compare the log(N_a) measurements for the 23 overlapping sight lines and find they agree within the errors. We note that and use a stricter 1σ criteria for defining matching doublet column density measurements,
whereas our threshold is 3σ. We also note that integrates over a variable velocity range that is determined by the visual identification of the thick disk component.
In the top panel of Figure <ref> we plot log(N_a sin|b|) vs. Galactic longitude for our sample and all three comparison samples. In the figure, the log(N_a sin|b|) values in the GC region stand out clearly against the rest of the Galactic halo, with GC sight lines having log(N_a sin|b|)≳13.8.
To see this effect in Galactic latitude, we plot log(N_a sin|b|) vs. b. Figure <ref> shows higher column densities at positive latitudes, demonstrating an asymmetry between the northern and southern hemispheres. The northern enhancement in log(N_a sin|b|) appears to occur within both the FB and eB latitude boundaries.
In the bottom panel of Figure <ref>, we plot the column density predictions of three models of the disk-halo interface. First, we plot the model which assumes that the disk column density decreases exponentially with a fixed scale height (plane-parallel slab model). Second, we plot the two-component model which adds a latitude-dependent CGM component to the plane-parallel slab model. Third we plot the <cit.> model which accounts for a two-dimensional radial distribution of the disk-halo interface with a latitude-dependent CGM. None of these models appear to account for the high columns in the northern sky. <cit.> and <cit.> also notice a north-south asymmetry in columns and attempt to account for it by artificially increasing the northern sight lines absorption, increasing their model limits from log(N_a sin|b|)≲13.55 to ≲13.75. Even with this increase in the modeled northern log(N_a sin|b|) values, Figure <ref> indicates that the GC northern enhancement in measured columns cannot be explained from the shape of the disk-halo interface.
To understand exactly where these enhancements arise, we plot the measured log(N_a) values of all samples on a map of the full sky X-ray and gamma-ray emission from <cit.> in Figure <ref>. This figure shows both the FBs and eBs. In Figure <ref>, the eBs and FBs visually appear to have much higher than the rest of the sky.
We discuss the statistics of potential enhancements in absorption below in detail in Section <ref>.
rclcccc|ccc[!ht]
Statistical analysis of N_a() values combined from this work, , , and
2c Basic Statistics 3c Statistical Test f
2cSpatially-Selected Count Count Standard
2cComparison Groups a Unsaturated b Saturated c log(N_a) d Error e χ^2 p-value Testg
6*Fermi { 2*1. Inside FBs (R 1+2) 12 10 13.93 0.03 2*20.1 2*7×10^-6 2*P
Outside FBs (R 3+4+5+6+7) 137 28 13.71 0.03
2-10
2*2. Inside FB: North (R 1) 6 6 13.94 0.04 2*8.5 2*0.004 2*P
Outside FB: North (R 3+5+7) 66 22 13.79 0.04
2-10
2*3. Inside FB: South (R 2) 6 4 13.89 0.04 2*14 2*2×10^-4 2*L
Outside FB: South (R 4+6) 71 6 13.60 0.03
2-10
6*eROSITA { 2*4. Inside eBs (R 1+2+3+4+7) 39 25 13.91 0.03 2*39 2*4×10^-10 2*L
Outside eBs (R 5+6) 110 13 13.62 0.03
2-10
2*5. Inside eB: North (R 1+3+7) 19 19 13.96 0.04 2*32.9 2*1×10^-8 2*P
Outside eB: North (R 5) 53 9 13.66 0.04
2-10
2*6. Inside eB: South (R 2+4) 20 6 13.78 0.06 2*8.6 2*0.003 2*L
Outside eB: South (R 6) 57 4 13.56 0.02
2-10
2*7. Inside eBs (R 3+4+7) 27 15 13.86 0.05 2*5.7 2*0.02 2*P
Inside FBs (R 1+2) 12 10 13.93 0.03
2-10
2*LNPS { 2*8. Inside LNPS (R 7) 8 12 14.01 0.05 2*1.7 2*0.2 2*L
Inside eB: North (R 1+3) 11 7 13.87 0.05
aSpatially-selected groups are compared two at a time with the comparison set bracketed by horizontal lines. The region (R) covered by these different group is denoted by a number shown in Figure <ref>.
bThe number of unsaturated sight lines.
cThe number of saturated sight lines.
dThe log of the restricted mean apparent column density. The restricted mean is calculated using an upper limit set to the maximum column density in each spatially selected group.
eThe standard error on the log restricted mean.
fComparison sets with statistics in bold are likely drawn from separate populations.
gSurvival test used in the function: P = Peto-Peto test and L = Log-rank test.
rclcccc|ccc[!ht]
Statistical analysis of disk-model-subtracted N_a() values combined from this work, , , and
2c Basic Statistics 3c Statistical Test f
2cSpatially-Selected Count Count Standard
2cComparison Groups a Unsaturated b Saturated c log(N_a, R) d Error e χ^2 p-value Testg
6*Fermi { 2*1. Inside FBs (R 1+2) 12 10 13.18 0.22 2*0.6 2*0.4 2*P
Outside FBs (R 3+4+5+6+7) 137 28 13.10 0.09
2-10
2*2. Inside FB: North (R 1) 6 6 13.16 0.37 2*0.1 2*0.7 2*P
Outside FB: North (R 3+5+7) 66 22 13.24 0.10
2-10
2*3. Inside FB: South (R 2) 6 4 13.18 0.22 2*1.2 2*0.3 2*L
Outside FB: South (R 4+6) 71 6 12.76 0.13
2-10
6*eROSITA { 2*4. Inside eBs (R 1+2+3+4+7) 39 25 13.42 0.10 2*2.9 2*0.09 2*P
Outside eBs (R 5+6) 110 13 12.91 0.11
2-10
2*5. Inside eB: North (R 1+3+7) 19 19 13.56 0.11 2*10.5 2*0.001 2*L
Outside eB: North (R 5) 53 9 12.93 0.17
2-10
2*6. Inside eB: South (R 2+4) 20 6 12.79 0.29 2*0.0 2*1 2*L
Outside eB: South (R 6) 57 4 12.83 0.12
2-10
2*7. Inside eBs (R 3+4+7) 27 15 13.38 0.12 2*0.2 2*0.6 2*L
Inside FBs (R 1+2) 12 10 13.18 0.22
2-10
2*LNPS { 2*8. Inside LNPS (R 7) 8 12 13.67 0.10 2*5.1 2*0.02 2*P
Inside eB: North (R 1+3) 11 7 12.98 0.39
aSee Table <ref> for table notes. Here the column densities have been disk-subtracted using the model of <cit.>.
§.§ Statistical Analysis of Spatially-Selected Regions
To quantitatively assess the enhancement in absorption towards the GC, we merge our sample with the , , and samples and then divide it into eight spatially-selected comparison sets, as shown in Tables <ref> and <ref> and Figure <ref>. For the overlapping sight lines between our sample and comparison samples, we use our sample's measurements. We then choose which of the overlapping sight lines in the comparison samples to use based on the following priority: (1) , (2) , and (3) . The only exception to this are the sight lines 3C273 and PKS0405-123, which appear in the as having unreliable column measurements due to a λ1393 measurement that is abnormally stronger than the λ1402 measurement. For those sight lines, we use the value, who identified a reliable measurement when Lyman α interference and specific velocity ranges were taken into account.
For our statistical analysis, we analyze two sets of combined data:
* The column density measurements: analysis of the observed columns (see Figure <ref>) allows us to directly look for GC features that stand out compared to other measurements in a model-independent manner,
without making assumptions about the shape of the disk. While statistically comparing the GC data to the entirety of the rest of disk-halo interface and CGM does not account for Galactic structure, our goal in this section is simply to measure the significance of the GC enhancement against the rest of the sky. This global approach is necessary because splitting the sky into different longitude or latitude bands for statistical tests provides samples that are too small to obtain statistically significant results.
* The disk-CGM model-subtracted residuals: we analyze the residuals formed by subtracting the <cit.> two-dimensional disk-CGM model from the column measurements (see the top panel of Figure <ref>). The <cit.> model contains a radial- and height-dependent disk component and a constant CGM component, with column densities determined by minimized χ^2 models using AGN and stellar sight line measurements. For each of our sight lines we determine the model column at that sky position and subtract it from the observed column to obtain a residual. This model-dependent analysis allows us to remove effects from the Galactic disk component on the column densities. These effects are particularly strong at low latitudes where sight lines pass through more of the ISM.
Throughout the remainder of the paper, we will refer to these two datasets as the column density measurements and model-subtracted residuals, respectively.
Since a significant portion of the sight lines towards the GC are lower limits, we use survival analysis to account for censored data.
Specifically, we use the package function, in the language r to calculate each spatial group's restricted mean and its associated standard error <cit.>. uses the Kaplan-Meier method to create a survival curve for the data and then estimates the mean value using a range of acceptable columns <cit.>. For this analysis, we calculated the restricted mean by setting the maximum allowed column density in each spatially-selected group to the maximum column density of that specific spatial group.
Additionally, we use function to determine if the two groups being compared are likely drawn from separate populations using either the log-rank or Peto-Peto test. The log-rank test gives equal weighting to all column density measurements and assumes proportional hazards <cit.>. The Peto-Peto test gives a larger weight to low column densities and is a more appropriate test for when the assumption proportional hazards is violated <cit.>. We decide which test to use by running the function for each set of spatially selected comparison groups, a function that tests whether proportional hazards can be assumed <cit.>. In the results of both the log-rank or Peto-Peto tests, the χ^2 value is an indication of how different the two survival curves are from what would be expected if they were drawn from the same population. The results also provide p-values which need to be <0.05 (95% confidence) in order to reject the null hypothesis that the two spatially selected groups are drawn from the same population. For examples of the survival curves used in this analysis, see Appendix <ref>.
All statistical tests were performed using the linear (not logarithmic) form of N_a. Survival analysis requires the values being evaluated to be positive, but the model-subtracted analysis results in both positive and negative residuals. As a solution, we found the minimum column residual in each pair of spatially compared groups and added the minimum residual to all residuals in both samples prior to running the statistical tests. This process ensures the survival analysis is conducted only on positive numbers and that the two compared groups are shifted by the same amount. After the tests, we then shift all restricted mean values by subtracting their respective minimum residual, accounting for our previous offset. The results of these tests are shown in Tables <ref> and <ref>, and spatially selected groups that are likely drawn from separate populations are highlighted in bold.
§.§.§ Sight Lines Inside vs. Outside the FBs
Survival analysis tests indicate different populations for measured columns in all spatially-compared groups associated with the FBs (p≤ 0.004). The sight lines passing though the FBs have a log(N_a) value 0.22±0.04 dex higher than the sight lines passing outside (Table <ref>; Figure <ref>: regions 1+2 vs. 3+4+5+6+7). This holds true in the northern Galactic hemisphere with a 0.15±0.06 dex difference (Figure <ref>: regions 1 vs. 3+5+7) and the southern Galactic hemisphere with a 0.39±0.05 dex difference (Figure <ref>: regions 2 vs. 4+6; see Table <ref>).
The survival analysis tests for the model-subtracted residual absorption do not show a difference in the populations for the Fermi Bubble (FB) spatially-selected comparison groups. Therefore, the location of the FB sight lines relative to the disk may be playing a large role in the excess of columns in FB sight lines.
§.§.§ Sight Lines Passing Though vs. Outside the eBs
The survival tests for the measured columns indicate a difference in the population for all sight lines inside vs. outside the eBs (Figure <ref>: regions 1+2+3+4+7 vs. 5+6; p= 4×10^-10) and eB sight lines vs. non-eB sight lines in the southern Galactic hemisphere (regions 2+4 vs. 6; p=0.003). On the other hand, their model-subtracted residual absorption do not show a difference in the populations of all and southern eB sight lines to non-eB sight lines. This could indicate that the disk is contributing to the differences in the populations.
For sight lines passing through the northern eB (Figure <ref>: regions 1+3+7) both the measured N_a and residual N_a, R values have significantly different populations than northern directions outside the eB (Figure <ref>: region 5), with p-values of 1×10^-8 and 1×10^-3 and differences of 0.30±0.06 and 0.63±0.20 dex, respectively.
Between the measured and model-subtracted column density analysis, only the statistical tests for inside vs. outside the northern eROSITA Bubble show a difference in population in both Tables <ref> and <ref>. This result indicates that there is strong absorption enhancement in the
northern eB.
The northern eB also has a model-subtracted log(N_a, R) value 0.77±0.39 dex higher than the southern eB (0.18± 0.07 dex higher in the measured log(N_a); see Tables <ref> and <ref>). We ran additional statistical tests for the eB north vs. south and found that the measured and model subtracted columns are likely drawn from different populations (χ^2=11.1 and 6.1, p = 0.0009 and 0.01, respectively),
further highlighting the asymmetry in the northern and southern GC features seen in Figure <ref>.
§.§.§ Sight Lines Passing Though the FBs vs. eBs
We continue to explore relationships of these different spatial regions by looking at sight lines that pass through the FBs (Figure <ref>: regions 1+2) and sight lines passing through the eBs, but not the FBs (Figure <ref>: regions 3+4+7). Only the measured
columns indicate that the populations are different (p=0.02).
However, this difference may depend on the location of the sight lines with respect to the disk as we do not see a population difference in the model-subtracted survival test.
§.§.§ LNPS and Northern eB Sight Lines
Strong absorption is detected along the sight lines associated with the northern eB and LNPS association (Figure <ref>). To explore the specific role of the LNPS in the northern eB's LIV enhancement, we separate out the sight lines passing through emission associated with LNPS.
In Figure <ref> we have plotted all of the sight lines selected as passing through the LNPS association in opaque symbols. These sight lines were selected based on the 408 MHz continuum emission <cit.> and the Spektr-RG–eROSITA all-sky survey <cit.>. We select the regions encompassed by LNPS based on definitions in the literature, such as <cit.>, <cit.>, <cit.>, and references therein. As such, we select this sample based on LNPS's multi-wavelength appearance so that we include all sight lines that may pass through the LNPS association. We chose not to include the faint southern structures that are at times associated with Loop I (Loop Is and the southern extension of Loop XII) because their connection with the northern LNPS association is not clear; it is possible that this structure is related to the southern eROSITA bubble or separate supernovae remnants <cit.>.
We compare the LNPS to the rest of the northern eB and do not find that the measured-column populations are different from one another (Figure <ref>: regions 7 vs. 1+3). However, the survival test for the model-subtracted analysis does show a difference in the populations with a p-value of 0.02 and χ^2=5.1. The LNPS has a model-subtracted log(N_a, R) enhancement of 0.69±0.40 dex compared to the rest of the northern eB (see Table <ref>).
§.§ Enhancements towards the GC
To better understand these enhancements in , we have plotted sight lines in the model-subtracted analysis with residuals greater than 1σ and 2σ away from the mean in the middle and bottom panels of Figure <ref>, respectively. For this simple visual, we calculate the mean and standard deviation assuming that all values (including saturated values) are measurements, giving us a mean of 6.0×10^12 cm^-2 and a standard deviation of 1.9×10^13 cm^-2. From these plots we see several 1σ enhancements (positive residuals) in sight lines near latitudes of -30 and the strongest outliers detected near longitudes of 0 with both enhancements and deficits (negative residuals) in sight lines. The 1σ enhancements near the -30 latitude line visually appear parallel to the disk and may reflect the inherent patchiness of the CGM and/or an underestimation of the disk contribution in the model at this latitude. The enhancements and deficits in sight lines close to the 0 longitude line are so strong that they persist in the 2σ outliers plot on the bottom of panel in Figure <ref>.
These 2σ enhancements and deficits near 0 longitude could be an indication that the model does not reproduce accurate columns at low longitudes.
To check how accurately the model reproduces the column densities at low longitudes, we use the measurement from the sight line toward HD 167402, a star at the distance of 7.0±1.7 kpc with l, b = 2.26,-6.39 that is towards the GC, but does not pass through GC features <cit.>. This sight line has a measured LIV log N() of 13.65 while the model predicts a log column of 13.90, a difference of 0.25 dex (3.5× 10^13 cm^-2). This indicates that the model appears to be overestimating the columns at low longitudes. If so, then the enhancements in the northern sight lines
that are close to 0 longitude may be even higher than indicated in Figure <ref>. This would strengthen the conclusion from the statistical tests that there is a strong enhancement towards the northern eB and LNPS.
If this excess absorption were associated with the Fermi or eBs alone, then we would expect these sight lines to be spread throughout both the northern and southern GC pointings. While a couple of enhanced southern GC sight lines are in the 2σ outlier plot, these two sight lines also fall close to the -30 latitude line, which shows a general enhancement,
possibly contributing to the enhancement in these two sight lines' residuals.
Figure <ref>, combined with the statistical results in Tables <ref> and <ref>, indicate that the northern eB has an enhancement of . Since the LNPS occurs in the northern Galactic hemisphere but not the southern Galactic hemisphere, and most of the excess absorption is contained in the north, the excess absorption may be associated with the LNPS.
Additionally, an enhancement towards the LNPS in FUV emission was found by <cit.>, supporting an enhancement in high ions towards LNPS. The enhancement in the LIV absorption suggests that the eB and LNPS can be detected and characterized through excess absorption.
§ CONCLUSIONS
We have used archival HST/COS spectra to measure the apparent column densities of LIV absorption along 61 AGN sight lines in the GC region (within ±30 in longitude of the GC). We have used these measurements to look for low-velocity signatures of high-ion absorption in three structures in the Galactic Center region: the FBs, eBs, and LNPS association. Using statistical tests that account for censored data,
we look for signatures of these features in the measured column densities
and in the residuals after the disk-CGM components are subtracted off,
using the models of <cit.>. Our results are as follows:
* Fermi Bubbles: We find larger measured LIV column densities
in sight lines passing through the FBs than in sight lines passing outside,
with a mean enhancement of 0.22 dex. However, when we subtract the disk-CGM components, a survival analysis test finds that the columns inside and outside the FBs are not drawn from different populations. This may indicate that the disk (foreground) absorption is creating the differential between the two regions.
* eROSITA Bubbles: Survival tests between directions through the northern eB and the rest of the northern hemisphere reveal a significant difference in the distribution of LIV column densities. This is true for both the measured and model-subtracted columns. The northern eB has columns 0.30 and 0.63 dex higher than the rest of the northern hemisphere in measured and model-subtracted columns, respectively.
* Northern enhancement: The northern eB has a significantly higher mean column than the southern eB for both the measured and model-subtracted analysis (0.18 and 0.77 dex higher, respectively), revealing a strong asymmetry between the northern and southern bubbles.
This can be seen in plots of the residuals (observed – model column densities), which show
that enhancements are clustered in the northern eB (Figure <ref>).
* Loop I North Polar Spur (LNPS):
If the enhancement were primarily related to the eBs, then it would be expected in both hemispheres. The lack of a southern counterpart to this enhancement suggests that the LNPS (which lies in the north) may be the underlying source of the excess LIV absorption. Statistical tests indicate an enhancement of 0.69 dex in the LNPS when compared to the rest of the northern eB. The LNPS enhancement is also supported by C4 emission measurements by <cit.>.
The results in this paper indicate a new method for detecting the northern eB and possibly the LNPS. The asymmetries detected in the absorption reflect those seen in the X-ray, gamma-ray, and radio emission of the GC features. The source of the GC north-south brightness asymmetry and east-west asymmetry caused by the LNPS, is still a subject of debate. Some suggested sources include: variations in halo gas densities <cit.>, overlapping emission from a GC plasma bubble and a nearby supernova remnant <cit.>, the outflow encountering the 3 kpc molecular ring in the GC <cit.>, and CGM winds impacting the outflow <cit.>. The measurements presented in this paper provide useful constraints for future models that attempt to understand the origins of the asymmetries in GC features.
§ ACKNOWLEDGEMENTS
We would like to thank the anonymous referee for their helpful comments. We would like to thank Zhijie Qu for providing us the disk-halo model from <cit.>. We gratefully acknowledge support from the NASA Astrophysics Data Analysis Program (ADAP) under grant 80NSSC20K0435, 3D Structure of the ISM toward the Galactic Center and from the STScI
Director's Discretionary Fund.
The HST COS data presented in this paper
were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed for the sample in this paper can be accessed via MAST: [10.17909/jeph-7j87]10.17909/jeph-7j87.
aasjournal
§ SAMPLE LIV MEASUREMENTS
We list basic information on the GC sample use for this study in Table <ref>. We also list the log(N_a) for the LIV λλ 1393, 1402 , the adopted log(N_a), and the N_a,R measurements.
lccccccccc
LIV Measurements in COS AGN sight lines in the Galactic Center Region
Sight Line l () b () Location log N_a,1393a log N_a,1402b log N_a, adop.c N_a,Rd PIDe Ref.f
RXJ1342.7+1844 0.24 75.52 eB, LNPS ≥13.70±0.03 ≥13.88±0.04 ≥13.88 ≥3.1×10^13 12248 2
HE2332-3556 0.59 -71.59 eB 13.31±0.06 13.44±0.09 13.38±0.12 -1.2×10^13 13444 3
RBS2023 0.61 -71.62 eB 13.40±0.04 NDg 13.40±0.04 -1.1×10^13 13444 3
SDSSJ151237.15+012846.0 1.80 47.50 FB, eB ≥13.98±0.03 ≥14.07±0.05 ≥14.07 ≥5.3×10^13 12603 2
MRK1392 2.80 50.30 FB, eB, LNPS ≥13.68±0.01 13.77±0.02 ≥13.77 ≥-2.2×10^12 13448 2
SDSSJ135712.60+170444.0 2.90 71.80 eB, LNPS ≤13.80±0.01h 13.79±0.02 13.79±0.02 1.6×10^13 12248 2
1H1613-097 3.50 28.50 FB, eB ≥13.76±0.02 13.84±0.03 13.84±0.03 -3.4×10^13 13448 2
PG1352+183 4.40 72.90 eB, LNPS 13.68±0.01 13.69±0.02 13.69±0.02 2.4×10^12 13448 2
RBS1768 4.51 -48.46 FB, eB 13.59±0.02 13.58±0.03 13.59±0.04 -1.5×10^12 12936 3
CTS487 5.54 -69.44 eB 13.36±0.03 13.32±0.07 13.34±0.07 -1.4×10^13 13448 3
RBS1454 5.60 52.90 FB, eB, LNPS ...h ≥13.95±0.03 ≥13.95 ≥3.1×10^13 12603 2
SDSSJ150928.30+070235.0 7.80 51.60 FB, eB, LNPS ≥13.85±0.02i ≥13.79±0.05 ≥13.79 ≥2.4×10^12 12603 2
UVQSJ191928.05-295808.0 8.18 -18.77 FB, eB ...h 13.97±0.01 13.97±0.01 4.3×10^11 15339 4
SDSSJ141542.90+163413.7 8.80 67.80 eB, LNPS ≥13.85±0.01 13.90±0.02 13.90±0.02 3.2×10^13 12486 2
SDSSJ151507.43+065708.3 9.00 50.40 FB, eB, LNPS ≥13.93±0.03 ≥13.99±0.04 ≥13.99 ≥3.8×10^13 12603 2
PDS456 10.40 11.20 FB, eB ≥14.05±0.01 14.02±0.02 14.02±0.02 -7.6×10^13 13448 1,2
MRK841 11.20 54.60 eB, LNPS ≥13.91±0.01 14.05±0.01 ≥14.05 ≥5.7×10^13 13448 2
ESO462-G09 11.33 -31.95 FB, eB ≥13.62±0.03 13.61±0.07 13.61±0.07 -1.6×10^13 13448 3
RBS2070 12.84 -78.04 eB 13.33±0.04 13.40±0.07 13.37±0.08 -1.3×10^13 12864 3
SDSSJ150952.19+111047.0 13.60 53.80 eB, LNPS ≥13.79±0.02 ≥14.18±0.02 ≥14.18 ≥9.6×10^13 12614 2
PG1522+101 14.90 50.10 FB, eB, LNPS ...h 13.70±0.02 13.70±0.02 -8.0×10^12 11741 2
PKS2155-304 17.73 -52.25 eB 13.36±0.02 13.35±0.03 13.36±0.04 -1.4×10^13 8125/12038 3
SDSSJ154553.48+093620.5 18.30 45.40 FB, eB, LNPS ≥13.98±0.02 ...h ≥13.98 ≥3.3×10^13 12248 2
UVQSJ192636.95-182553.0 20.00 -15.86 eB ≥14.14±0.01 ≥14.14±0.01 ≥14.14 ≥4.5×10^13 15339 4
RXJ1303.7+2633 21.80 87.20 - ≥13.90±0.03 ...h ≥13.90 ≥3.7×10^13 13382 2
PG1553+113 21.90 44.00 FB, eB, LNPS ...h 13.91±0.01 13.91±0.01 1.9×10^13 11520/12025 2
SDSSJ141038.39+230447.1 24.60 71.60 eB ...h ...h ... ... 12958 2
SDSSJ135424.90+243006.3 25.90 75.60 - ≥13.68±0.04 ...h ≥13.68 ≥3.5×10^12 12603 2
SDSSJ134822.31+245650.1 26.40 77.00 - ≥14.00±0.03 ≥14.09±0.05 ≥14.09 ≥7.9×10^13 12603 2
RXJ1605.3+1448 27.80 43.40 eB, LNPS ≥13.82±0.01 ...h ≥13.82 ≥6.8×10^12 12614 2
SDSSJ131802.01+262830.3 28.20 84.00 - ...h ≥13.53±0.15 ≥13.53 ≥-9.0×10^12 12603 2
RXJ1356.4+2515 29.30 75.30 - ≥13.83±0.02 ≥13.91±0.02 ≥13.91 ≥3.7×10^13 12248 2
PG1424+240 29.50 68.20 eB, LNPS ...h 13.68±0.03 13.68±0.03 2.3×10^12 12612 2
NGC5548 32.00 70.50 - ≥13.77±0.01 13.83±0.01 ≥13.83 ≥2.3×10^13 7572 2
MRK877 32.90 41.10 eB, LNPS 13.84±0.01 13.87±0.02 13.86±0.02 1.2×10^13 12569 2
3C323.1 33.90 49.50 eB, LNPS ≥13.83±0.01 13.85±0.03 13.85±0.03 1.9×10^13 13398 2
QSO1503-4152 327.70 14.60 eB ≥13.91±0.01 ≥14.01±0.03 ≥14.01 ≥-1.1×10^13 11659 2
HE1340-0038 328.80 59.40 eB 13.46±0.05 13.69±0.06 13.59±0.08 -9.3×10^12 11598/13033 2
SDSSJ131545.21+152556.3 329.90 77.00 eB, LNPS ...h ≥13.66±0.08 ≥13.66 ≥1.7×10^12 12603 2
HE2259-5524 330.64 -55.72 eB 13.51±0.03i 13.42±0.07i 13.47±0.07 -5.8×10^12 13444 3
HE2258-5524 330.72 -55.67 eB 13.36±0.03i 13.41±0.05i 13.39±0.06 -1.1×10^13 13444 3
IRASF21325-6237 331.14 -45.52 eB 13.64±0.01 13.67±0.02 13.66±0.02 7.1×10^12 12936 3
RXJ1342.1+0505 333.90 64.90 eB ≥13.58±0.04 13.62±0.07 13.62±0.07 -5.4×10^12 12248 2
SDSSJ125846.66+242739.1 335.10 86.90 - ≥13.65±0.03 ...h ≥13.65 ≥2.2×10^12 13382 2
ESO141-G55 338.18 -26.71 FB, eB 13.79±0.01 13.78±0.01 13.79±0.01 1.5×10^12 12936 3
RBS1897 338.51 -56.63 eB ≥13.30±0.02 13.42±0.03 ≥13.42 ≥-9.4×10^12 11686 3
SDSSJ135726.27+043541.4 340.80 62.50 eB ...h ≥13.97±0.02 ≥13.97 ≥4.4×10^13 12264 2
UVQSJ185649.37-544229.9 341.66 -22.60 FB, eB ≥13.88±0.02 ≥13.89±0.04 ≥13.89 ≥5.5×10^12 15339 4
SDSSJ140655.66+015712.8 341.80 59.00 eB ...h ≥13.64±0.10 ≥13.64 ≥-7.2×10^12 12603 2
PG1435-067 344.00 47.20 FB, eB 13.57±0.02 13.61±0.04 13.59±0.05 -2.3×10^13 12569/13448 2
RBS1892 345.90 -58.37 eB 13.48±0.02i 13.53±0.04i 13.51±0.05 -4.0×10^12 12604 3
SDSSJ142614.79+004159.4 347.60 55.10 eB ≥13.96±0.03 ≥13.97±0.04 ≥13.97 ≥3.9×10^13 13473 2
RBS2000 350.20 -67.58 eB 13.55±0.02 13.61±0.04 13.58±0.05 2.4×10^12 13448 3
PKS2005-489 350.37 -32.60 FB, eB ≥13.97±0.01 14.02±0.01 ≥14.02 ≥4.9×10^13 11520 3
RXJ1429.6+0321 351.80 56.60 eB ≥13.77±0.04 ≥13.84±0.06 ≥13.84 ≥1.5×10^13 12603 2
SDSSJ141949.39+060654.0 351.90 60.30 eB ≥13.78±0.03 ...h ≥13.78 ≥8.7×10^12 13473 2
UVQSJ185302.65-415839.6 354.36 -18.04 FB, eB ≥14.03±0.01 13.92±0.02 ≥13.92 ≥-1.7×10^13 15339 4
RBS1795 355.18 -50.86 FB, eB ≥13.54±0.01 13.70±0.02 ≥13.70 ≥1.1×10^13 11541 3
UVQSJ193819.59-432646.3 355.47 -26.41 FB, eB ≥13.86±0.03h 13.87±0.02h 13.87±0.02 3.8×10^12 15339 4
HS1302+2510 357.40 86.30 - ≥13.74±0.02 13.69±0.06 13.69±0.06 6.4×10^12 13382 2
RBS1666 335.73 -31.00 FB, eB ≥13.89±0.01 13.89±0.02 13.89±0.02 2.5×10^13 13448 3
aThe log of the apparent column density (N_a in cm^-2) measured from λ1393 absorption in the range -100<v_ LSR<100 .
bThe log of the apparent column density (N_a in cm^-2) measured from λ1402 absorption in the range -100<v_ LSR<100 .
cThe adopted log apparent column density used for statistical analysis. See Section <ref> for a discussion on how log N_a, adop. was calculated.
cThe residual apparent column density used for statistical analysis. See Section <ref> for a discussion on how N_a,R was calculated.
eHubble Space Telescope Proposal ID number.
fReferences: (1) <cit.> (2) <cit.> (3) <cit.> (4) <cit.>.
gND = not detected.
hThese measurements are affected by interfering emission (e.g. Lyman-α or intergalactic absorption) and were therefore set as an upper limit. If the line was saturated and affected by interfering absorption, then the line could not be measured and `...' was placed in the measurement.
iHVC absorption identified in the associated reference is likely contributing to the LIV absorption in the |v_ LSR|<100 .
§ SURVIVAL ANALYSIS
In Figure <ref> we show two examples of survival curves created using for the model-subtracted residual analysis. The plots start on the y-axis at 100% “survival” and drop by the percent of sight lines with an unsaturated detection at the specified column density on the x-axis. The function determines if these two curves are drawn from different populations.
|
http://arxiv.org/abs/2307.05601v1 | 20230710202858 | Unsupervised Domain Adaptation with Deep Neural-Network | [
"Artem Bituitskii"
] | cs.CV | [
"cs.CV",
"cs.LG"
] |
/
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§ INTRODUCTION
We start with necessary theory and motivation to understand some essential definitions and ideas associated with domain adaptation.
§.§ Theory
Domain adaptation is a subfield of machine learning that deals with the problem of transferring knowledge learned from one domain to another related but different domain. In real-world scenarios, it is common to encounter situations where the data distribution of the target domain differs significantly from the source domain used to train a model. This can lead to a significant drop in the performance of the model on the target domain.
To understand clearly what domain adaptation is about, we should start with transfer learning. For this purpose, we should dig deeper into the theory. In these articles <cit.> and <cit.> you can find a high level overview of the theory that connects with domain adaptation. Let's start with the transfer learning definition and types that it consists of.
(Transfer learning) We consider a source data distribution S called the source domain, and a target
data distribution T called the target domain. Let X_S × Y_S be the source input and output spaces associated to S, and X_T × Y_T be the target input and output spaces associated to T. We use S_X and T_X to denote the marginal distributions of X_S and X_T , t_S and t_T to denote the source and target learning tasks depending on Y_S and Y_T, respectively. Then, transfer learning aims to help to improve the learning of the target predictive function f_T : X_T ⟶ Y_T for t_T using
knowledge gained from S and t_S , where S = T.
According to these papers (<cit.>, <cit.>), transfer learning algorithms can be classified into three categories based on the differences between the source and target tasks and domains: inductive, transductive, and unsupervised transfer learning.
* Inductive transfer learning involves using labeled data from the source domain to train a model for a different, but related, target task in the target domain. In this case, some labeled data from the target domain is required to fine-tune the model.
* Transductive transfer learning, on the other hand, refers to using both labeled data from the source domain and unlabeled data from the target domain to improve the model's performance on the target domain. In this case, the tasks remain the same while the domains are different.
* Unsupervised transfer learning involves adapting a model trained on the source task to perform well on a related, but different target task in the target domain, without any labeled data in either the source or target domains.
Domain adaptation is a type of transfer learning where the target task remains the same as the source task, but the domain differs (the second type – transductive transfer learning). Depending on whether the feature spaces remain the same or differ, domain adaptation is categorized into homogeneous and heterogeneous domain adaptation. Machine learning techniques are commonly categorized based on the availability of labeled training data, such as supervised, semi-supervised, and unsupervised learning. However, domain adaptation assumes the availability of data from both the source and target domains, making it ambiguous to append one of these three terms to "domain adaptation". There are different ways how these terms can be applied to domain adaptation, but we use the same as in <cit.>.
* Unsupervised domain adaptation refers to the case where both labeled source data and unlabeled target data are available
* Semi-supervised domain adaptation refers to the case where labeled source data and some labeled target data are available
* Supervised domain adaptation refers to the case where both labeled source and target data are available.
Unsupervised Domain adaptation could be applied to a wide range of tasks in NLP <cit.>, in vision <cit.> and in many other applications where assigning labels to examples is tedious or impossible.
This report is more focused on studying unsupervised domain adaptation with using deep neural-networks. Before move on to the practical part, it is important to discuss theoretical analysis and guarantees that can be used in fields associated with transfer learning. Thus, there are several methods that allow you to analyze the generalization gap in machine learning <cit.>. One of the most popular approaches is the model complexity approach, which estimates the generalization bound by measuring the complexity of the hypothesis set, such as Vapnik-Chervonenkis (VC) dimension and Rademacher complexity. Another approach is to use the stability of the supervised learning algorithm in relation to the datasets. Stability is a measure of how much a change in a data point in the training set can affect the output of the algorithm. Both of these approaches have been used to analyze the generalization bounds of transfer learning algorithms.
It is equally important to discuss distributions and what experts mean by shift when analyzing transfer learning algorithms. Distribution refers to the set of all possible values of a random variable, and a shift refers to a change in the distribution of the data between the source and target domains. Understanding the shift in the distribution of the data is crucial in developing effective transfer learning algorithms, as it enables the selection of appropriate techniques for adapting the model to the target domain.
Unsupervised domain adaptation (UDA) is a type of supervised learning that involves training a model using labeled source data and applying it to unlabeled target data, where the distributions of the two domains differ. Let the source domain be represented by (x^S , y^S ) = (x^S_k , y^S_k)_k=1^m_S , and the target domain be represented by x^T = (x_k^T)_k=1^m_T. The number of observations in the source and target domains are denoted by m_S and m_T respectively. The main challenge of domain adaptation is to develop a predictor that performs well in the target domain by leveraging the similarities between the two domains. One way to accomplish this is by making assumptions about how the joint distribution P(X, Y) changes across the domains. In the case of covariate shift, the marginal distribution P(X) changes while the conditional distribution P(Y|X) remains the same. However, in real-world scenarios, P(Y|X) may also change, requiring further assumptions. One such assumption is that the joint distribution can be factored into P(Y) and P(X|Y), allowing changes in P(Y) and P(X|Y) to be addressed independently. The problem is then brokee tudied a cial10.5555/3241691.324170810.5555/3241691.3241ution of features and labels.
§.§ Motivation
Unsupervised domain adaptation (UDA) is a technique used in machine learning where a model is trained on labeled data from a source domain that has similar characteristics to the target domain, but where the target domain lacks labeled data. The goal is to create a model that will perform well on the target domain despite not having labeled data from that domain. In UDA, the source and target domains are not directly related, so the model has to learn how to generalize across domains.
The first reason to be engaged in this field is a scarcity of data. It is known that collecting labeled data in the target domain can be expensive and time-consuming. UDA allows us to use the available labeled data in the associated source domain to learn representations that generalize well to the target domain without requiring additional labeled data. Minimizing the discrepancy between domains, the model can learn more robust and transferable representations, which leads us to the second reason – improved generalization and domain robustness. The last reason is that UDA allows models to adapt to new environments. I reckon that this is a common situation in real applications, when models are trained on specially prepared data, and then applied to all other data types.
§ STATE-OF-THE-ART
In this section, we discuss main purposes, approaches and algorithms that specialists in the field of domain adaptation use in their research. In this section all figures are taken from the articles.
§.§ UDA by Backpropagation
The purpose of the article "Unsupervised Domain Adaptation by Backpropagation" written by Yaroslav Ganin and Victor Lempitsky <cit.> is to tackle the problem of domain shift in machine learning and to propose a solution to this problem using a neural-network model with few standard layers and gradient reversal layer (GRL). The GRL makes the network to learn domain-invariant features by minimizing the difference between the distributions of the source and target domains. The architecture of the model is shown below (see Figure <ref>)
The authors introduce an architecture that predicts both the label y ∈ Y and the domain label d ∈{0, 1} for each input x. The architecture consists of three parts: feature extractor f = G_f(x, θ_f), where θ_f is a vector that represents the parameters of all its layers; label predictor G_y that maps the features obtained after feature extractor to the label y, with θ_y representing its parameters; domain classifier G_d maps the same feature vector f to the domain label d, with θ_d representing its parameters. The purpose to minimize the label prediction loss for the source domain and simultaneously make the features f invariant to the domain. To achieve this, the authors optimize the parameters θ_f of the feature mapping to maximize the loss of the domain classifier, however the parameters θ_d are optimized to minimize the loss of the domain classifier. The authors consider the loss
E(θ_f, θ_y, θ_d) = ∑_i=1..N
d_i = 0 L_y(G_y(G_f(x_i; θ_f); θ_y), y_i) - λ∑_i = 1..N L_d(G_d(G_f(x_i; θ_f); θ_d), y_i)
= ∑_i=0..N
d_i = 0 L_y^i(θ_f, θ_y) - λ∑_i=1..N L_d^i(θ_f, θ_d)
where L_y and L_d are label prediction and domain classification losses, respectively. (index i means the i-th example). It is considered the parameters θ̂_f, θ̂_y, θ̂_d to gain a saddle point
(θ̂_f , θ̂_y ) = min_θ_f, θ_y E(θ_f, θ_y, θ̂_d)
θ̂_d = max_θ_d E(θ̂_f, θ̂_y, θ_d)
During learning, the trade-off between the two objectives that shape the features is controlled by the parameter λ. The following stochastic updates can find a saddle point
θ_f ←θ_f - μ( ∂ L_y^i∂θ_f - λ∂ L_d^i∂θ_f)
θ_y ←θ_y - μ∂ L_y^i∂θ_y
θ_d ←θ_d - μ∂ L_d^i∂θ_d
where μ is a learning rate. These updates are similar to SGD but with a -λ factor in the first update to prevent dissimilar features across domains. Therefore, the authors introduce a GRL that acts as an identity transform during forward propagation but multiplies the gradient by -λ during backpropagation.
§.§ Semantic Representations for UDA
Next, we continue with the article "Learning Semantic Representations for Unsupervised Domain Adaptation" written by Xie, Shaoan, et al. <cit.> The main purpose of the article is to propose a new method for unsupervised domain adaptation that utilizes semantic information to learn domain-invariant representations. The authors propose a domain adaptation algorithm which is based on the idea of using an adversarial learning to learn a feature representation that is invariant to domain shifts.
The authors train a feature extractor and then use it to map the input data to a high-dimensional feature space, and a domain classifier that predicts the domain label of the input data (see Figure <ref>). The feature extractor G is trained to confuse the domain classifier D, while the domain classifier is trained to correctly predict the domain label. In this way, the feature extractor is encouraged to learn features that are invariant to domain shifts, while still being discriminative for the task.
First, the authors denote the cross entropy loss for the source domain as L_C(X_S, Y_S). Then, the discrepancy between source domain and target domain is supposed to be
L_DC(X_S, X_T) = d(X_S, X_T) = 𝔼_x ∼ D_S [log(1 - D ∘ G(x))] + 𝔼_x ∼ D_T [log(D ∘ G(x))]
Moreover, the authors introduce one more loss, which targets the semantic representation. Centroid alignment is used for this purpose. By computing the centroid for each class, both correct and incorrect pseudo-labeled samples are utilized together:
L_SM(X_S, Y_S, X_T) = ∑_k=1^K Φ(C_S^k, C_T^k)
where C_S^k, C_T^k are centroids for each class and Φ(x, x') = x - x' ^2. This approach aims to cancel out the negative effects caused by inaccurate pseudo labels with accurate ones. Thus, the authors get the following total loss
L(X_S, Y_S, X_T) = L_C(X_S, Y_S) + λ L_DC(X_S, X_T) + γ L_SM(X_S, Y_S, X_T)
where λ and γ are responsible for the balance between the classification
loss, domain confusion loss and semantic loss. In the article, algorithm of moving average centroid alignment is presented that allows to align the centroids in same class
but different domains to achieve semantic transfer for UDA.
§.§ Fixbi for UDA
The purpose of the article "Fixbi: Bridging domain spaces for unsupervised domain adaptation" written by Jaemin Na, Heechul Jung et al. <cit.> is to propose a fixed ratio-based mixup method to address the problem of large domain discrepancies. The authors mix up images and then fed them into neural networks to achieve greater reliability in learning from corrupted labels. It is proposed to use two predetermined mixup ratios λ_sd and λ_td for the source and target domain respectively.
Denote input samples and their labels for source and target domain as (x_i^s, y_i^s) and (x_i^t, ŷ_i^t), the authors define mixup configurations in the following way:
x̃^st_i = λ x_i^s + (1 - λ)x_i^t
ỹ^st_i = λ y_i^s + (1 - λ)ŷ_i^t,
where λ∈{λ_sd, λ_td} and λ_sd + λ_td = 1, ŷ_i^t is the pseudo-labels for the target samples. By leveraging the fixed ratio-based mixup, it is constructed two neural networks with different perspectives: the "source-dominant model" (SDM) and the "target-dominant model" (TDM) (see Figure <ref>).
The SDM provides robust supervision for the source domain but relatively weak supervision for the target domain, while the TDM has strong supervision for the target domain but weaker supervision for the source domain. Thus, denoting p(y| x̃^st_i) as a predicted class distribution, it is defined fixed ratio-based mixup function
L_fm = 1B∑_i = 1^B ŷ^st_i log (p(y|x̃^st_i)),
where ŷ^st_i = max p(y|x̃^st_i) and B is the size of a mini-batch. In order to have connections between source and target domains, it is suggested to use a confidence-based learning approach whereby one model educates the other using positive pseudo-labels, or penalties itself using negative pseudo-labels. Positive pseudo-labels means labels which predictions are above a specific threshold, then the authors use them in training the second model by utilizing a conventional cross-entropy loss. Thus, denote p and q as distributions of two models, the authors get the following loss function
L_bim = 1B∑_i=1^B 1(max (p(y|x_i^t) > τ)ŷ_i^t log (q(y|x_i^t)),
where ŷ_i^t = max p(y|x_i^t). In contrast, a negative pseudo-label refers to the top-1 label predicted by the network with a confidence below the threshold τ. The function of self-penalization is defined as follows:
L_sp = 1B∑_i=1^B 1(max (p(y|x_i^t) < τ)ŷ_i^t log (1 - p(y|x_i^t))
Furthermore, the threshold is changed adaptively during training. In addition, it is introduced the following expression:
L_cr = 1B∑_i=1^B p(y|x̃^st_i) - q(y|x̃^st_i)^2_2
that represents consistency regularization to guarantee a stable convergence during the training of both models.
§.§ Spherical Space DA with Pseudo-label Loss
Now we are going to the next article "Spherical Space Domain Adaptation with Robust Pseudo-label Loss" written by Xiang Gu, Jian Sun, and Zongben Xu. <cit.> The authors propose a spherical space representation of data, which allows them to get more effective feature extraction and better adaptation across domains. One approach associated with increasing performance with differences in data distribution between source and target domain is to use pseudo-labels. However, the use of pseudo-labels can be problematic in the presence of noisy or incorrect labels. To tackle this problem, the authors map the data to a high-dimensional sphere and introduce a new loss function, called the robust pseudo-label loss, which is designed to address the problem of noisy or incorrect labels in the target domain.
In the Figure <ref> we can see spherical domain adaptation method. Domain invariant features are learned by adversarial training, entirely in the spherical feature space. The feature extractor F is utilized to normalize the features to map onto a sphere. The classifier C and discriminator D are defined in the spherical feature space, consisting of spherical perceptron layers and a spherical logistic regression layer (see Figure <ref>).
Although the use of a spherical element reduces feature dimension by one, it simplifies the domain adaptation process by eliminating differences in norms. The authors define spherical adversarial training loss as follows:
L = L_bas(F, C, D) + L_rob(F, C, ϕ) + γ L_ent(F)
This lost consists of three parts: basic loss, robust pseudo-label loss and conditional entropy loss. Let's start with the first one. To align features, the authors utilize basic loss which is defined as an adversarial domain adaptation loss:
L_bas(F, C, D) = L_src(F, C) + λ L_adv(F, D) + λ' L_sm(F),
where L_src is a cross-entropy loss for the source domain, L_adv is an adversarial training loss and L_sm is a semantic loss. The second one is conditional entropy loss which is used to keep the learned features away from the classification boundary:
L_ent(F) = 1N_t∑_j=1^N_t H(C(F(x_j^t))),
where H(·) denotes the entropy of a distribution. Additionally, the authors propose robust pseudo-label loss to increase robustness of the model. Denote ỹ_j^t = max_k [C(F(x_i^s))]_k as a pseudo-lebel of x_j^t where [·]_k means the k-th element. To be ensured in precision of pseudo-labels, it is assumed to use new random variable z_j ∈{0, 1} for each pair (x_j^t, ỹ_j^t) that specify the correctness of the data (1 is correct, 0 is not). Let the probability of correct labeling be P_ϕ (z_j = 1| x_j^t, ỹ_j^t) and ϕ to its parameters, then robust loss is defined as follows:
L_rob(F, C, ϕ) = 1N_0∑_j=1^N_t w_ϕ(x_j^t) 𝒥(C(F(x_j^t)), ỹ_j^t),
where N_0 = ∑_j=1^N_t w_ϕ(x_j^t) and 𝒥(·, ·) is mean absolute error (MAE). The function w_ϕ (x_j^t) is defined using the posterior probability of correct labeling
w_ϕ(x_j^t) = {γ_j, if γ_j ≥ 0.5,
0, otherwise,
.
where γ_j = P_ϕ(z_j = 1| x_j^t, ỹ_j^t). By utilizing a Gaussian-uniform mixture model in spherical space based on pseudo-labels, the authors model the probability P_ϕ(z_j = 1| x_j^t, ỹ_j^t) as a function of the feature distance between the data and the center of the corresponding class. Thus, samples from target domain with a probability of correct labeling below 0.5 can be discarded. For further details of computing posterior probability, please refer to the article <cit.>.
§.§ DA with Invariant Representation Learning
Next, we move on to the article "Domain Adaptation with Invariant Representation Learning: What Transformations to Learn?" written by Stojanov, Petar, et al. <cit.> The researchers focus on the conditional shift scenario, where the data-generating process is utilized to (i) explain why two distinct encoding functions are required to infer the latent representation, (ii) improve an implementation of these functions, and (iii) impose meaningful structure on the latent representation Z to increase prediction accuracy in the target domain.
Let's consider the data-generating process shown in the Figure <ref> to understand what information is required for learning. The label Y is generated first from its prior distribution P(Y). Then, the invariant representation Z is generated from Y through P(Z|Y), and X is generated from P(X|Z; θ_X), where θ_X represents the changing parameters of P(X|Y) across domains. We can consider Z as a latent representation of our data. The variable θ_X may correspond to environment-specific changes that are irrelevant for predicting the class Y. Generally speaking, Z is conditionally dependent on θ_X given X, although they may be marginally independent. Therefore, to recover Z given X, the information of θ_X should also be considered in the transformation (see detailed in the article to understand clearly how authors measure the influence of θ_X). The authors made two key observation associated with the data-generating process. Firstly, the encoder function ϕ requires θ_X as an input in addition to X. Secondly, assuming that θ_X has minimal influence on the relationship between X and Z, allowing us to use a single encoder ϕ(X, θ_X) instead of two separate encoders. A decoder function ϕ that restricts the influence of θ_X, acting as a regularizer on the encoder ϕ, in order to retain important semantic information.
Thus, the authors proposed a domain-adaptation network, which is shown in the Figure <ref>, where θ_X ∈{θ_X^S, θ_X^T} parameters for source and target domains respectively.
§.§ Domain Adaptation for Segmentation with CBST
The main purpose of the paper "Domain Adaptation for Semantic Segmentation via Class-Balanced Self-Training" written by Zou, Yang, et al. <cit.> propose a new UDA framework for semantic segmentation based on iterative self-training procedure. A novel technique, referred to as Class-Balanced Self-Training (CBST), has been suggested by the authors, which aims to adapt the segmentation model from the source domain to the target domain by leveraging unlabeled target data. In the Figure <ref>, the authors present a structure and the results of their deep self-training framework using two datasets: GTA 5 <cit.> and Cityscapes <cit.>.
The CBST approach is based on two main components: a class-balancing strategy and a self-training algorithm. The class-balancing strategy aims to address the problem of class imbalance between the source and target domains, which can negatively impact the performance of the segmentation model. The authors change the loss function using parameters that determine the proportion of selected pseudo-labels due to balance the class distribution during the self-training process. Furthermore, when the images in the source and target domains are similar, spatial prior knowledge can be effectively utilized to adapt models. For this purpose, the authors count the class frequencies in the source domain using Gaussian kernel. The experimental results show that the CBST approach outperforms several state-of-the-art unsupervised domain adaptation methods for semantic segmentation.
§ CONTRIBUTION
The "Contribution" section of the thesis highlights the developed new method called DannFixbi, which combines the Fixbi approach and the backpropagation approach. In an attempt to implement the state-of-the-art method, extensive research has been conducted, and existing approaches have been implemented. The decision to combine these two approaches was based on their respective strengths and the potential for mutually beneficial interaction between them. By incorporating the Fixbi technique, which addresses domain shift, and leveraging the benefits of backpropagation, DannFixbi aims to enhance the performance and robustness of domain adaptation in the field of images. The development of DannFixbi represents an original contribution to the field.
This new method consists of two neural networks, which are trained using a modified version of the Fixbi approach. To enhance the performance of this method, two domain classifiers are added to each of these two networks. Two different approaches have been explored for incorporating these domain classifiers.
The first approach involves adding a domain classifier to each neural network (see Figure <ref>). During training, images obtained by mixing from the source and target domains with predefined mixup ratios are fed into these classifiers.
For mixed images, the following loss functions is used:
L_dom = α L_ds(X̂, Y_s) + (1 - α) L_dt(X̂, Y_t) ,
where X̂ is mixed images, Y_s is source domain labels, Y_t is target domain labels and α∈{λ_sd, λ_td}. L_ds and L_dt presents cross entropy loss for source and target domains, respectively.
The second approach is to use a domain classifier for each net with images from the source and target domains without any mixing, similar to the backpropagation method (see Figure <ref>).
The second approach utilizes the following loss function for domain classification:
L_dom = L_d(X_st, Y_st) ,
where X_st, Y_st denotes source and target images and domain labels, respectively, and L_d is a cross entropy loss.
The total loss for the new method is calculated as the sum of the loss from the Fixbi method and the domain loss described earlier:
L_total = β L_fixbi + γ L_dom
Here, β and γ represent the weights assigned to the Fixbi loss and the domain loss, respectively. The values of these weights determine the relative importance of each component in the overall loss calculation. Further, unless otherwise stated, the values of alpha and beta are assumed to be equal to 1. The Fixbi loss, denoted as L_fixbi, is composed of several summands described in the equations <ref> – <ref>:
L_fixbi = L_fm + L_sp + 1{e > k}(L_bim + L_cr)
where e denotes a current epoch, and k is warm-up epochs. To establish independent characteristics for the two networks, it is introduced a warm-up period of k epochs. During this phase, each network is trained separately using the fixed ratio-based mixup and self-penalization techniques. Once an enough amount of training has been completed, bidirectional matching loss is added, which helps networks train collaboratively, exchanging knowledge and benefiting from each other's insights.
The new method, referred to as DannFixbi, demonstrates improved performance and robustness in unsupervised domain adaptation for image analysis. This contribution represents a unique approach to UDA, offering valuable insights and potential for future developments in this field. All the results obtained are presented in the "Experiments" section.
§ EXPERIMENTAL SETUP
In this part, we start with description of different datasets that are commonly used in transfer learning. Then, we will continue with implementation details and experiments that have been conducted. Code is available at https://github.com/Jetwev/domain-adaptationhttps://github.com/Jetwev/domain-adaptation.
§.§ Datasets
The most popular datasets are Office-31, ImageCLEF-DA, Office-Home, DomainNet and VisDA-2017. Detailed discussion of each of them is given below:
* Office-31 <cit.> consists of 4,110 images categorized into 31 classes, which are distributed across three separate domains: Amazon (A), Webcam (W), and Dslr (D).
* ImageCLEF-DA, utilized in <cit.>, includes three distinct domains: Caltech-256 (C), ImageNet ILSVRC 2012 (I), and Pascal VOC 2012 (P). There are 600 images in each domain and 50 images for each category.
* Office-Home <cit.>, includes four absolutely different domains: Artistic images (Ar), Clip Art (Cl), Product images (Pr) and Real-World images (Rw). This dataset contains 15 500 images in 65 object classes, which makes it more complex than Office-31.
* VisDA-2017 <cit.> consists of 12 classes shared between two very different domains: Synthetic and Real. It contains synthetic images (training set) and real-world images (test set). The dataset was designed to have a large domain gap, which makes it a challenging benchmark for domain adaptation methods.
* DomainNet <cit.> is a large-scale visual recognition dataset designed to evaluate domain adaptation algorithms, which consists of almost 600 thousand images and includes 345 classes.
§.§ Implementation details
At the beginning, it was necessary to start with some approaches to check their performance and have a possibility to compare results. Four different methods described in the papers have been chosen for the study:
* Source only is a method where a model is trained solely on the source domain data without any adaptation to the target domain.
* Domain-Adversarial Neural Network (Dann) is domain adaptation technique that aims to learn a domain-invariant feature representation by aligning the feature distributions of the source and target domains. The architecture of Dann consists of three components: a feature extractor network, a label predictor network, and a domain classifier network, and is described in more details in section 2.1.
* Moving Semantic Transfer Network (Mstn) The key idea behind Mstn is to add semantic transfer loss to the Dann approach. In the section 2.2, it is proposed to use average centroid alignment for aligning the feature distributions of the source and target domains. The architecture is the same as in the Dann method.
* Fixbi is the approach described in details in the section 2.3. The main idea is to train two neural networks, allowing models to learn from each other or on their own results. For this purpose, the authors add bidirectional matching and self-penalization losses.
CNN architectures. For all approaches, pretrained Resnet50 <cit.> is utilized as the backbone network. The weights for the neural network can be downloaded from this https://download.pytorch.org/models/resnet50-19c8e357.pthlink. Resnet50 has been pretrained on large image datasets such as ImageNet, which means that the network has already learned to recognize a wide range of features in images. Resnet50 is a convolutional neural network architecture consisting of 50 layers. This is a variant of the Resnet family of neural networks, which are designed to solve the vanishing gradient problem in deep neural networks. Resnet networks achieve this by using short connections between layers, which allow gradients to move more easily during backpropagation. Resnet50 is a widely used architecture in many articles, which makes it a good choice for research.
Resnet50 is used as a Feature extractor in all considering methods. Label predictor is a simple network that consists of two fully connected layers (2048 → 256 →number of classes). Domain classifier architecture represents several fully connected layers with a ReLU activation function and dropouts between each two fully connected layers. Using dropouts can reduce the sensitivity of the model to specific features in the input data and encourage the model to learn more generalizable features. This can lead to better performance on new, unseen data and can prevent overfitting.
Learning rate schedulers. Learning rate is an important hyperparameter that determines the step size at which the optimizer updates the model's parameters during training. There are many of them and it can be challenging to find the optimal learning rate, as setting it too high can cause the model to diverge, while setting it too low can slow down the learning process. Thus, in this study it is utilized two different learning rate schedulers: CustomLRScheduler and CosineAnnealingLR. The implementation of the first one follows the rules that are described in <cit.>
η_p = η_0(1 + α· p)^β,
where p linearly increases from 0 to 1, and the values η_0, α, and β are set to 0.01, 10, and 0.75, respectively. The second one, CosineAnnealingLR is a popular learning rate scheduler utilized in deep learning. It systematically reduces the learning rate over multiple epochs in a cyclical manner. Initially, the learning rate starts at its maximum value and then gradually decreases until it reaches the minimum value. Upon reaching the minimum value, the cycle restarts, and the learning rate returns to its maximum value. This process continues until the end of the training, which is usually determined by the total number of epochs or a predefined stop criterion. By starting with a higher learning rate and gradually decreasing it, the model can avoid getting stuck in local minima and converge to a better global minimum. The formula for the CosineAnnealingLR scheduler is:
η_t = η_min + 12(η_max - η_min) (1 + cos( T_curT_maxπ)),
where η_max is your initial learning rate, η_min – minimum learning rate value, T_cur is the number of epochs since the last start, T_max – the total number of epochs. More detailed information about CosineAnnealingLR can be found https://pytorch.org/docs/stable/generated/torch.optim.lr_scheduler.CosineAnnealingLR.htmlhere.
Optimizers. This study uses two popular optimization algorithms - stochastic gradient descent (SGD) and adaptive moment estimation (Adam). Both algorithms are commonly employed in deep learning to optimize the parameters of a neural network and improve its performance.
SGD is a simple and popular optimization algorithm that updates the weights of a model in the direction of the negative gradient of the loss function. One limitation of SGD is that it can get stuck in local minima and struggle with noisy or sparse gradients. To tackle this problem, several modifications can be used. In PyTorch, the SGD optimizer has several hyperparameters that can be tuned to improve its performance. The following parameters (except learning rate) are considered in this study:
* Momentum is a hyperparameter that determines how much past gradients affect the current gradient update. It helps to minimize the impact of the noise and fluctuations in the gradient updates. However, setting the momentum too high can also lead to slower convergence.
* Weight decay is a form of L2 regularization that adds a penalty term to the loss function during training. This penalty term is proportional to the square of the weights in the network, which encourages the model to use smaller weights and reduce overfitting.
* Nesterov momentum is a variant of momentum that takes into account the momentum term in the calculation of the gradient. This can help to reduce oscillations and improve convergence rates, especially in high-dimensional optimization problems.
Adam is another optimization algorithm that is commonly used in deep learning. It is an extension of SGD. The key idea behind Adam is to maintain a separate adaptive learning rate for each parameter in the network, based on estimates of the first and second moments of the gradients. This makes Adam more effective than SGD for optimization problems with noisy or sparse gradients. However, it may not always be the best choice for every task and model architecture, so it's important to experiment with different optimization algorithms and settings to find the best approach for your specific problem. Adam is considered in this study with default parameters, more information about the implementation and usage can be found at this https://pytorch.org/docs/stable/generated/torch.optim.Adam.htmllink.
Pytorch Lightning. PyTorch Lightning is a lightweight PyTorch wrapper that allows users to focus on the high-level design of their experiments and models, instead of dealing with the low-level implementation details. It provides a structured way to organize PyTorch code, making it easier to read and maintain.
PyTorch Lightning offers a range of benefits that make it a good choice for deep learning researches. Firstly, it offers a modular design that makes it easy to organize code. It gives you a convenient and user-friendly interface to manage and run experiments. Moreover, all these benefits can help to improve your productivity. Secondly, PyTorch Lightning makes it easier to scale models to multiple GPUs, which can significantly reduce training times for large models. Finally, it is flexible and can be easily integrated with other PyTorch libraries. Overall, PyTorch Lightning is an excellent choice for researchers who want to focus on the research aspect of deep learning and leave the engineering components to the library. More information can be found on the official https://www.pytorchlightning.ai/index.htmlwebsite.
Weights and Biases. Weights and Biases (WandB) is a platform that provides a suite of tools to help developers and data scientists track and visualize their machine learning experiments. WandB makes it easy to log, keep track of your progress and compare different experiments, visualize model performance, and collaborate with team members. One of the main advantages of WandB is its integration with popular machine learning frameworks such as TensorFlow, PyTorch, and Keras. This means that you can easily log and track your model's hyperparameters and performance metrics during training and evaluation. Moreover, WandB is a cloud-based platform, which means that users can access their experiments and data from anywhere with an internet connection and also share them with colleagues and co-workers. For more detailed information, it is recommended to visit the official https://wandb.ai/sitewebsite.
Batch size. Different domains in your dataset can contain different number of images that makes your training process more complicated. To tackle this problem, it is proposed two approaches.
The first one is to find the ratio of the smaller dataset size to the larger one and concatenate the smaller dataset multiple times to ensure that the number of batches is aligned during the training loop. However, it is important to emphasize that with this approach, overfitting can occur if the appropriate number of epochs is not established. This is because the smaller dataset will be fed into the model more times than the larger one (depends on the ratio).
The second approach involves varying the number of images taken per batch for each domain. Applying this approach, it becomes possible to avoid concatenating the smaller dataset multiple times, which effectively reduces the amount of memory consumed. It is crucial to carefully consider the number of images per batch, as choosing a value that is either too high or too low can have negative consequences.
Augmentation techniques. Augmentation techniques for images are used to create variations and increase the size of the training dataset by applying a series of transformations. These techniques are widely employed in computer vision tasks, including image classification, object detection, semantic segmentation, etc. Augmentation helps increase the diversity of the dataset, leading to improved model generalization and robustness. PyTorch provides a variety of image augmentation techniques. The following transformations are used in this research:
* Normalize – normalizes the image by subtracting the mean value and dividing by the standard deviation.
* Resize is a function that allows you to resize an image to a specific size.
* RandomCrop is a function that randomly crops a portion of the image.
* CenterCrop is a transformation that allows you to perform a center crop on an image.
* RandomHorizontalFlip – randomly flips the image horizontally with a specified probability.
* RandomVerticalFlip - randomly flips the image vertically with a specified probability.
* RandomRotation is a function that randomly rotates the image by a given angle.
* ColorJitter is a transformation that allows you to adjust the brightness, contrast, saturation, and hue of the image.
* ToTensor is a specific function in PyTorch that is used to convert an image into a tensor.
These augmentation techniques can be applied individually or combined sequentially using the transforms.Compose function. More detailed information about transformations and their usage can be found https://pytorch.org/vision/stable/transforms.htmlhere. It's important to emphasize that the choice and combination of augmentation techniques depend on the specific task and dataset characteristics, and careful selection of them are crucial to achieve optimal results.
OmegaConf. OmegaConf is a library for Python that provides a convenient and flexible way to manage complex configurations in machine learning projects. It is designed to provides a number of features that can help to simplify the configuration process. Here are some reasons why OmegaConf can be a good choice:
* It is easy to use and allows developers to define nested configurations and easily access and modify configuration values.
* OmegaConf supports a wide range of configuration formats, including YAML and JSON. This makes it flexible and easy to integrate in your project.
* It supports type checking, which can help to catch configuration errors and improve code quality.
To sum up, OmegaConf can be a good choice for Python developers who work on large and complex projects and want a flexible and powerful configuration system for their applications. Additional details can be found https://omegaconf.readthedocs.io/en/2.3_branch/here.
§.§ Experiments
The dataset Office-31 is used to test the approaches. This dataset consists of three domains: Amazon (A) - 2817 images, Dslr (D) - 498 images and Webcam (W) - 795 images (see Figure <ref>).
The Table <ref> below highlights various key features of the dataset, including the number of classes, image resolution, task and evaluation metric.
The existence of dissimilar image quantities ensures us in the importance of utilizing one of the approaches discussed in the previous section in order to avoid any information loss. In the first approach, where the smaller domain is concatenated, a batch size of 32 or 64 is utilized for all experiments. The second approach takes into account the size of each domain, and as a result, the batch sizes are utilized in the experiments according to the Table <ref>:
For the all methods, two kinds of optimizers are used: SGD and Adam. However, the second one shows worse results with default parameters than SGD with lr = 0.001, momentum = 0.9, weight decay = 0.0005. The CustomLRScheduler and CosineAnnealingLR are both used as schedulers, but it has been found that the model performs better when using the second one. Thus, all the following results have been obtained using the CosineAnnealingLR scheduler. Furthermore, all methods give the best results with an approach using a different number of images in each batch. As a result, this approach will be assumed by default, unless otherwise stated. For each two domains, at least three experiments were conducted for all methods, and the best results were selected.
Let's start with the first method – Source only. Here, the model is trained on the source domain and then tested on the target domain. The obtained results are shown in Table <ref> (at the end of the 60th epoch).
Dann is an architecture that consists not only of a feature extractor and label predictor, but also a domain classifier. This domain classifier helps to identify the domain of the input data and allows the model to learn domain-invariant features. The Table <ref> below clearly demonstrates that the results for each of the two domains are superior to those obtained using the simple Source only method. The results are obtained at the end of the 60th epoch.
Mstn method is a complication of Dann by adding a semantic loss. To get this loss, we add centroids for each class and utilize the algorithm described in the section 1.2.2. In the Table <ref>, you can see the results that are acquired at the end of the 60th epoch. The quality of the results tends to suffer due to the significant influence of randomness. The selection of pictures that are included in a batch determines the movements of the centroids, ultimately influencing the overall quality to a significant extent.
Fixbi is a method that addresses the domain adaptation problem by training two neural networks that can help each other. As it is described in the article <cit.>, we define λ_sd = 0.7 and λ_td = 0.3. In this method, we cannot use the approach with different batch sizes due to the need to mix up images from source and target domain. Therefore, the second approach with concatenation is utilized. The model is trained for a combined duration of 150 epochs, with the first 100 epochs designated as the warm-up period. It is important to note that 150 epochs are used, not 200, because after the warm-up period the validation score stabilizes and almost does not change. After the warm-up period, L_bim starts to be applied, which leads to a critical changing in the total accuracy. The sudden improvement in accuracy can occur in either a positive or negative direction, and is often heavily influenced by randomness. One possible explanation for this phenomenon is that the model may have already found a local minimum prior to the introduction of L_bim, and the application of L_bim causes a sudden shift in gradients that propels the model out of the current minimum and into a new one. Depending on the new minimum, this can result in either an improvement or a deterioration in the model's performance. As we can see in the Figure <ref>, for Amazon (source) and Webcam (target) domains this method gives significant increase in accuracy, while for DSLR (source) and Amazon (target) it shows the worst results.
In the Figure <ref>, you can see the separate accuracy of “source-dominant model” (SDM) and “target-dominant model” (TDM) in case of Amazon (source) and Webcam (target) domains.
The results for each two domains of Office-31 dataset for Fixbi method are shown in Table <ref>.
The Fixbi method was selected for modification, wherein the ratios for SDM and TDM were adjusted, and a domain classifier was added for mixup images. This modified method is named DannFixbi. λ_sd and λ_td are set as 0.9 and 0.7, respectively. As it was mentioned before, it is used one of the two approaches described in "Contribution" section with domain classifiers for mixup images or separately for each domain. The Table <ref> indicates that certain domains exhibit an increase in accuracy as a result of these changes.
Table <ref> presents all the obtained results. The DannFixbi method yields the highest accuracy for the A→D, A→W, D→A and D→W tasks, while the Dann method achieves the best results for the W→D, and W→A tasks.
Additionally, it is worth noting that an overall assessment of the methods across all domains can be obtained by calculating the average accuracy (see Table <ref>).
To sum up, the new introduced method called DannFixbi outperforms all other methods in visual recognition. The <ref> provides additional results for each domain, allowing for comparisons using the Wilcoxon signed-rank test to determine the statistical significance of the findings (see Tables <ref> – <ref>). The new method demonstrates statistically significant results in three tasks: A→D, A→W, and D→W, while outperforming all other methods on average (Tables <ref> – <ref>).
§ CONCLUSION AND PERSPECTIVES
In this thesis, the focus was on exploring and implementing methods related to unsupervised domain adaptation. The Office-31 dataset was utilized for evaluating these methods and conducting a comprehensive comparison. The results obtained from the experiments were analyzed, leading to the development of the new DannFixbi method that demonstrated the best performance compared to all the other methods presented.
The Office-31 dataset provided a suitable benchmark for evaluating the effectiveness of various unsupervised domain adaptation techniques. By conducting experiments on this dataset, the performance of different methods could be objectively assessed and compared. The analysis of the results shows the strengths and weaknesses of each method, allowing a deeper understanding of their capabilities.
Based on the comparative analysis, it was observed that the newly developed method showcased the best results among all the presented methods. The success of the new method can be attributed to its ability to leverage the strengths of existing techniques. By combining the back propagation method with domain classifiers and applying the Fixbi approach, it is possible to identify common features in different domains and share knowledge and insights between networks. This collaborative approach to learning has led to higher performance and increased the overall effectiveness of the method.
Overall, this thesis contributes to the field of unsupervised domain adaptation by providing an analysis of existing methods, introducing a new approach, and demonstrating the potential for improving visual recognition tasks across different domains. The results of this study open up opportunities for further study and development of advanced methods in the field of domain adaptation.
By addressing the challenge of distribution mismatch between the labeled and unlabeled data, we can note that advances in domain adaptation can significantly benefit other related domains specially semi-supervised learning. One line of research would be to study the generalization performance of semi-supervised learning models that have been studied under the cluster assumption <cit.>. Indeed, by explicitly considering the differences between the source and target domains, domain adaptation techniques can enhance the model's ability to adapt to new, unseen data in the target domain and can hence provide strategies to handle domain shift and improve the generalization performance of the semi-supervised learning model. Furthermore, by reducing the distribution mismatch between labeled and unlabeled data, domain adaptation methods can enable semi-supervised learning algorithms to leverage the unlabeled data more effectively. Moreover, domain adaptation methods often focus on learning robust representations that are less sensitive to noise and domain shifts. By leveraging such robust representations, semi-supervised learning algorithms can become more resilient to label noise and improve their accuracy even with limited labeled data. Domain adaptation is essentially a form of transfer learning, where knowledge learned from a source domain is transferred to a target domain. By studying domain adaptation, we can gain insights into transfer learning techniques that can be beneficial for semi-supervised learning scenarios like ranking <cit.>. These techniques can help leverage knowledge from a labeled source domain to improve the performance of a semi-supervised learning model in the target domain.
unsrt
§ APPENDIX
This appendix presents the results for each domain. In order to make comparisons, 15 experiments were conducted for each method within each area. The Wilcoxon rank test was employed to analyze and assess the performance of the methods. The best-performing method for each experiment is denoted by bold values and statistical significance is denoted with a star (*).
|
http://arxiv.org/abs/2307.07545v1 | 20230714180000 | Prescaling relaxation to nonthermal attractors | [
"Michal P. Heller",
"Aleksas Mazeliauskas",
"Thimo Preis"
] | nucl-th | [
"nucl-th",
"cond-mat.quant-gas",
"hep-ph",
"hep-th"
] |
[email protected]
Department of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium
[email protected]
Institut für Theoretische Physik, Universität Heidelberg, 69120 Heidelberg, Germany
[email protected]
Institut für Theoretische Physik, Universität Heidelberg, 69120 Heidelberg, Germany
We study how isotropic and homogeneous far-from-equilibrium quantum systems relax to nonthermal attractors, which are of interest for cold atoms and nuclear collisions. We demonstrate that a first-order ordinary differential equation governs the self-similar approach to nonthermal attractors, i.e., the prescaling. We also show that certain natural scaling-breaking terms induce logarithmically slow corrections that prevent the scaling exponents from reaching the constant values during the system's lifetime. We propose that, analogously to hydrodynamic attractors, the appropriate mathematical structure to describe such dynamics is the transseries. We verify our analytic predictions with state-of-the-art 2PI simulations of the large-N vector model and QCD kinetic theory.
Prescaling relaxation to nonthermal attractors
Thimo Preis
received: ** 2023, accepted: * 2023
==============================================
Introduction
Thermalization of isolated quantum many-body systems is an important contemporary research problem of a broad scope. Its relevance ranges from cold atom systems, through QCD in ultrarelativistic nuclear collisions all the way to gravity and black hole physics <cit.>.
Given the complexity of modeling quantum many-body dynamics and the richness of non-equilibrium phenomena, emergent regularities that form a basis for a quantitative understanding are of particular interest.
In this work we are concerned with an important instance of such an emergent regularity: far-from-equilibrium self-similar time evolution of nonthermal attractors, also known as nonthermal fixed points. These phenomena are transient stages in the thermalization dynamics, whose defining feature is self-similar scaling behavior in time. Consider a momentum distribution function f(t,) of a homogeneous and isotropic system, where t is time and spatial momentum. The system reaches a nonthermal attractor, when f scales with time
f(t,) = (t/t_ref)^α_∞ f_S((t/t_ref)^β_∞ ||)
with constant scaling exponents α_∞ and β_∞.
Indeed, such behavior corresponds to a vast reduction in the complexity, as the knowledge of the distribution function at some time allows one to determine the distribution function at a different time by a simple rescaling.
Nonthermal attractors appear in the studies of isolated quantum systems across a wide range of energy scales: ultracold quantum gases <cit.>, ultrarelativistic nuclear collisions <cit.> and early universe cosmology <cit.>. Despite significant interest in nonthermal attractors, a quantitative understanding of how a system approaches a nonthermal fixed point remains elusive <cit.>.
In <cit.> it was proposed that even prior to reaching the nonthermal attractor (<ref>) the system can exhibit prescaling with time dependent scaling exponents α(t) and β(t)
f(t,) =A(t) f_S(B(t)||),
where the prescaling factor A(t)=exp[∫_t_0^t dt^'α(t^')/t^'] reduces to the fixed-point scaling of Eq. (<ref>) when α(t) approaches α_∞. The same holds for B(t) in terms of β(t).
Given that scaling (<ref>) is an asymptotic late times statement known to be reached slowly, the systems of interest might in fact spend a much greater fraction of their lifetime prescaling (<ref>) rather than scaling (<ref>). Therefore, a quantitative understanding of prescaling is as important as understanding scaling itself.
In our work, we develop a simple theoretical description of prescaling dynamics that uses the same assumptions as the ones used to derive scaling. We test our predictions using strongly-correlated large-N vector model and weak coupling
QCD kinetic theory simulations.
Scaling implies prescaling
Understanding prescaling requires identifying laws governing time evolution of A(t) and B(t) (or, alternatively, α(t) and β(t)). As we show, these laws have a surprisingly simple origin and form.
The key role in deriving scaling (<ref>) is played by conserved quantities: particle number density n = ∫ d^d f/(2π)^d or energy density ℰ = ∫ d^d ω_ f/(2π)^d, where d is the number of spatial dimensions and ω_ is the dispersion relation of particles. We focus on ω_∼ ||^z.
Requiring conservation of n or ℰ is known to impose the relation between the scaling exponents: α_∞ = σ β_∞ <cit.>. When n=const, then σ = d, while ℰ=const gives σ = (d+z). The conserved quantities are local in time, which means that they in fact constrain also prescaling exponents in exactly the same way: α(t) = σ β(t). Equivalently, A(t) = B(t)^σ. This implies that there is only one independent degree of freedom in the isotropic and homogeneous prescaling, which we will choose to be B(t).
The time evolution for the independent prescaling factor B(t) is still subject to the equation of motion for f. In the case of a kinetic theory it is given by the Boltzmann equation with collision kernel C[f]
∂_t f(t,) = C[f](t,).
In the present section, we assume the collision kernel to be a homogeneous functional of particles momenta, i.e., to simply scale under Eq. (<ref>) by
A(t)^μ_α B(t)^μ_β≡ B(t)^σμ_α+μ_β for some real numbers μ_α,β.
This assumption applies to many (but not all) collision kernels describing nonthermal attractors. Typically overoccupation singles out terms with the highest power of the distribution function and associated matrix elements often happen to scale homogeneously under rescalings of momenta.
For such collision kernels we can separate time-dependent and (rescaled) momentum-dependent contributions in the Boltzmann equation
B(t)^1-1/β_∞/∂_t B(t)=1/D_1=[ σ + ·∂_] f_S()/ C[f_S](),
with =B(t) being the rescaled momentum, 1/β_∞=(1-μ_α)σ -μ_β, and D_1 being a separation of variables constant. The intrinsic time dependence of our setup implies nonzero D_1. The right-hand side of Eq. (<ref>) has the same form for constant scaling exponents but the left-hand side is more general, and is an exact equation of motion for prescaling.
The idea of separation of variables in the context of nonthermal attractors appeared already in <cit.>, but only solutions with constant scaling exponents were considered. In particular, the late time form (<ref>) fixes D_1 = β_∞/t_ref. Our key observation here is that prescaling is encapsulated by the general solution of Eq. (<ref>),
B(t) = (t-t_*t_ref)^β_∞≈( tt_ref)^β_∞( 1 -β_∞t_*t + …)
β(t) = β_∞tt-t_*≈β_∞ + β_∞t_*t + … .
From Eq. (<ref>) it is clear that prescaling induces power-law corrections to scaling. The prescaling originates from the presence of nonzero t_*. Its appearance comes as no surprise: the dynamics of the system in question is time translationally invariant and therefore t appearing in formulas needs to be measured with respect to some time, t_*. The reason why it does not appear in Eq. (<ref>) is because the exact scaling is an asymptotic late time statement and dependence on t_* drops. Note that while β_∞ and α_∞ are theory specific and independent of initial conditions, t_* is going to depend on a chosen initial state.
Before we move to testing Eq. (<ref>) using ab initio solutions of quantum dynamics, let us reiterate that this result originates from the Boltzmann equation and pertinent conservation laws. These are exactly the same constraints as used in a conventional scaling analysis <cit.>. Prescaling in the case of collision kernels being homogeneous functionals of momenta can therefore be understood as a direct consequence of the existence of scaling.
Prescaling in large-N vector model
We begin by benchmarking Eq. (<ref>) against the full quantum dynamics of a large-N vector model at small coupling λ. The resulting nonthermal fixed point, residing in the infrared, is characterized by (α_∞,β_∞)=(d/2,1/2) and was also realized in cold quantum gases <cit.>. This fixed point arises for large occupations f∼ 1/λ≫ 1 such that its description requires going beyond a standard kinetic theory analysis. Large-N kinetic theory addresses this regime due to inclusion of relevant resummations <cit.>. The corresponding collision kernel scales homogeneously with μ_α=1 and μ_β=-2, with the associated scattering matrix element void of scaling breaking terms.
We perform ab initio studies of this fixed point using the 2PI formalism following <cit.>.
Below the mass gap the equal-time statistical function F(t,t,|𝐩|) reduces to f(t,|𝐩|) <cit.>.
Following <cit.> we extract the prescaling factors A(t) and B(t) from the time evolution of integral moments of F(t,t,|𝐩|), e.g., B(t) = n(t)ℰ(t_0)/(ℰ(t)n(t_0)) (see Eq. (<ref>) in the Appendix).
In the upper right panel of Fig. <ref> we show how rescaling with A(t) and B(t)
leads to an early collapse of distributions at different times, while a considerable spread remains when rescaling with the fixed point exponents (left panel). The evolution of the extracted B(t) is shown in the lower panel to be well described by Eq. (<ref>) already at early times (dashed black line), and only asymptotes to the corresponding fixed point scaling behavior (<ref>) (solid green line).
Prescaling in isotropic QCD kinetic theory
We move now to studying prescaling dynamics in QCD, whose nonthermal fixed point plays an important role in our understanding of thermalization dynamics in weakly-coupled models of ultrarelativistic nuclear collisions <cit.>. We use QCD kinetic theory, where the evolution of the color and polarization averaged gluon distribution function f(t,) is described by 2↔ 2 and 1↔ 2 processes <cit.>: ∂_t f(t,) = C^2↔ 2[f](t,)+ C^1↔ 2[f](t,). Explicit expressions can be found in the Appendix, see Eq. (<ref>), and in <cit.>.
Nonthermal fixed points can be reached from a wide range of initial conditions including large occupation numbers <cit.>, which we implement via
f(t_i,) = n_0/g^2 exp[-^2/Q^2 ].
Here g^2 is the square of the coupling and n_0 is the initial occupation. We consider n_0=1 and g^2=10^-8. To obtain the precise late time behavior we initialize at t_iQ=0 and evolve for very long times until t_fQ=10^8. Results will be given in units of the characteristic energy scale Q as given by the maximum of ||^2 f(t_i,). We discuss explicitly only the pure gluon simulations where the scaling phenomenon is encountered after checking that our results do not change under the inclusion of quark/anti-quark dynamics.
A scaling analysis for the vacuum QCD collision kernel together with energy conservation σ=d+z=4 reveals the direct energy cascade fixed point (α_∞,β_∞)=(-4/7,-1/7), see <cit.>. However, the overall scaling of the elastic collision kernel is broken by the presence of the Debye mass m^2_D(t)∼∫ d^3 f(t,)/p ∼ A(t) B(t)^-2m̅^2_D that regulate soft elastic scatterings where m̅^2_D≡ m^2_D(t_0), see Eq. (<ref>).
The time-dependent (scaling) Debye mass can not generally be factored from the elastic scattering matrix element,
which breaks the assumption of overall scaling behavior of the collision kernel.
This breaking of scaling is visualized in the top panel of Fig. <ref>, where we show ||^2 f(t,||) rescaled with fixed point exponents. The rescaling for simulations with only inelastic scatterings (top right) shows a very clear collapse of all curves from shortly after initialization over an evolution of six orders of magnitude, but a spread cannot be removed for all times even by time dependent rescalings if one includes elastic scatterings (top left) due to the presence of the Debye mass. In the lower panel we extract prescaling exponents for only inelastic scatterings
C^1↔ 2 from different moments of the distribution function (see Eq. (<ref>)).
The resulting approach of β(t) to the fixed point value β_∞=-1/7 (dashed gray line) is demonstrated to be well-described over more than seven orders of magnitude in time by Eq. (<ref>) (dashed black line) with μ_α=3 and μ_β=-1, see Eq. (<ref>), even at surprisingly early times shortly after initialization. In the inset, we show that this solution initialized at a very early time t_0 Q∼ 2 with t_* obtained from β(t_0) ≡β_0 captures the evolution at intermediate times t Q∼ 10^4 only qualitatively. If we obtain t_* in the same way at a later time t_0 Q∼ 10^4, Eq. (<ref>) describes the late-time evolution quantitatively well as given by the solid black line. We do not display α(t) explicitly here and in the following as we find the scaling relation α(t)/β(t)=4 realized to a very good accuracy.
The conclusion here is that the separation of variables, see Eq. (<ref>), in QCD kinetic theory can thus not generally be performed, as m_D(t) leads to a mixing of time and (rescaled) momentum scales in the prescaling regime. With α_∞,β_∞ <0, the Debye mass decreases over time such that the violation of scaling becomes increasingly small. These two effects lead to the expectation that different momenta of the distribution function approach the fixed point on different timescales. This is visible already in the top left of Fig. <ref>, where (rescaled) momenta ≳ 4 are observed to scale well with fixed point exponents whilst smaller ones do not. We then expect scaling exponents from different moments to converge only asymptotically in the approach to the fixed point. This we corroborate in Fig. <ref>, where a spread in prescaling exponents obtained from different moments is observed to remain for the latest times reached.
Effect of scaling breaking terms in the Fokker-Planck approximation
The breaking of scaling inhibits prescaling exponents extracted from different moments to share the same universal prescaling dynamics. Nevertheless, at qualitative level the scaling dynamics can be reasonably modeled
via the Fokker-Planck (FP) approximation <cit.>. This approach assumes the dominance of small angle scatterings and has previously been used in the context of nonthermal attractors <cit.> and prescaling <cit.>.
We will compare our analytical results from the FP approximation to simulations using the full QCD collision kernel. The corresponding FP collision kernel allows us to factorize the
scaling-breaking Coulomb logarithm, which involves the ratio of the UV scale, the characteristic gluon energy ⟨ p⟩, and the IR scale, the Debye mass m_D
C^FP[f](t,)
=A(t)^3/B(t)log[⟨p̅⟩/A(t)^1/2m̅_D]C̃^FP[f_S](),
where we only display terms relevant for the scaling analysis and refer to Eq. (<ref>) for details.
Upon separating variables with associated constant D_2, see Eq. (<ref>), and relating α(t)=σβ(t) as before, we now obtain
∂_t B(t) = β_0/t_0 B(t)^2σ[1-σ/2log[B(t)]/log[⟨p̅⟩/m̅_D]].
We identified D_2=β_0/(t_0log[⟨p̅⟩/m̅_D]), which is an intricate self-consistent equation as log[⟨p̅⟩/m̅_D] depends implicitly on D_2 via f_S, see also the discussion below Eq. (<ref>).
We avoid the need to solve it as we utilize our simulations to obtain ⟨p̅⟩/m̅_D as a function of β_0/t_0.
Equation (<ref>) can be directly integrated, but then B(t) appears as an argument of a nontrivial transcendental function. A more useful approach is to derive from Eq. (<ref>) a second order differential equation for β(t)
β̈(t)/β(t) =β̇(t)^2/β(t)^2+14 β̇/t- (7β(t)+1)^2/t^2.
This equation is also subtle, as its second order character stays in contrast with the number of parameters needed to solve Eq. (<ref>), which requires specifying only β_0 at t_0. Indeed, there is a nontrivial constraint on initial data for Eq. (<ref>) directly following from a derivative of Eq. (<ref>):
β̇(t_0) ≡β̇_0 =β_0t_0[ 1 - β_0/β_∞ - 1/ log√(⟨p̅⟩/m̅_D)].
Its complexity arises from the dependence of ⟨p̅⟩/m̅_D on β_0 and t_0. What one can already see quickly is that β(t)=β_∞=-1/7 is a consistent late time solution of (<ref>), as it satisfies the constraint in the limit t_0→∞.
In Fig. <ref> the evolution of β(t) extracted from QCD kinetic theory (solid color lines) is shown to be captured well by Eq. (<ref>) (dashed black line) from times shortly after initialization of the system over more than seven orders of magnitude. We obtain this result by solving Eq. (<ref>) with initial condition at t_0 for β̇_0 determined by the constraint and β_0 (as well as log√(⟨p̅⟩/m̅_D)) extracted from the data. In the inset we show that solving Eq. (<ref>) shortly after initialization also describes the late-time dynamics qualitatively well, where the prescaling exponents from different moments retain a finite spread for the latest simulated times.
We now want to understand the prescaling dynamics in the vicinity of the fixed point. We can linearize Eq. (<ref>) in perturbations δβ(t) around the fixed point value β_∞=-1/7, which yields a power-law decay from below δβ(t) ∼ -1/t.
On top of this, the slow part of the solution to Eq. (<ref>) is
β(t) ≈ β_∞ + ∑_m=1^∞∑_n =0^m-1β_m, nlog(log(t Q))^n/log(t Q)^m
≈ β_∞[ 1+ 1/log(t Q)],
where we used Q as the reference scale but emphasize that the choice of a constant does not matter at late enough times. This logarithmic correction is a non-linear effect not captured by the linearization procedure. Similar late-time power-law <cit.> corrections from linearization and late-time logarithmic <cit.> corrections induced by the temporal evolution of the Coulomb logarithm were found for the Baier-Mueller-Schiff-Son <cit.> fixed point in longitudinal expanding plasma.
The simple power-law approach to the fixed point found in the absence of scaling breaking terms in Eq. (<ref>) is therefore enriched to involve both fast (power-law) and slow (inverse powers of logarithms and slower) behavior. This discussion is reminiscent of the transseries form <cit.> for late time dynamics of the energy-momentum tensor of matter undergoing longitudinal boost-invariant expansion <cit.>. There slow modes came from relativistic hydrodynamics and exponentially faster modes from transient excitations. Here slow modes come from the Debye mass breaking the homogeneity of the collision kernel with respect to rescalings of momenta and fast modes are the original prescaling excitations encountered already in (<ref>). Similarly to <cit.>, it is not difficult to gather finite order indications that the series containing slow modes (<ref>) is likely to have a vanishing radius of convergence with β_m+1,0/β_m,0∼ m. Curiously, the leading (at each m) doubly logarithmic term behaves geometrically: β_m+1,m/β_m,m-1 = -1. It would be interesting to develop systematic understanding of this behavior, including resummations of the resulting transseries.
In Fig. <ref> we visualize the attractive nature of the prescaling dynamics by extracting prescaling exponents from different initial conditions (solid color lines) and corresponding solutions to Eq. (<ref>) (dotted color lines). All simulations are initialized with variations in parameters of the class of initial condition used in this work apart from the data represented by green, which uses box initial conditions f(t_i,)=n_0/g^2 θ(Q-||). Comparing solid and dotted lines of the same color, we observe that the slow approach to the fixed point value from below is described well by Eq. (<ref>) in accordance with the discussion for Fig. <ref>. Deviations are observed at early times with β(t) approaching the fixed point value initially from above, where Eq. (<ref>) captures the fast dynamics in the fall-off region only qualitatively. Finally, the prescaling exponents extracted from different simulations and the corresponding solutions to Eq. (<ref>) are all found to converge to a universal late-time behavior, which we additionally show is well described by Eq. (<ref>) (solid black line). Furthermore, we want to emphasize the similarity between the behavior shown in Fig. <ref> and hydrodynamic attractors, where different solutions converge to a single universal curve which at sufficiently late times is described by relativistic hydrodynamics <cit.>.
The above analysis has an important bearing on the appearance of scaling. The regime when the highest order terms in the collision kernel dominate parametrically ends when the typical occupancy becomes of 𝒪(1). This is realized if t_f Q ≥α^-7/4_S <cit.>. At that time, we have a deviation of δβ(t)/β∼ 1/log( α^-7/4_S) ≲ 0.03 with g^2=10^-8. As a consequence of this, the system will therefore still show percent deviations from the fixed point values when the direct energy cascade ceases and ultraviolet modes |𝐩|/Q≥ 1 start to thermalize.
Conclusions
We studied the approach of isotropic and spatially homogeneous quantum many-body systems to nonthermal attractors.
Our results demonstrate that the prescaling is governed by a simple first-order ordinary differential equation obtained from the underlying dynamics via emergent conservation laws.
Our analytical prediction implies that prescaling entails infinitely many power-law corrections to constant scaling exponents. They conspire to a simple time off-set in the fixed point scaling.
We have successfully tested our simple formula for prescaling against ab initio simulations of a vector model relativistic QFT using 2PI formalism and QCD kinetic theory simulations. Our QCD kinetic theory simulations span eight orders of magnitude in time and provide the most accurate extraction of scaling exponents to date.
The exact scaling associated with nonthermal attractors requires the collision term to be a homogeneous functional of particle momenta at large occupations. For QCD kinetic theory this property is violated by the Debye mass term that regulates the Coulomb divergence in the elastic scattering matrix element. We demonstrate that exact scaling exponents are not reached during the lifetime of the system.
Using the Fokker-Planck approximation to QCD kinetic theory we show that the scaling-breaking Couloumb logarithm significantly enriches the prescaling dynamics. The late-time behavior is given by a factorial divergent series that includes inverse powers of logarithms and positive powers of double logarithms of time. This constitutes a striking structural similarity with theoretical descriptions of hydrodynamic attractors in the boost-invariant models of nuclear collisions.
Our work shows that prescaling is an unavoidable consequence of nonthermal attractors. Therefore our analytical predictions for prescaling can be verified experimentally in cold atom systems. Furthermore, we uncovered that scaling breaking terms generate rich prescaling dynamics that bares similarities to transseries in the context of hydrodynamic attractors. It would be fascinating to utilize the enormous degree of control in cold atom systems to induce scaling breaking terms and experimentally discover the phenomenology of transseries.
Acknowledgments
We thank
I. Aniceto,
J. Berges,
K. Boguslavski,
J. Brewer,
A. Kurkela,
A. Mikheev,
J. Noronha,
B. Scheihing-Hitschfeld,
S. Schlichting,
A. Serantes,
R. Venugopalan,
and
Y. Yin
for useful discussions and comments on the draft.
The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no
INST 40/575-1 FUGG (JUSTUS 2 cluster), and DFG under the Collaborative Research Center SFB 1225 ISOQUANT (Project-ID 27381115) and the Heidelberg STRUCTURES Excellence Cluster under Germany's Excellence Strategy EXC2181/1-390900948.
The work of AM is funded by DFG – Project number 496831614. We would like to thank KITP for its hospitality during the program “The Many Faces of Relativistic Fluid Dynamics" supported by the National Science Foundation under Grant No. NSF PHY-1748958.
§ APPENDIX
§.§ Extraction of scaling exponents
Under the prescaling ansatz for an isotropic system, Eq. (<ref>), we introduce moments of the distribution function
n_m(t) = ∫d^d /(2π)^d ||^m f(t,) = A(t) B(t)^-(d+m)n̅_m,
whose evolutions are given by the dynamics of prescaling exponents <cit.>
d/dlog tlog(n_m(t)) =α(t)-(d+m)β(t).
We can thus extract prescaling exponents from the evolution of the moments, for example,
α_m(t) =(m+d)(m+1+d) d/dlog tlog[ n^1/m+d_m(t)/n^1/m+1+d_m+1(t)], β_m(t) = d/dlog tlog(n_m(t)/n_m+1(t)).
We can similarly obtain the general A(t) and B(t) from the moments of the distribution function (or equivalently from moments of the equal-time statistical function F(t,t,|𝐩|) for the inverse cascade fixed point) according to
A(t) =[n^1/(d+m)_m(t)/n^1/(d+m+1)_m+1(t)n̅^1/(d+m+1)_m+1(t)/n̅^1/(d+m)_m]^(d+m)(d+m+1), B(t) = n_m(t)/n_m+1(t)n̅_m+1/n̅_m,
where n̅_m≡ n_m(t_0) such that B(t_0)=1.
§.§ QCD kinetic theory and prescaling
In this work we study QCD kinetic theory which includes 2↔ 2 and 1↔ 2 collinear scattering terms. We will give the corresponding equations for the gluon sector here and refer to <cit.> for the complete expressions including other particle species. The gluon collision kernels are parametrized as
C^2↔ 2[f](t,) = -1/8p(N^2_c-1)∫d^3 d^3 ^' d^3 ^'/(2π)^9 2k2p^' 2k^' |ℳ^gg_gg|^2(p,k,p^',k^') (2π)^4 δ^(4)(p^μ+k^μ-p^'μ-k^'μ)
×[f_ f_ (1+f_^')(1+f_^')-f_^' f_^' (1+f_)(1+f_) ]
C^1↔ 2[f](t,) = -(2π)^3/8 p^2(N^2_c-1)∫_0^∞ dp^' dk^'{γ^g_gg(p;p^',k^') δ^(1)(p-p^'-k^') [f_p(1+f_p^')(1+f_k^')-f_p^' f_k^' (1+f_p) ].
. -2γ^g_gg(p^';p,k^') δ^(1)(p^'-p-k^') [f_p^'(1+f_p)(1+f_k^' )- f_p f_k^' (1+f_p^') ]},
where we used the abbreviations f_=f(t,), p=||, and =/p is the direction of collinear splitting with corresponding rate γ^g_gg. |ℳ^gg_gg|^2 is the 2↔ 2 scattering matrix element averaged over spin and color degrees of freedom. In vacuum, the corresponding expression <cit.>
|ℳ^gg_gg|^2(p,k,p^',k^')=16(N^2_c-1) N^2_c g^4 (3-s_M t_M/u^2_M - s_M u_M/t^2_M-t_M u_M/s^2_M)
contains an infrared divergent term (u_M-s_M)/t_M∼ 1/q^2 (also in the u_M-channel) with q the t_M-channel momentum transfer for a soft gluon exchange and u_M,s_M,t_M denote the usual Mandelstam variables. This is regulated by inclusion of necessary physical interactions with the medium, where the leading thermal corrections are obtained as <cit.>
u_M-s_M/t_M→u_M-s_M/t_Mq^2/q^2+ξ^2_g m^2_D(t)= u̅_M-s̅_M/t̅_Mq̅^2/q̅^2+ξ^2_g A(t)m̅^2_D
with ξ_g = e^5/6/2. The inclusion of screening effects however makes the scattering matrix element not invariant under rescaling as indicated due to different scaling of the Debye mass
m^2_D = 2g^2N_c∫d^3 /(2π)^3 p f(t,)=A(t)B(t)^-2m̅^2_D.
Moreover, the time-dependent Debye mass enters the scattering matrix element via Eq. (<ref>) in the denominator such that it can not be factored out thereby violating the assumption of overall scaling for the elastic collision kernel.
We now show that the inelastic collision kernel (<ref>) scales with μ_α=3 and μ_β=-1 under prescaling in the non-expanding system (<ref>).
We consider only the first term since both have the same scaling behavior. Crucially, we need to know the scaling behavior of the splitting rate, which can be extracted from its two prevalent limiting regimes for soft gluon radition z=p^'/p≪ 1: the Bethe-Heitler (BH) limit in which interferences between successive scatterings are negligible and the Landau-Pomeranchuk-Migdal (LPM) limit in which successive scattering events by the medium interfere destructively. The rate in the respective limit reads
γ^g_gg(p;p^',k^')|^z≪ 1_BH∼q̂(μ) p/m^2_D |_μ = e m_D, γ^g_gg(p;p^',k^')|^z≪ 1_LPM∼√(q̂(μ) p),
where we only included those terms relevant for a scaling analysis with diffusion coefficient
q̂(μ) ∼logμ^2/2m^2_D∫d^3 /(2π)^3 f_(1+f_)≃logμ^2/2 A(t)B(t)^-2m̅^2_D A(t)^2 B(t)^-3∫d^3 /(2π)^3 f_S, f_S,
in this overoccupied scenario. For q̂(μ)|_μ=em_D in the BH limit, we thus have q̂(μ)=A(t)^2 B(t)^-3q̅̂̅ and
γ^g_gg(p;p^',k^')|^z≪ 1_BH∼A(t)^2 B(t)^-3q̅̂̅ B(t)^-1p̅/A(t) B(t)^-2m̅_D∼ A(t) B(t)^-2(t) γ̅^g_gg|^z≪ 1_BH.
For the LPM limit, μ is to next-to-leading-logarithmic order given self-consistently via
μ^2∼√(q̂(μ)p)∼√(A(t)^2 B(t)^-3log( μ^2/2A(t)B(t)^-2m̅^2_D) B(t)^-1)∼ A(t) B(t)^-2√(log( μ^2/2A(t)B(t)^-2m̅^2_D)),
which show that μ scales self-consistently as μ^2 ∼ A(t) B(t)^-2 such that the time dependence drops out of log(μ^2/(2m^2_D)). Accordingly, for the LPM limit we thus find again
γ^g_gg(p;p^',k^')|^z≪ 1_LPM∼√(q̂(μ) p)∼√(A(t)^2 B(t)^-3q̅̂̅(μ) B(t)^-1p̅)∼ A(t) B(t)^-2γ̂^g_gg |^z≪ 1_LPM
The rate therefore scales like the Debye mass in both limits and we will adopt this scaling for the complete rate γ^g_gg∼ A(t) B(t)^-2γ̅^g_gg. This leads to the overall scaling prediction
C^1↔ 2[f](t,) ∼1/B(t)^-2p̅^2∫_0^∞ B(t)^-2 dp̅^' dk̅^' A(t) B(t)^-2γ̅^g_gg(p̅;p̅^',k̅^') B(t) δ^(1)(p̅-p̅^'-k̅^')
× A(t)^2[f_S,p̅(f_S,p̅^'+f_k̅^') - f_S,p̅^' f_S,k̅^']
∼ A(t)^3 B(t)^-1 C^1↔ 2[f_S]()
such that we can identify μ_α=3 and μ_β=-1 as anticipated. C^1↔ 2 will therefore lead to the direct energy cascade fixed point (α_∞,β_∞)=(-4/7,-1/7) as we demonstrate in Fig. <ref> of the main text.
The scaling analysis for the elastic collision kernel in the absence of the Debye mass has been performed in <cit.> and can simply be generalized to the prescaling case with again μ_α=3 and μ_β=-1. This scaling analysis however needs to be augmented by the inclusion of the Debye mass as we discussed above, which leads to an effective regulation of the divergent soft contributions to the elastic collision kernel and prevents one from extracting an overall scaling behavior thereof.
§.§ Fokker-Planck approximation
We assume the gluons to interact via elastic small-angle scatterings such that the collision kernel takes a FP form <cit.>
C^FP(t,) ∼ln⟨ p ⟩(t)/m_D(t)[∫d^3 /(2π)^3 f_ (1+f_) ∂^2_p f_ + ∫d^3 /(2π)^3 k 2 f_ ∇⃗_·( /p (1+f_)f_) ]
≡ln⟨ p ⟩(t)/m_D(t)C̃^FP[f](t,),
where we highlighted the contributions relevant for a scaling analysis and here ⟨…⟩≡∫d^3 /(2π)^3… f(t,)/∫d^3 /(2π)^3 f(t,). The advantage of the FP approximation for a prescaling analysis becomes apparent here, as the contribution due to the Debye mass in the elastic QCD scattering matrix element is simply factorized into a logarithm of the characteristic UV and IR scale.
Plugging in the prescaling ansatz, we then find directly that
C^FP[f](t,) = ln[ ⟨p̅⟩/A(t)^12m̅_D] A(t)^3 B(t)^-1C̃^FP[f_S]().
|
http://arxiv.org/abs/2307.03873v1 | 20230708012434 | Why does dissolving salt in water decrease its dielectric permittivity | [
"Chunyi Zhang",
"Shuwen Yue",
"Athanassios Z. Panagiotopoulos",
"Michael L. Klein",
"Xifan Wu"
] | cond-mat.soft | [
"cond-mat.soft",
"cond-mat.dis-nn",
"physics.chem-ph"
] |
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
[email protected]
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
Institute for Computational Molecular Science, Temple University, Philadelphia, Pennsylvania 19122, USA
Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, USA
[email protected]
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
Institute for Computational Molecular Science, Temple University, Philadelphia, Pennsylvania 19122, USA
The dielectric permittivity of salt water decreases on dissolving more salt. For nearly a century, this phenomenon has been explained by invoking saturation in the dielectric response of the solvent water molecules. Herein, we employ an advanced deep neural network (DNN), built using data from density functional theory, to study the dielectric permittivity of sodium chloride solutions. Notably, the decrease in the dielectric permittivity as a function of concentration, computed using the DNN approach, agrees well with experiments. Detailed analysis of the computations reveals that the dominant effect, caused by the intrusion of ionic hydration shells into the solvent hydrogen-bond network, is the disruption of dipolar correlations among water molecules. Accordingly, the observed decrease in the dielectric permittivity is mostly due to increasing suppression of the collective response of solvent waters.
Why does dissolving salt in water decrease its dielectric permittivity
Xifan Wu
======================================================================
In chemistry and biology, water is widely referred to as the universal solvent <cit.>. As salts dissolve in water, the anomalously large dielectric permittivity of water promotes the solubilization of salt by screening interionic Coulomb interactions. At the same time, the dielectric response of water is influenced by the presence of dissolved salts <cit.>. Almost 100 years ago, it was found that the static dielectric permittivity of sodium chloride (NaCl) solution decreases as more salt is dissolved <cit.>. Later, more sophisticated experiments revealed a nonlinear behavior in which dielectric decrement slows down at high solute concentrations <cit.>. A theoretical explanation of this phenomenon was conceived soon after the first experiment. As stated in their dielectric saturation theory, Debye <cit.> and Sack <cit.> envisioned the formation of hydration shells due to the tendency of water dipoles to be aligned along electric fields of dissociated ions. Debye further estimated that ionic electric fields are strong enough to saturate the polarizability of water molecules near the ions and therefore lower the dielectric response <cit.>. Because of its built-in physical intuition, dielectric saturation has been, to date, the most adopted theory to explain dielectric decrement in salt water <cit.>.
The past half-century has witnessed significant progress in understanding water through principles of quantum mechanics and statistical physics <cit.>. This progress calls into question the dielectric saturation explanation. Indeed, consensus has been reached that the high dielectric permittivity of water is closely associated with correlated dipole fluctuations of water molecules on the underlying hydrogen(H)-bond network <cit.>. However, this collective dipolar response is missing in the picture of dielectric saturation which mainly concerns the suppressed dielectric response of individual water molecules <cit.>. More disturbingly, based on classical electrodynamics, dielectric saturation is estimated to occur on water molecules that are a few angstroms away from ions <cit.>. The above length scale is comparable to the estimated de Broglie wavelength of electrons at room temperature <cit.>. Physical interactions at such length scales are governed by quantum mechanics rather than a classical description. In this regard, density functional theory (DFT)-based <cit.> ab initio molecular dynamics (AIMD) <cit.> provides an ideal framework to predict properties of liquids from quantum mechanical principles. Indeed, recent AIMD simulations found that polarizabilities of water molecules in ionic first hydration shells are only slightly different from that in neat water <cit.>, which contradicts the dielectric saturation hypothesis.
Due to the long-range nature of the dipole-dipole interaction and the disordered liquid structure, the prediction of dielectric response in water demands both a spatially extensive model containing many hundreds of water molecules and a simulation time beyond nanoseconds <cit.>. However, AIMD simulations of such large timescale and system size are simply not feasible using current computer architectures. Thus, to date, dielectric decrement has been mostly studied using molecular dynamics with classical force fields, and the effect of electronic polarizability has been neglected <cit.>.
Herein, we overcome the challenge by studying dielectric decrement by combining AIMD and deep neural networks (DNNs) <cit.>. The liquid structures of NaCl solutions are simulated by a DNN that explicitly incorporates long-range electrostatic interactions <cit.> with periodic simulation cells containing about 4000 water molecules. Importantly, the potential is trained on DFT calculations based on the strongly constrained appropriately normed (SCAN) functional <cit.>. In addition, a second DNN <cit.> is trained separately for centers of electronic orbitals, in terms of maximally localized Wannier functions <cit.>. Notably, this second DNN allows us to rigorously partition the electronic charge density into contributions from dipole moments of individual water molecules. The dual DNNs enable efficient computations of dielectric permittivity at the DFT accuracy. (See Supplemental Material <cit.> for more details on this methodology.)
Based on linear response theory, the static dielectric permittivity of NaCl solutions, ε_NaCl(aq), is related to the fluctuation of the total dipole moment, M, by <cit.>
ε_NaCl(aq) =⟨M^2⟩/3 V k_B T ε_0+ε_∞
=⟨(M_W(aq)+M_I(aq))^2⟩/3 V k_B T ε_0+ε_∞
=ε_W(aq)+ε_W(aq)-I(aq)+ε_I(aq)+ε_∞
where V, k_B, T, and ε_0 are the system volume, Boltzmann constant, temperature, and vacuum permittivity, respectively. ε_∞ is the electronic contribution in the high-frequency limit. As expected, the theoretical ε_∞ are small values around 1.88-1.99 at concentrations under consideration. We report the computed dielectric permittivity of NaCl solutions in Fig. <ref> together with experimental data. Note that both results have been normalized to enable a better comparison of dielectric decrement behavior. There is good agreement between experiments and present calculations. In particular, the nonlinear behavior in dielectric decrement observed in experiment is well reproduced. The dielectric permittivity drops steeply at low concentrations, but its slope becomes gradually flattened as solute concentration increases. Notably, the nonlinearity generates a bowing feature in dielectric decrement. Absolute values of the computed dielectric permittivity are reported in Supplemental Material Table 1 <cit.>. It should be noted that the predicted dielectric permittivity of neat water by SCAN functional is 102.5, which is larger than the experimental value of 78. The overestimation of the dielectric permittivity is consistent with a previous study employing the SCAN functional <cit.>, and this overestimation is particularly attributed to the self-interaction error in the SCAN functional that over-strengthens H-bonds. The slightly overstructured liquid water has been widely reported in literature <cit.> and its effects on observables can be approximated by the effects of decreasing the temperature, which does not affect our conclusions.
In NaCl solutions, the fluctuation of the overall dipole moment, M, involves contributions from both water molecules, M_W(aq), and ions, M_I(aq). Therefore, the dielectric permittivity, ε_NaCl(aq) in Eq. <ref> is composed of the self-terms, ε_W(aq) and ε_I(aq) whose dipole fluctuations are restricted to water molecules and solvated ions only, and the cross-coupling term ε_W(aq)-I(aq) reflecting dipole fluctuations in water induced by the movements of ions or vice versa. The computed values of above terms are presented in the inset of Fig. <ref>. Notably, ε_NaCl(aq) is dominated by ε_W(aq) at all concentrations, which agrees with previous findings <cit.>. Thus, dielectric decrement observed in NaCl solutions is due to the weakened dielectric response of solvent water molecules.
The dielectric component ε_W(aq) due to solvent water can be further evaluated via the dipolar correlation formalism proposed by Kirkwood <cit.> as
ε_W(aq)=ρμ^2 G_K/3 k_B T ε_0,
where ρ and μ denote water number density and average dipole moment per water molecule respectively, and G_K is the so-called correlation factor that measures the total angular correlations among water dipoles. In polar liquids, G_K is obtained by the integration of the dipolar correlation function as G_K=∫𝒞(r)dr=1/N∑_i=1^N∑_j=1^N μ̂_i·μ̂_j, where μ̂_i is the unit vector of the ith molecular dipole and N is the number of water molecules. The dipolar correlation is defined as 𝒞(r)=⟨d(0)·d(r)⟩, accounting for the spatial correlation between the dipolar density as a function of distance, r. Because of the discretized nature of water molecules, the dipolar density is defined as d(r)=∑_i=1^N μ̂_i δ(r-r_i) with r_i denoting the position vector of the ith water molecule.
In neat water, both the dipole moment, μ, and the correlation factor, G_K, are largely enhanced by the underlying H-bond network, leading to the anomalously large dielectric permittivity <cit.>. In NaCl solutions, as shown in Fig. <ref> (a), the correlation factor, G_K, the water number density, ρ, and the water dipole moment, μ, all decrease as increasing amounts of salt dissolved, which according to Eq. <ref> leads to dielectric decrement.
The effect from the disrupted H-bond network
As seen in Fig. <ref> (a), dielectric decrement of NaCl solutions is mostly attributed to the decreased correlation factor, G_K, relative to that of neat water. Thus, the strong correlation among dipole moments in neat water is significantly suppressed in salt solutions. In neat water, the large G_K is closely associated with the tetrahedral H-bond structure, in which a water molecule at the center of a tetrahedron is H-bonded with four neighboring water molecules. The directions of dipole moments of any two H-bonded water molecules, therefore, point in a similar direction, resulting in a positive μ̂_i·μ̂_j, which gives rise to the first positive sharp peak at 2.7 Å in the dipolar correlation function in Fig. <ref> (b). Under the influence of the directional H-bonding, dipole moments on vertices of a tetrahedron also prefer to be aligned in a similar direction to some extent, which yields a second positive peak around 5.1 Å in Fig. <ref> (b). In the same fashion, the dipolar correlation propagates to the third coordination shell and beyond.
The H-bond network is disrupted increasingly as more salt is dissolved. Salt ions exert electrostatic fields that can attract water molecules by competing with the H-bonding. In the close vicinity of ions, water molecules hydrate the ions by orienting their electric dipole moments towards the ions, thereby lowering the electrostatic energy of the system, as schematically shown in Fig. <ref> (b). For a sodium cation, the first hydration shell can be described as a relatively tight sphere comprised of about 5 or 6 water molecules, whose oxygen is attractive to the cation at the center <cit.>. On the other hand, the first hydration shell of a chloride ion is a relatively large sphere composed of as many as 6-8 water molecules whose protons are attracted to the chloride lone pair electrons <cit.>.
Because of the intrusion of the hydration shells, water molecules in the solvent are now divided into two distinct categories: the “hydration (H) water” inside the ionic hydration shells and the “bulk (B) water” outside it. As such, the pattern of dipolar correlation is fundamentally revised. As shown in Eq. <ref>,
G_K =∫[𝒞^B(r)+𝒞^H(r)+𝒞^BH(r)]dr
=G_K^B+G_K^H+G_K^BH,
the total correlation factor G_K involves the self-terms of G_K^B (G_K^H) by dipolar correlation restricted to “bulk water” (“hydration water”) only, and the coupling term G_K^BH due to the dipolar correlation between “bulk water” and “hydration water”. The above components in correlation factors, relative to neat water, are presented in Fig. <ref> (a). (See: Supplemental Material <cit.> for more details.)
As seen in Fig. <ref> (a), the reduction in the overall correlation factor, G_K, is mostly from G_K^H, which describes the correlation among “hydration water”. This is because water molecules in hydration shells are constricted by the ion-water attraction instead of H-bonding. Within a hydration shell, the cation (anion)-water attraction reorientates the dipole moments from an H-bonding direction to a central-force direction pointing outwards (towards) ions. As such, the dipolar correlation between two neighboring “hydration water” molecules is thereby significantly suppressed. This is evidenced by the sharp negative peak at ∼ 2.7 Å in the dipolar correlation function Δ𝒞^H(r) as plotted relative to neat water in Fig. <ref> (c). Moreover, the absence of H-bonding even causes anti-correlations between two “hydration water” molecules located on the opposite sides of a single ion as schematically shown by opposite directions of water molecular dipoles in the inset of Fig. <ref> (c). Therefore, the aforementioned positive peak of neat water in Fig. <ref> (b) due to correlated dipole moments on vertices of a tetrahedron at 5.1 Å disappears. Instead, it is replaced by two negative peaks at 4.8 and 6.1 Å, which are caused by the anti-correlated water dipoles in hydration shells of Na^+ and Cl^- ions, respectively. At long range, water molecules in a hydration shell, in principle, should be correlated to those in another hydration shell. However, such correlations are also weaker than those in neat water as expected in Fig. <ref> (c). As concentration increases, the loss of G_K^H should accumulate linearly, which is responsible for most of the linear dielectric decrement in salt water.
Of course, “hydration water” is H-bonded to “bulk water”, and in this way, the H-bond network is partially restored. Nevertheless, the reconstructed H-bond structure deviates from that found in neat water. Within a hydration shell, two water molecules located on opposite sides of a single ion are anti-correlated, as mentioned above. Because of the highly directional nature of H-bonding, the anti-correlation extends to the correlation between one “hydration water” molecule and one “bulk water” molecule that is H-bonded to another “hydration water” molecule at the other side of the ion, as schematically shown by the opposite direction of green arrows in the inset of Fig. <ref> (d). Again, these anticorrelations can be identified as a broad negative peak centered at 8 Å, which weakens the dipolar correlation. As a result, G_K^BH also contributes to the decreased overall correlation factor of G_K relative to neat water, as shown in Fig. <ref> (a). Moreover, G_K^BH plays a surprisingly key role in the nonlinear dielectric decrement as evidenced by its arc shape in Fig. <ref> (a). This nonlinearity is an intrinsic property because G_K^BH describes the correlation between the dipolar density of “bulk water” d^B(r) and the dipolar density of “hydration water” d^H(r), and its value depends on the existence of both types of water, i.e., ⟨d^B(0)·d^H(r)⟩. In neat water, G_K^BH=0 since the dipolar density of “hydration water” d^H(r) is zero. As salt dissolve in water, hydration shells appear in the solution, and the absolute value of G_K^BH starts to increase, reaching its maximum at about 2.3 M, in which the NaCl solution is roughly equally occupied by “bulk water” and “hydration water”. After the maximum, G_K^BH decreases with further elevated concentrations. In principle, it will vanish again at d^B(r)=0, when the entire solution is completely occupied by hydration shells.
The tetrahedral H-bond network is expected to recover in the “bulk water” outside the hydration shell. The dipolar correlation among “bulk water” molecules is captured by the G_K^B component of the correlation factor. Indeed, the analysis in Fig. <ref> (a) shows that G_K^B of NaCl solutions at all concentrations is little different from neat water. Thus, the large decrease in the correlation factor, G_K, in salt water is mostly due to the disrupted H-bond network in the “hydration water”.
Excluded volume effect
Due to short-range repulsion, ions and water molecules are separated by 2-4 Å. This extra volume demanded by ions is no longer accessible to water molecules, and the water number density is therefore decreased. In the literature, this is referred to as the excluded volume effect <cit.>. According to Eq. <ref>, this effect should lead to the decreased dielectric permittivity. Indeed, the present computations show that the excluded volume effect makes a small contribution to dielectric decrement, in which the water number density decreases slightly with increasing solute concentration as shown in Fig. <ref> (a). Since the repelled volume by ions is proportional to the salt concentration, dielectric decrement due to the excluded volume effect is indeed linear as expected.
Local field effect
Hydrated ions, like all charged defects, change the electrostatic potential profile throughout the solution. As expected, water molecules nearby an ion are polarized in a different manner from neat water. In condensed matter physics, related phenomena have been already identified, for example around defects in semiconductors or at interfaces in solid materials, and they have long been recognized as the local field effect <cit.>. There is consensus that a proper description of local field effects, particularly for regions close to charged defects, demands electronic structure details computed from quantum mechanics. Based on DFT, the present DNN simulations yield a dipole moment, μ=2.85 (2.91) Debye for the “hydration water” of the cation (anion), which is only slightly smaller than the value of 2.99 Debye in neat water. This suggests that the capability of ions to polarize the water dipole is comparable to that of H-bonding. Indeed, it is also consistent with the recent theoretical discovery that molecular polarizabilities of the “hydration water” are only marginally different from that in neat water <cit.>. Since H-bonding is mostly electrostatic in nature, it strongly indicates that water molecules nearby ions are far from being saturated by ions’ local fields. Nevertheless, the local field effect also contributes slightly to dielectric decrement as indicated by Eq. <ref>. Because the μ of the “hydration water” is only a little smaller than in neat water, μ^2 of NaCl solutions drops slowly as a function of concentration, as shown in Fig. <ref> (a).
In addition to the SCAN ab initio simulations, we also simulated the dielectric permittivity using the classical OPC water model <cit.>. As shown in Supplemental Material <cit.>, the results obtained using the OPC model agree well with those from the SCAN-DFT approach. A notable distinction between the OPC model and the SCAN-DFT model is that the OPC model is a rigid model with a fixed dipole moment of 2.48 D, indicating that the DFT approach is necessary for accurately capturing the local field effect.
In conclusion, dielectric decrement, as a century-old problem, has been extensively studied over decades.
However, a critical question remains unresolved in the field regarding the main origin behind the dielectric decrement—whether it is the dielectric saturation effect <cit.> or the loss of dipolar correlation on the H-bond network <cit.>.
To provide an unambiguous answer, theoretical simulations must explicitly include both a polarizable model of water molecules and an accurate model of H-bonding, which can account for the dielectric saturation effect and correlation effect simultaneously. Importantly, the polarizable models of water molecules should be described from first principles at the quantum mechanics level, because the length scale of dielectric saturation effect is about a few angstroms which is comparable to the de Broglie wavelength of electrons at room temperature. In this work, we achieve the above goal by reproducing dielectric decrement in NaCl solutions on the DFT level using advanced DNNs. The results unambiguously determine that the dielectric decrement in NaCl solutions is dominated by the loss of correlations between water molecules due to the intrusion of ionic hydration shells into the H-bond network, while the contribution from dielectric saturation effect is small.
Importantly, the present computations provide a quantitative explanation of dielectric decrement in salt water; we found that the linear dielectric decrement is due to the loss of correlation within hydration shells, while nonlinear dielectric decrement is due to the loss of correlation between water in hydration shells and bulk water.
We thank Roberto Car, Linfeng Zhang, and Han Wang for fruitful discussions. This work was supported by National Science Foundation through Awards No. DMR-2053195. We also acknowledge support from the “Chemistry in Solution and at Interfaces” (CSI) Center funded by the U.S. Department of Energy through Award No. DE-SC0019394. This research used resources of the National Energy Research Scientific Computing Center (NERSC), which is supported by the U.S. Department of Energy (DOE), Office of Science under Contract No. DE-AC02-05CH11231. This research includes calculations carried out on HPC resources supported in part by the National Science Foundation through major research instrumentation grant number 1625061 and by the U.S. Army Research Laboratory under contract No. W911NF-16-2-0189. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.
|
http://arxiv.org/abs/2307.04094v1 | 20230709043319 | Class-Incremental Mixture of Gaussians for Deep Continual Learning | [
"Lukasz Korycki",
"Bartosz Krawczyk"
] | cs.LG | [
"cs.LG",
"I.5.0; I.5.1"
] |
Class-Incremental Mixture of Gaussians for Deep Continual Learning
Lukasz Korycki
Virginia Commonwealth University
[email protected]
Bartosz Krawczyk
Virginia Commonwealth University
[email protected]
August 12, 2023
==================================================================================================================================================
Continual learning models for stationary data focus on learning and retaining concepts coming to them in a sequential manner. In the most generic class-incremental environment, we have to be ready to deal with classes coming one by one, without any higher-level grouping. This requirement invalidates many previously proposed methods and forces researchers to look for more flexible alternative approaches. In this work, we follow the idea of centroid-driven methods and propose end-to-end incorporation of the mixture of Gaussians model into the continual learning framework. By employing the gradient-based approach and designing losses capable of learning discriminative features while avoiding degenerate solutions, we successfully combine the mixture model with a deep feature extractor allowing for joint optimization and adjustments in the latent space. Additionally, we show that our model can effectively learn in memory-free scenarios with fixed extractors. In the conducted experiments, we empirically demonstrate the effectiveness of the proposed solutions and exhibit the competitiveness of our model when compared with state-of-the-art continual learning baselines evaluated in the context of image classification problems.
§ INTRODUCTION
While the initial research done in the domain of continual learning from stationary data was, in large part, oriented towards task-incremental solutions, more recent works attempt to address generalized cases consisting of purely class-incremental and data-incremental (also known as domain-incremental) settings <cit.>. These scenarios are usually more universal but also more challenging and restrictive mainly due to the lack of task or even class labels. Such settings make many of the previously proposed solutions practically useless, for example, the methods based on memory-free regularization <cit.>, which are not capable of discriminating between older and new classes, even if they address the catastrophic forgetting problem <cit.>. Although the most standard experience replay methods can be effectively applied in the class-incremental scenarios <cit.>, there has been also a search for alternative approaches that could provide natural capabilities required for such cases. A significant group of methods can be identified based on their reliance on centroids (or prototypes) combined with the nearest-centroid classification methods <cit.>. Since centroids can be independently added to the classifier, they are examples of methods that can be very smoothly incorporated into class-incremental scenarios, offering almost no interference in the latent space.
In this work, we explore an advanced version of these alternatives by proposing integration of the gradient-based Gaussian mixture model with a class-incremental deep continual learning framework, called MIX. In fact, it requires us to tackle three major problems at the same time: (i) gradient-based mixture training, (ii) combining it with a trainable deep feature extractor and, finally, (iii) making it suitable for class-incremental scenarios. To achieve these goals, we introduce a set of dedicated losses, configurations and methods, providing a probabilistic classifier on top of a feature extractor and within a model capable of learning end-to-end. This opens many potential research directions that could exploit the well-modeled statistical properties of Gaussians. In addition to that, we show that our class-incremental mixture model, analogously to the centroid-driven algorithms, is characterized by some inherent properties useful in continual learning scenarios. They allow it for much better separation of concepts at the level of the classification module, leading to significant improvements in memory-free scenarios when pre-trained extractors are used. Through an extensive empirical study, we analyze different configurations of our method, provide the reader with some intuition about its parameters and show its competitiveness in the context of other continual learning algorithms.
§ RELATED WORKS
Continual learning: In continual learning, our focus should be on effective incorporation of the arriving data and retention of the acquired knowledge <cit.>. The main problem that learning algorithms will encounter here is catastrophic forgetting <cit.>. The most straightforward approaches involve replaying instances of previously seen tasks or classes while learning new ones <cit.>. Instead of putting instance-level constraints on the learning directions, we can apply direct adjustments to the loss using dedicated regularization terms. The most commonly used approach involves utilizing the knowledge-distillation loss <cit.> combined with standard cross-entropy <cit.> or maintaining importance weights to distinguish parameters that are crucial for the retention <cit.>. These methods generally cannot be used in more realistic class-incremental or data-incremental scenarios (if they do not use memory buffers), since they cannot learn how to discriminate new instances from the older ones <cit.>. Other approaches may employ masking to isolate parameters per task to keep them static when learning new ones <cit.>, use dynamic structures to expand the network for new concepts <cit.>, utilize ensemble techniques <cit.> or meta-learning and hypernetworks <cit.>. Finally, interesting alternative approaches focus on hybridizing the neural networks with different machine learning methods, e.g. decision trees <cit.> or centroid-driven algorithms <cit.>. The latter group of methods has been found especially useful in one-class-incremental scenarios, since, as mentioned in the introduction, centroids can be stored independently per class, allowing for natural class-incremental learning without additional interference at the level of a classifier. In this work, we follow these approaches and replace basic centroids learned separately from the feature extractor with more complex end-to-end mixture models.
Mixture optimization: Various techniques can be applied for the task of fitting the mixture model to given data. The most standard approach utilizes the EM algorithm, which can be realized in both offline and online settings <cit.>. While EM provides a stable framework for learning the mixtures – in terms of mathematical constraints and convergence – it is critically limited when it comes to working with high-dimensional data and feasible memory consumption <cit.>. On top of that, this algorithm is intrinsically incapable of being fully integrated with neural networks, preventing it from achieving joint end-to-end deep learning and benefiting from dedicated features. An alternative approach involves gradient-based optimization <cit.>. This method has been proved to be able to provide more scalable and flexible algorithms capable of working in challenging scenarios with high-dimensional data and in online settings. Most importantly, the gradient-based approach naturally enables combining the model as a classifier with a trainable deep feature extractor <cit.>, allowing for extending the optimization process with the input space adjustments. Methods utilizing such a compound learning process showed much evidence of its usability in offline and unsupervised scenarios, while at the same time encouraging researchers to develop further extensions and improvements <cit.>. Given all of the characteristics, we decided to use this approach in our scenario of continual learning.
§ MIXTURE OF GAUSSIANS FOR CLASS-INCREMENTAL LEARNING
Formally, the general goal of our work is to incrementally learn a classification model defined as ϕ^(t): 𝒳→𝒞 that can effectively incorporate subsequent class batches ⟨ (X^(1), c=1), (X^(2), c=2), ..., (X^(t), c=t)⟩, where X^(t) contains instances x only for a given class c. After t classes the model ϕ^(t) should aim at minimizing the loss for the current class c=t and all previously observed ones:
ℒ^(t) = ∑_c=1^t∑_n=1^N_cℒ^(c)(ϕ^(t)(x_n^(c))),
where x_n^(c)∈X^(c) and ℒ^(c) can be any supervised loss.
Additionally, since we are interested in deep learning, we define the whole trainable model as a tuple ϕ^(t)=⟨ℱ^(t), 𝒢^(t)⟩ consisting of a feature extractor ℱ^(t) and a classifier 𝒢^(t) jointly aggregating knowledge from t classes. The model makes prediction by classifying the features provided from the extractor ϕ^(t)(x)=𝒢^(t)(ℱ^(t)(x))=𝒢^(t)(x̂). In this work, we aim at employing the mixture of Gaussians as a jointly trained incremental classifier. Although the model learns from dedicated features x̂, in the next section, we use x for the sake of simplicity of notation.
§.§ Generic supervised mixture model
Formally, in a standard unsupervised setting the density for a given point 𝐱 can be expressed using a multivariate normal distribution defined as:
𝒩(𝐱|μ_k, Σ_k) = 1/√((2π)^D|Σ_k|)
× exp(-1/2(𝐱-μ_k)^TΣ_k^-1(𝐱-μ_k)),
where μ and Σ are its mean and covariance, and D is the size of the input (number of dimensions). The Gaussian mixture models (GMM) have been designed to approximate more complex multivariate densities by decomposing them into K components:
p(𝐱) = ∑^K_k=1ω_k𝒩(𝐱|μ_k,Σ_k),
where each of them is defined using a single Gaussian defined above and ω_k are their weights. The combined model, equipped with more degrees of freedom, should be capable of providing more accurate expressions of the overall observed distributions than a simpler approach utilizing only a single component. In such a framework, the fitting of the mixture to given data X is based on minimizing the loss defined using the log-likelihood function:
ℒ̅ = -log p(X|ω,μ,Σ)
= -1/N∑^N_n=1log∑^K_k=1ω_k
𝒩(𝐱_n|μ_k,Σ_k),
where we adjust the free parameters of the model – means μ, covariance matrices Σ and weights ω. To adapt the given framework to supervised scenarios we can simply specify a separate mixture model for each class c:
p(𝐱 | c) = ∑^K_k=1ω_k^(c)𝒩(𝐱|μ_k^(c),Σ_k^(c)),
and focus on minimizing the aforementioned loss also per class ℒ̅^(c):
ℒ̂ = ∑_c=1^Cℒ̅^(c) = -log∑_c=1^C p(X^(c)|ω^(c),μ^(c),Σ^(c)),
where X^(c) are N_c class-specific observations.
In continual learning we should aim at minimizing the interference of current updates with previously created models to alleviate the detrimental effect of catastrophic forgetting. Therefore, it is worth mentioning here that GMMs create such an opportunity by allowing for maximizing the log-likelihood only for a currently learned class through ℒ̅^(c). It provides a perfect separation at the level of the classification model.
§.§ Mixture optimization for class-incremental deep learning
In order to apply gradient-based learning to GMM in class-incremental deep learning scenarios, we have to address several different issues. Some of them are common for all GMM models using gradient-based learning, while others are specific for the class-incremental deep learning settings.
In general, we say that our goal is to optimize the class-incremental joint model ϕ^(t)=⟨ℱ^(t), 𝒢^(t)⟩, defined at the beginning of Sec. <ref>, using some supervised loss ℒ. Since we set 𝒢^(t) = 𝒩^(t), where 𝒩^(t) is a whole GMM model, we have ϕ^(t)(x)=𝒩^(t)(ℱ^(t)(x)). The trainable parameters are weights ∂ℒ / ∂W and biases ∂ℒ / ∂b for the extractor, and means ∂ℒ / ∂μ, covariance matrices ∂ℒ / ∂Σ and component weights ∂ℒ / ∂ω for the classifier. All of the subsequent paragraphs focus on designing optimization in the classifier (mixture) space, as it was introduced in Sec. <ref>.
§.§.§ Loss design
Max-component: It has been shown that optimizing the full loss ℒ̅^(c) given in Eq. <ref> may lead to some numerical instabilities, especially for high-dimensional data <cit.>. To address this issue a max-component approximation can be used. This approach is very straightforward. Since all p(x|c,k) in Eq. <ref> are positive, any component provides a lower bound for the whole sum used in ℒ̅^(c). If for every point x_n we find a component providing the highest log-likelihood and sum all of them, we will get the largest (max-component) lower bound <cit.>:
ℒ^(c)_max = -1/N_c∑^N_c_n=1max_klog(ω_k^(c)𝒩(𝐱^(c)_n|μ_k^(c),Σ_k^(c))).
Since we can state that ℒ^(c)_max≥ℒ̅^(c), we are able to minimize ℒ̅^(c) by focusing only on ℒ^(c)_max. It is also worth mentioning that just like the general formula given in Eq. <ref> may eliminate the interference with previously learned classes, the max-component approximation can limit the same issue at the level of class components, for example, in data-incremental scenarios <cit.>, making this approach a natural candidate for continual learning settings.
Inter-contrastive loss: All of the introduced losses are limited to scenarios either without a feature extractor or with a fixed pre-trained one. Unfortunately, if we operate in a setting where we can modify the input space of the mixture model and we utilize any of the aforementioned metrics relying entirely on maximizing log-likelihood, we will inevitably end up with a local minimum that for a joint model ϕ^(t) exists for ∀x(𝒢^(t)(x) = 0). This issue can be solved by incorporating an inter-contrastive loss that will distance representations for different classes. We define the loss as:
ℒ^(c)_ie = 1/N_cmax_j ≠ c∑^N_c_n=1max_klog(ω_k^(j)𝒩(𝐱^(c)_n|μ_k^(j),Σ_k^(j))),
which boils down to finding the closest component in other classes, and then optimizing against the class that on average is the closest to the one currently being considered. We keep the log-likelihood to ensure a similar numerical space of loss values as the one for the positive part given in Eq. <ref>. However, now one should notice that minimizing such a loss may very easily destabilize learning since optimization will gravitate towards ℒ̅^(c)_ie→ -∞ preventing the model from actually fitting to the class examples. To avoid it we introduce a tightness bound τ that clips the contrastive loss value at some pre-defined point ℒ^(c)_ie(τ) = max(τ, ℒ^(c)_ie). This basically means that we stop the decrease of the contrastive loss below the given bound, allowing for a more significant contribution of the actual fitting part ℒ^(c)_max. We parametrize the τ value with a simple linear transformation τ = p̅_max^(c) - 1/τ_p, where p̅_max^(c) is the average maximum density value observed across all class components (can be obtained on-the-fly) and τ_p is a tunable hyperparameter that takes values between ( 0,1 ⟩. Such a loss can provide effective discrimination between components of different classes, as shown for an example in Appendix A.
Diverse components: While all of the introduced techniques and modifications ensure reliable discrimination between components of different classes, they do not consider differentiation between components of the same class or their quality. In fact, even in offline gradient-driven settings without dynamic feature extraction it is common to obtain mixtures reduced to a single component per class with all the others practically meaningless, e.g., due to zeroed weights <cit.>. In scenarios with a trainable extractor, this problem becomes even more significant as it is very easy for the optimizer to focus on maximizing log-likelihood from a single component, as both mixture model and flexible extractor lack constraints to prevent this. While in standard scenarios this problem can be successfully addressed with a good initialization method, e.g., using k-means <cit.>, we observed that it was not enough in our case. As a consequence, we introduced two elements to the learning process.
Regionalization – before learning each class, we first divide it into K clusters using the k-means clustering. Then we force each component to fit only to the data from its cluster called a region ℛ^(c)_k. This replaces the max-component loss ℒ^(c)_max defined in Eq. <ref> with:
ℒ^(c)_reg = -∑^K_k=11/N_k∑_x∈ℛ^(c)_klog(ω_k^(c)𝒩(𝐱|μ_k^(c),Σ_k^(c))).
Intra-contrastive loss – the regionalization approach is necessary yet not sufficient to provide sufficient diversification between same-class components. The reason for it is the same as for discrimination between different classes, as described in the previous paragraph. Analogously to the inter-contrastive loss, we add the intra-contrastive loss with the tightness bound τ:
ℒ^(c)_ia(τ) = ∑^K_k=1max( τ, max_m ≠ k1/N_k
×∑_x∈ℛ_klog(ω_m^(c)𝒩(𝐱|μ_m^(c),Σ_m^(c))),
which for each class region pushes away other same-class components that on average are closest to the currently considered one, based on the regionalization conducted in the previous step. Obviously, one can define separate τ for the inter- and intra-contrastive loss.
Such an approach can effectively increase the diversity of the same-class components, as given for an example in Appendix A. However, this approach imposes a hard constraint on how the representation and mixture may look, which limits the flexibility of the whole model. Regardless of these concerns, this method can still effectively improve the overall performance of a multi-component model over a method without the proposed improvement, as we will show in our extensive experiments.
Final component-based losses: To summarize, we distinguish two component-based losses. One uses the max-component approach (MC):
ℒ_mc = ∑_c=1^tℒ_max^(c) + ℒ_ie^(c)(τ_ie),
while the second loss adds the regionalization technique with the intra-contrastive part (MCR):
ℒ_mcr = ∑_c=1^tℒ_reg^(c) + β(ℒ_ie^(c)(τ_ie) + ℒ_ia^(c)(τ_ia)).
Cross-entropy loss: Last but not least, we can also attempt to directly optimize the whole standard loss ℒ̂ given in Eq. <ref>, using a high-level supervised wrapper loss, e.g., based on cross-entropy (CE). In such a case, our loss is defined as:
ℒ_ce = -∑_c=1^t∑_n=1^N_cy_n^(c)logŷ_n^(c),
where y is a one-hot target vector and ŷ_n^(c) comes from the softmax function ŷ_n^(c) = e^p_n^(c)/∑_c=1^te^p_n^(c) and p_n^(c)=p(x_n|c) is a density value for a given class produced by the mixture model accordingly to Eq. <ref>.
§.§.§ Constraints
Other issues that have to be addressed when using gradient-based mixture training are the mathematical constraints that have to be enforced to preserve a valid mixture model. This is required since gradient-based learning does not constrain the possible values for means, covariance matrices and weights, and the last two have to remain in a specific range of values.
Component weights: For the GMM model its component weights ω_k have to sum up to one: ∑_k=1^Kω_k=1. To ensure that the effective weights satisfy this requirement we simply train auxiliary free parameters ω̂_k and use the softmax-based normalization ω_k = e^ω̂_k/∑_j=1^Ke^ω̂_̂ĵ to obtain required values <cit.>.
Covariance matrices: For a general case, the covariance matrices of the GMM model should be symmetric positive definite v^TΣv > 0 for all nonzero vectors v. This can be enforced using the Cholesky decomposition <cit.> Σ = AA^T, where A is a triangular matrix with positive diagonal values a_ii > 0 and, at the same time, our trainable proxy parameter. To enforce positive diagonal values, after each gradient-based update we clamp them with a_ii = min(a_ii, d_min) using some predefined d_min value. Finally, we also consider a case of a mixture using only the diagonal of the covariance – variance σ, which we control using the same clamp-based approach σ_i = min(σ_i, d_min).
§.§ Memory buffer
In our work, we consider the class-incremental scenario with strictly limited access to previously seen observations (classes). Therefore, in all of the introduced losses we use all available data for the currently learned class t, while for the others we sample from the memory buffers ℳ_c that store an equal number of examples per each previously seen class. On the other hand, if the feature extractor is pre-trained and static we could remove the inter-contrastive loss and even get rid of the memory buffer, allowing for memory-free training, as we will show in our experimental study. The memory buffer is needed in a general case when we assume the joint training of the whole model.
§.§ Classification
Finally, in the presented model, the classification of an instance x_n can be performed using two approaches, either utilizing the softmax function ŷ_n^(c) = e^p_n^(c)/∑_c=1^te^p_n^(c), where p_n^(c) = p(x_n|c), or by taking the weighted support of the closest component ŷ_n^(c) = max_kω_k^(c)𝒩(𝐱_n|μ_k^(c),Σ_k^(c)). We will empirically show that these methods work best with specific losses designed in the previous sections.
§ EXPERIMENTAL STUDY
In our experiments, we empirically explore all of the introduced methods and parameters and put our method in the performance context of different state-of-the-art baselines. We show how our model performs in end-to-end scenarios and with a pre-trained extractor, compared with other solutions. For more specific details regarding data, configurations and results, please refer to Appendix A and B, as well as to our repository containing source code for our methods and all experiments: (please check the source code provided in the supplementary materials, a public URL will be added later). All of the experiments were conducted using 4 GPUs (Tesla V100) that were part of an internal cluster.
§.§ Setup
For the purpose of the evaluation we selected commonly utilized image classification datasets that were turned into class-incremental sequences by presenting their classes subsequently to the models <cit.>. We used: MNIST, FASHION, SVHN, CIFAR and IMAGENET datasets using various variants (number of classes, pre-trained features). For the analysis of different configurations of our model we used shorter sequences. We extended them with the longer benchmarks for the comparison with baselines.
In the final section of this work, we compared our class-incremental Gaussian mixture model (MIX-MCR, MIX-CE) with other classifiers dedicated for continual learning scenarios. We considered: standard experience replay (ER) <cit.>, experience replay with subspaces (ERSB) <cit.>, centroid-based iCaRL <cit.>, two gradient-based sample selection methods (GSS and A-GEM) <cit.>, experience replay combined with knowledge distillation and regularization (DER) <cit.>, and two purely regularization-based approaches – LWF <cit.> and SI <cit.>. Most of the algorithms were implemented as wrappers of the source code provided in <cit.> under MIT License. For the last two we used their modifications adjusted for single-task learning <cit.>. As our lower bound we used a naively learning net (NAIVE), and for the upper bound we present results for the offline model (OFFLINE).
We evaluated the presented methods in a class-incremental setting, where all of the classes were presented to the models subsequently and were not shown again after their initial appearance. We measured the accuracy of a given algorithm after each class batch, utilizing holdout testing sets, and then, based on <cit.>, used it to calculate the average incremental accuracy over the whole sequence:
Ω_all = 1/T∑_t=1^Tα_t,
where α_t is the model performance after t classes and T=C is the total number of classes. In addition to the whole aggregation, for the final comparison, we provided these values after each batch to present a more complete perspective of the obtained results.
§.§ Results
In this section, we present and describe all of the results that were obtained for the experiments introduced in the previous paragraphs. The first part consists of the analysis of different configurations of MIX, while the second one focuses on a comparison with other class-incremental algorithms.
Loss and classification:
We analyzed different combinations of the proposed losses and classification methods. Based on Fig. <ref>, we can make three major observations. Firstly, the softmax classification works significantly better with the CE loss, and max-component can be more efficiently paired with MC and MCR than softmax. It was evident for almost all cases (except for MC on CIFAR10) and resulted in almost 0.15 difference on average between softmax and max-component for CE, and about 0.05 for MC and MCR.
Secondly, the MCR loss performed better than MC, showing consistent improvements, especially for more complex datasets like SVHN, CIFAR10 or IMAGENET10, which resulted in more than 0.1 for a difference on average. This demonstrate that the regionalization and intra-contrastive loss are capable of providing meaningful improvements over simpler MC loss utilizing only max-component and inter-contrastive elements, and that ensuring higher diversity among class components can be beneficial to the model.
Finally, we can see that CE with softmax could provide very similar results as MCR with max-component, which means that the general GMM learning formula, wrapped with a high-level supervised loss, can be sometimes as useful as more complex MCR without the need for tuning additional parameters. One drawback of using CE, however, is the fact that it does not model the Gaussian mixtures well (see Appendix B for additional visualizations). The CE loss does not really have to fit the mixtures to the data since it is enough for it to ensure high classification quality. We can also observe a similar behavior for the MC loss. It may be prohibitive if one wants to obtain a reliable description of the latent space. The MCR loss achieves both objectives at the same time: high classification accuracy and high quality of the mixture models for features. This may be important if someone requires interpretable models or would like to extend the proposed algorithm with some Gaussian-oriented techniques that MCR may enable. Furthermore, we believe that analyzing its probabilistic properties in detail could be a part of incremental works built on top of the mixture model. They could utilize its well-defined characteristics, e.g. by proposing new mixture-based losses.
Tightness:
In Fig. <ref>, we presented a grid of values for the average incremental accuracy per each pair of inter- and intra-tightness for every dataset. One can clearly see that imposing the constraint (tightness) on the inter- and intra-contrastive loss values is beneficial to the learning process. Most of the benchmarks required τ_p, ie at the level of 0.0001 or 0.001 and slightly higher intra-tightness τ_p, ia around 0.001 or 0.01 to achieve the best results. At the same time, one should notice that imposing too high inter-tightness (0.01) leads to abrupt deterioration of quality, which is a result of blocking the contrastive part of the loss from pushing components of different classes from each other. The influence of setting too high intra-tightness is less important since we may simply end up with a single component that can still be effectively used for classification.
The examples for FASHION, given in Fig. <ref> and <ref>, show how increasing the inter-tightness (the first one) and intra-tightness (the second one) affects learned representations and mixture models. We can observe the positive impact of the constraint and the potential for sweet spots providing a good balance between differentiating components between each other and fitting them to the actual data. It is evident that too low values introduce critical instabilities to the learning process (very high contrastive loss values overwhelming the fitting part), while too high thresholds lead either to the decline of discriminative properties of the model or degenerate solutions.
Baseline comparison:
In the second section of our experimental study, we placed our algorithm in the class-incremental performance context by comparing it with the introduced baselines (Fig. <ref>). First of all, we can see that the MIX-MCR variant performed better than the MIX-CE for most of the datasets, while being very close to it for the longer sequences (difference between less than 0.01 and 0.03). This proves that MIX-MCR is capable of providing not only a better representation (mixture) model but also that it is more reliable from the accuracy perspective. This also means that it is worth trying to maximize the quality of the produced Gaussian models as an alternative to high-level cross-entropy for classification. Secondly, although our model cannot be distinguished as the best classifier (being worse than iCaRL on average, with a difference equal to about 0.04), it is, at the same time, reliably competitive when compared with the remaining baselines (ER, GSS, DER) with a difference about 0.01 and less than 0.03. Also, it does not fall into the same pitfalls as either the weakest replay method (A-GEM) or the regularization-based ones (LWF, SI), outperforming them by almost 0.4 for accuracy on average. We can see that MIX could be found among the best models for MNIST, FASHION, IMAGENET10, IMAGENET20A and IMAGENET20B, especially at the end of the datasets, providing relatively reliable performance throughout the whole sequences. On the other hand, it struggled with catching up with the best replay methods for SVHN and CIFAR-based datasets showing that there is still a potential for improvements when it comes to predictive accuracy.
The overall very poor performance of LWF and SI (but also A-GEM), which were not much better than the NAIVE approach, confirms the observations made in other publications that the regularization-based methods cannot handle the most challenging 1-class-incremental scenarios without memory buffers <cit.> even after improvements proposed in <cit.>.
We can also see that the for the scenarios with end-to-end training the models were much closer (0.01-0.3) to the OFFLINE upper bound for the shorter sequences (MNIST, FASHION, SVHN and IMAGENET10, except for CIFAR10) than for the longer ones (IMAGENET20A, IMAGENET20B, CIFAR20) with differences between 0.4-0.5, which shows that all of the state-of-the-art methods still struggle with bridging the gap between incremental learning and offline optimum.
Finally, the results for the memory-free scenarios with pre-trained models, given in the last row of Fig. <ref>, exhibit the main strength of the MIX algorithm. Since in these scenarios, it does not use the inter-contrastive loss, it can perfectly separate the incremental learning process for each class, preventing catastrophic forgetting at the level of the classifier. As a result, it does not have to rehearse the previous concepts at all (ℳ_c=0) while still being able to conduct very effective learning producing results very close to the OFFLINE upper bound (difference between about 0 and 0.1), regardless of the quality of the extractor (pre-trained on 10 and 20 or 100 and 200 classes). The MIX-MCR method outperforms all of the baselines for all cases except for IMAGENET200-PRE20, for which only iCaRL was able to provide slightly higher accuracy, even though they had a small advantage of having approximately one example per class in the buffer. It is not a coincidence that practically only iCaRL is close to our method on average (worse by about 0.1), since it uses a similar paradigm in the classification layer by storing prototypes/centroids that are used for classification. All of the remaining algorithms cannot handle the memory-free scenario effectively, producing solutions worse by at least 0.2 on average. This can be a crucial property when one has to consider, for example, data privacy issues or mobile and edge computing.
All of the presented observations, conclusions and recommendations can be also found in a condensed form at the end of Appendix B.
§ SUMMARY
In this work, we introduced a class-incremental mixture of Gaussians model (MIX) for deep continual learning. We proposed different variants of the algorithm to make it suitable for gradient-based optimization and, through an extensive experimental study, we exhibited its practical configurations and capabilities in the context of other state-of-the-art continual learning models.
In our future research, we will focus on replacing the regionalization approach with a more flexible method that do not assume any pre-training structure and allows the gradient-based procedure to fully explore potential solutions, e.g. annealing <cit.>, and on removing the static tightness hyperparameter to increase flexibility even more – it could be more beneficial to either find a better (parameter-free) distance function or propose an adaptive threshold. It is also an open question whether we can effectively train a gradient-based mixture using a full covariance matrix. Finally, we could consider some kind of hybridization of the mixture models with the feature extractor to benefit from the capabilities of the former to limit interference with previously learned concepts by utilizing max-component losses. All of these potential improvements combined could provide significant performance gains in the class-incremental continual learning scenarios.
ieee_fullname
§ APPENDIX
§.§ Data
We used: MNIST, FASHION, SVHN, CIFAR10 and IMAGENET10 – a subset of the tiny IMAGENET200, to gain deeper insights into our method while conducting experiments with hundreds of different configurations. Then, we extended this set with CIFAR20 – the coarse-grained version of CIFAR100, IMAGENET20A and IMAGENET20B – larger subsets of IMAGENET200 – to benchmark our method against other algorithms.
For the experiments involving fixed extractors, we used pre-trained features to construct four additional sequences – CIFAR100-PRE10, CIFAR100-PRE100, IMAGENET200-PRE20 and IMAGENET200-PRE200, which consisted of features extracted for CIAFR100 and IMAGENET200, using extractors trained on 10, 20, 100 and 200 classes of the original datasets. The summary of the used benchmarks is given in Tab. <ref>. Details of the feature extractors can be found in the next section.
§.§ Model configurations
In the first section of our experiments, we explored different configurations of our algorithm, which can be mostly seen as an ablation study. Firstly, we evaluated different losses (CE, MC and MCR) combined with different classification methods (softmax, max-component). Secondly, we checked different settings for the tightness bound parameter τ_p by evaluating a grid of values for inter-tightness and intra-tightness – we considered τ_p ∈⟨1e-06, 1e-05, 0.0001, 0.001, 0.01⟩ for both. Thirdly, we analyzed how assuming different numbers of components affects the classification performance on different datasets. We used K ∈⟨1, 3, 5, 10, 20⟩. Then we checked if it is better to maintain a whole covariance matrix or only its variance (FULL, VAR). Finally, we evaluated different learning rates for the extractor and GMM part, using α_ℱ∈⟨1e-07, 1e-06, 1e-05, 0.0001, 0.001⟩ and α_𝒢∈⟨1e-05, 0.0001, 0.001, 0.01, 0.1⟩, to check whether it may be beneficial to configure them separately, and different memory sizes ℳ_c ∈⟨8, 64, 128, 256, 512⟩ to analyze how our method exploits limited access to class examples.
While evaluating specific parameters we kept others fixed. For our base configuration we chose a setup that was capable of providing performance comparable with a standard experience replay. We used the MCR with max-component as our loss and classification method, K=3, τ_p,ie=0.002, τ_p,ia=0.01, β=0.5, α_ℱ=0.0001, α_𝒢=0.001 and d_min=0.001 with only variance stored per each component. We assumed a modest memory buffer per class ℳ_c=256 and matched the size of a memory sample per class with the training batch size. The model was trained for 10 (MNIST, FASHION) or 25 epochs per class, with 32 (IMAGENET) or 64 instances in a mini-batch.
§.§ Algorithms
Based on the observations made in the first section of the experiments, in the final evaluation we used two variants of our algorithm: MIX-CE and MIX-MCR with τ_p,ie=0.0001, τ_p,ia=0.001, α_ℱ=0.0001, α_𝒢=1e-05 and, once again, d_min=0.001 with only variance maintained per each component. The only parameter that we tuned per each dataset was the number of components K. We used Adam as the optimizer. For the memory-free scenarios with pre-trained extractors, we turned off the inter-contrastive loss to minimize interference with previously learned classes.
The main parameters of the baselines methods were set based on the original papers and other literature, including empirical surveys or works containing vast empirical studies <cit.>. For all memory sampling methods we matched the memory sampling size with the training batch size. For ERSB we used 10 centroids per class each containing up to either 25 or 15 instances to match the total memory size. DER used α_d=0.5, for LWF we set the softmax temperature T=2 and progressively increased its distillation coefficient as suggested in <cit.>, and SI used λ =0.0001. All of the methods utilized the Adam optimizer with a learning rate α=0.0001 as we did not observe any significant differences when changing this parameter.
Analogously to the configuration section, all of the algorithms, including ours, were trained for 10 (MNIST, FASHION) or 25 epochs per class, using 32 (IMAGENET) or 64 instances per mini-batch. The offline models were trained for either 50 or 100 epochs, until they achieved a saturation level. The memory buffer was set to ℳ_c=128 (IMAGENET) or ℳ_c=256 for methods supporting memory per class (ER, ERSB, iCaRL), and ℳ=C·128 or ℳ=C·256 for the remaining ones (GSS, A-GEM, DER), where C was the total number of classes. The latter group was equipped with reservoir buffers <cit.>. For the experiments with pre-trained extractors we wanted to check the memory-free scenario, therefore we set ℳ_c=0 for our methods and ℳ_c=1 or ℳ=C for others, since most of them could not be run without storing any examples.
All of the algorithms, including different configurations of our method, were combined with feature extractors. For MNIST and FASHION we used a simple CNN with two convolutional layers consisting of 32 (5x5) and 64 (3x3) filters, interleaved with ReLU, batch normalization and max pooling (2x2). For SVHN and IMAGENET we utilized ResNet18, its modified version for CIFAR10 and CIFAR20, and ResNeXt29 for CIFAR100 <cit.>. The classification layers consisted of the default configurations.
Finally, for our method, ER, ERSB, A-GEM and DER we disabled batch normalization, since, consistently with <cit.>, we observed a significant difference in performance when those layers were turned off for the given methods. As mentioned in Sec. <ref>, for the memory-free scenarios, the extractors were pre-trained on either 10, 20, 100 or 200 classes of CIFAR100 and IMAGENET200. For this setting we trained all the models for 20 epochs per class.
Results for the offline model were either obtained by us (learned from scratch for IMAGENET20A, IMAGENET20B and fine-tuned models for IMAGENET200), or by referring to other publications <cit.>.
§ APPENDIX
§.§ Additional visualizations
Fig. <ref> presents an example of a single-component class-incremental mixture model learned with the inter-contrastive loss. Fig. <ref> demonstrates the effectiveness of training a multi-component model with the intra-contrastive loss and regionalization.
As mentioned in the main document, the CE loss can often achieve similar predictive performance even if its mixture models are not really fitting the data (Fig. <ref>). We can see it when compared with MC for K=1 or MCR for both K (Fig. <ref> and <ref>). Furthermore, the model produced for MC with K=3 clearly shows that it is incapable of effectively utilizing multiple components for the same class. Please notice that only the Gaussians in the middle actually cover some data points, while the remaining components are completely unrelated to the observed data. These are examples of the degenerate solutions. While for FASHION this loss could still, analogously to CE, provide similar performance as MCR (the components in the middle are fitted to the data and they are sufficient to model it), the observed desynchronization of components results in its weaknesses for more complex problems. The MCR loss can provide high quality of predictive performance and of the produced mixture models.
§.§ Additional configurations
Number of components: Tab. <ref> presents how many components were required to obtain the best solutions per each dataset for the given settings. We can observe that for simpler datasets (MNIST, FASHION) using a single component per class for sufficient and that introducing additional ones led to slightly worse performance, most likely due to the fact of fitting to simple concepts and overcomplicating the optimization problem. On the other hand, more complex benchmarks (SVHN, CIFAR10, IMAGENET10) preferred access to more components per class, which could provide significant improvements, e.g., for SVHN the difference between K=1 and K=10 was almost 0.3. While for these experiments we set the learning rate slightly higher for the GMM model (0.001) than for the extractor (0.0001), we observed that when the former used rate lower than the latter (as suggested by the results for learning rates that will be presented below), the optimal K tended to be lower on average. It is possible that if GMM is dominant it prefers having more flexibility (components), while when the extractor has a higher learning rate it may be more effective in adjusting representations to lower numbers of components.
Covariance: Results presented in Tab. <ref>, unequivocally show that our gradient-based MIX can much better adapt to data if it maintains only the variance of the covariance matrix (better by almost 0.3 when compared with full covariance). It is not surprising since previous publications related to the gradient-based GMMs for offline settings suggested a similar thing <cit.>. Most likely, working with a full covariance matrix leads to less stable loss values, and many more free parameters (especially if the feature space is high-dimensional) likely cause problems with convergence.
Learning rates: Analogously to the experiments for tightness, in Fig. <ref> we presented the grid of results for different extractor (horizontal) and mixture (vertical) learning rates. The obtained results suggest that the former part is more important – once the optimal rate is set (0.0001 for the given settings) tuning the latter seems less significant, although overall it should be set to a similar or slightly lower value.
Memory size: Finally, if we look at the results of class-incremental learning using different memory sizes, given in Fig. <ref>, we will see that MIX can effectively utilize larger buffers and that it seems to be quite memory-dependent, especially for SVHN where the difference between subsequent sizes ranged from 0.1 to 0.2. Still, the gap was much smaller for all of the remaining datasets. While this characteristic of the algorithm may be problematic (the fewer examples we need, the better), it is still valid that if we can use a pre-trained extractor, the whole model does not need to use the memory buffer at all.
§.§ Lessons learned
Based on the theoretical and empirical analysis presented for this work we can conclude the following.
* Class-incremental learner. Regardless of many combined challenges, it is possible to successfully hybridize the gradient-based mixture models on top of convolutional feature extractors, and use them in class-incremental end-to-end continual learning scenarios. The presented results show that MIX is capable of providing competitive results when compared with well-known incremental baselines.
* Dedicated losses. It has been shown that the training of the mixture models combined with dynamic feature extractors requires the inter-contrastive loss to effectively distinguish components of different classes from each other. In addition to that, to ensure diversity among same-class components and avoid degenerate solutions, such techniques as regionalization combined with the intra-contrastive loss are required. We showed that not only do the proposed approaches deliver what was intended, but also that they can translate into significant performance gains for more complex datasets. Finally, although the more generic high-level cross-entropy loss may provide good solutions in many cases, only the most advanced variant (MIX-MCR) delivers both high predictive performance and high quality of generated mixture models, which may be important from the perspective of interpretability or potential Gaussian-based extensions.
* Effective tightness. The tightness bound plays a crucial role in stabilizing the mixture learning procedure. Setting the optimal values of inter- and intra-tightness leads to striking a balance between pushing different components from each other and actually fitting them to the data. Intuitively, the inter-tightness prefers slightly lower values than intra-tightness.
* Recommended configurations. By analyzing other different hyperparameter settings and combinations of our methods we could observe that: (i) the CE loss works much better with the softmax classification method, while MC and MCR should be combined with the max-component approach, (ii) different numbers of components may be required for different data and different learning rates may also affect the optimal number, (iii) maintaining only the diagonal of the covariance matrices leads to more stable optimization and better results, (iv) the learning rate for the feature extractor dominates over the one for the mixture model, and that (v) MIX is quite memory-dependent in general end-to-end scenarios.
* Memory-free scenarios. At the same time, MIX is capable of learning without a memory buffer if we use a fixed pre-trained extractor and disable the contrastive loss that is not needed in this case. Our method stands out as the best model for such class-incremental scenarios which can be very important if there are any data privacy concerns or strict memory limits.
|
http://arxiv.org/abs/2307.04000v1 | 20230708155126 | Synthesis of resonant modes in electromagnetics | [
"Antonello Tamburrino",
"Carlo Forestiere",
"Giovanni Miano",
"Guglielmo Rubinacci",
"Salvatore Ventre"
] | physics.optics | [
"physics.optics",
"physics.class-ph"
] |
Department of Electrical and Information Engineering M. Scarano, Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy.
Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI-48824, USA.
e-mail: [email protected]
Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21, Napoli, 80125, Italy
Department of Electrical and Engineering Information M. Scarano, Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy.
Resonant modes determine the response of electromagnetic devices, including dielectric and plasmonic resonators. Relying on the degrees of freedom that metamaterials provide, this contribution shows how to design, at will, the resonant modes of a dielectric object placed in an unbounded space. Specifically, the proposed method returns in analytical form the spatial distribution of the dielectric susceptibility tensor for which the object exhibits resonances at prescribed frequencies and spatial distribution of the polarization. Together with the synthesis of the material, two key concepts are introduced: the controlled tunability of the resonant modes and the number of essential modes, i.e. the number of modes that uniquely characterize the spatial distribution of the dielectric susceptibility.
Moreover, this approach can be applied to design the resonant modes of any system where the constitutive relationship is linear and local.
Synthesis of resonant modes in electromagnetics
Salvatore Ventre
August 12, 2023
================================================
Media with spatially inhomogeneous refractive index have fascinated the humankind for millennia, exhibiting counter-intuitive effects such as mirages, or fata morgana. Archaeological evidence indicates that humans learned how to engineer the refractive index variations to make lenses in antiquity, spanning several millennia. More recently, nano-fabrication techniques, the discovery of materials with tunable permittivity, and the introduction of the metamaterial concept <cit.> have greatly expanded the landscape of feasible permittivity distributions for the electromagnetic design. Anisotropic and even continuous effective variations of the permittivity can be now implemented.
Using the degrees of freedom in the choice of the materials, it is possible to control the electromagnetic field as shown by Pendry et al <cit.> by introducing trasformation optics <cit.>. They showed that, the permittivity and permeability effectively determine a curved spatial geometry for the electromagnetic field. Thus, leveraging on this analogy, they showed how the anisotropic and inhomogeneous permittivity and permeability profiles to redirect the electromagnetic field in a prescribed way. Recently, several optimization methods have been introduced to design materials to achieve a prescribed electromagnetic response, incorporating at the same time fabrication constraints
<cit.>.
In this manuscript, we take a fresh path to the design of electromagnetic resonances of a scatterer, which plays a central role in electromagnetic devices, e.g. <cit.>. Plasmonic and dielectric nano-resonators are an interesting example. When the resonance condition is met, the near-field and far-field characteristics of the device are dominated by the corresponding resonant mode.
We introduce a theoretical framework that enables the synthesis of the spatial distribution of the permittivity profile of a dielectric object, to design its resonant modes, i.e. polarization current density distributions. The designer preliminary specifies, in the spatial domain occupied by the object, one or several modes, together with the corresponding resonant frequencies. Then, the synthesis process returns the possibly inhomogeneous and anisotropic permittivity profile which guarantees that the dielectric object exhibits the prescribed modes at the specified resonance frequencies. It is a direct method: it does not require the use of any optimization approaches, but explicitly returns the analytical solution in a single step. The syntheses approach leverages on a formulation of the generalized eigenvalue problem where the contributions of the material and of the electromagnetic field are separated. Yet, this approach is very general: it can be applied to any system where the constitutive relationship is linear and non-spatially dispersive. For instance, it can be used to design the properties of an elastic material to control its vibrational modes.
In addition, the proposed framework allows one to clearly identify the physical feasibility and limitations inherent to the problem of the design of the modes. The main outcome is that the maximum number of modes (essential modes) that can be prescribed at a given resonance frequency, is equal to the dimension of the problem (two for a 2D problem and three for a 3D problem). These are inherent physical limits unveiled by the proposed framework.
Finally, we also address the problem of the tunability where, by scaling the dielectric susceptibility, we can change completely the resonance property in a controlled way. This feature enables the design of tunable materials, where one can adapt the response of the material dynamically, according to specific needs.
§ MODES AND EIGENVALUE PROBLEM
We consider a linear, nonmagnetic and non-spatially dispersive dielectric of finite size, shown in Fig. <ref>. We denote the space occupied by the dielectric with Ω, its boundary by ∂Ω, the (unit vector) normal to ∂ V that points outward by 𝐧.
Under these assumptions, the polarization density 𝐏 is given by 𝐏( 𝐫,ω)
= ε_0χ( 𝐫,ω)
·𝐄( 𝐫,ω), where is
the dielectric susceptibility tensor, ω is the angular frequency
(the e^jω t time behavior is assumed), ε_0 is the vacuum permittivity, and · corresponds to the
usual dot product between tensors and vectors.
When the dielectric scatterer is excited by an external electric field 𝐄^i, the total electric field 𝐄 can be written as the sum of 𝐄^i and of the reaction field 𝐄^𝙿 due to the presence
of the polarization current density jω𝐏. The constitutive
relation can be written as
1/ε_0( 𝐫, ω)
·𝐏( 𝐫, ω) - (
𝐫, ω) = 𝐄^i( 𝐫, ω)
in Ω,
where tensor is the pointwise inverse of
, i.e. ( 𝐫,ω)
=^-1( 𝐫,ω).
Let ℰ( ω)
be the operator giving the electric field produced by a prescribed polarization
density field 𝐏 radiating in the free space at frequency ω <cit.>:
𝐄^P( 𝐫) =jω∫_Ω𝐆
( 𝐫-𝐫^') 𝐏( 𝐫^') dS^'
where 𝐆 is the proper electric-electric dyadic Green function.
For any prescribed angular frequency ω, the electromagnetic scattering is
governed by the integral equation
1/ε_0·𝐏 - ℰ(
ω) 𝐏=𝐄^i in Ω.
Two particularly significant auxiliary eigenvalue problems can be defined starting from Eq. <ref>, setting the exciting field to zero, and assigning the material tensor .
Quasi Normal Modes <cit.> (QNM) are nontrivial solutions ω and 𝐏 of
ℰ( ω) 𝐏=1/ε_0
·𝐏 in Ω.
QNM are often used to characterize micro- and nano- resonators <cit.>, enabling the calculation of synthetic parameters such as the quality factor, the mode volume <cit.>, and the Purcell factor. QNM are also used to expand the response of micro-nanoresonators by <cit.> highlighting the contribution of the individual modes in the overall scattering response. The eigen-frequencies ω are complex numbers, i.e. ω∈ℂ, and (ω, 𝐏) forms a (generalized)
eigenvalue/eigenvector pair.
Material Modes are nontrivial solution ξ∈ℂ and 𝐏 of
ℰ( ω) 𝐏=ξ1/ε_0
·𝐏 in Ω,
where the frequency ω∈ℂ is prescribed.
ξ and 𝐏 form a (generalized)
eigenvalue/eigenvector pair.
These modes for ω∈ℝ and uniform and isotropic
material ((𝐫) = χ scalar constant in Ω) have been already investigated in <cit.>, and have been used to expand the electromagnetic response of nano-resonators <cit.>, and also to design the scalar permittivity of a homogeneous object to achieve a prescribed scattering response, such as scattering cancellation or maximization <cit.>.
In this work χ may be non uniform and/or non isotropic, and ω may be complex. The characteristic feature of the eigenvalue/eigenvector pair for (<ref>) is to be a
homogeneous function of , i.e. if ^'=α then
𝐏^' =𝐏; 1/ξ^' =α1/ξ
is an eigenvalue/eigenvector pair for ^'. Specifically, the eigenvector 𝐏 is a 0-degree homogeneous function, whereas the reciprocal of the eigenvalue ξ is a 1-degree homogeneous function.
After this property, we term these modes as Homogeneous Material Modes. Homogeneous Material Modes have been successfully introduced in low-frequency electromagnetism for eddy current tomography <cit.>.
A unique feature of Material Modes and, more in general, of Homogeneous Material Modes, is that since the eigenvalue ξ and the eigenvector are homogeneous function of χ, it is possible to tune on different resonant modes the electromagnetic system by scaling the susceptibility. This feature, which we call tunabilty, opens the door to a systematic design of reconfigurable materials and will be discussed in detail in a subsequent Section.
§ SYNTHESIS OF MODES (SOM)
In this Section, we introduce a theoretical framework enabling the synthesis of the dielectric permittivity tensor =
( 𝐫, ω) of the object, such that it exhibits the set of resonance modes {(ω_k,ξ_k,𝐏_k) }_k=1… N at prescribed frequencies ω_k. Each individual mode is described by the triplet ( ω_k,ξ_k,𝐏_k). Hereafter, ω_k is referred as the frequency eigenvalue, ξ_k as the material eigenvalue, and 𝐏_k as the spatial mode. The problem consists in solving for a proper γ_k ( 𝐫) = γ( 𝐫, ω_k ), the set of equations imposing the modes
ℰ( ω_k ) 𝐏_k=ξ_k 1/ε_0_k ·𝐏_k in Ω, for k=1, …, N.
The synthesis is carried out in two steps. First, we solve the problem at each prescribed angular frequency ω_k, by evaluating γ_k, as solution of (<ref>). Then, we interpolate in the frequency domain the collection of tensors χ_1, …, χ_N, being χ_k = γ_k^-1
Hereafter, we consider the _z scenario where the electromagnetic problem is x_3- invariant and
the electric field is transverse to the x_3-axis. This is a 2D case where
the tensor is of the type ( 𝐫, ω) =∑_l,m=1^2χ_lm( 𝐫, ω) 𝐞
_l 𝐞_m, the electric field is 𝐄( 𝐫, ω) =E_1( 𝐫, ω) 𝐞_1+E_2(
𝐫, ω) 𝐞_2, 𝐫=x_1𝐞_1
+x_2𝐞_2 and 𝐞_1 and 𝐞_2 are the unit
vectors along the x_1 and x_2 directions, respectively. The elements of the Green function are given in Appendix <ref>.
§.§ Synthesis of Modes at a prescribed angular frequency
Given a prescribed angular frequency ω_k, we distinguish two cases:
(i) a single mode is prescribed or (ii) two modes are prescribed. In a 3D setting, one have to include also the third case when three modes are prescribed. The treatment of this case is nothing but a straightforward extension of the one needed when two modes are prescribed.
Single mode case. Let ( ω_k,ξ_k,𝐏_k) be an individual prescribed resonances modes at frequency ω_k, where ω_j
≠ω_k for j≠ k. The solution of equation (<ref>) can
be expressed in explicit form as
_k( 𝐫) = ε_0
_k( 𝐫) /ξ_k|𝐏_k( 𝐫) | ^2𝐏_k
^∗( 𝐫) +α_k( 𝐫)
𝐯_k( 𝐫) 𝐩_k^∗(
𝐫),
where ∗ is the complex conjugate operation, _k=ℰ( ω_k) 𝐏_k, 𝐩
_k( 𝐫) ⊥𝐏_k( 𝐫)
for almost everywhere (a.e.) 𝐫∈Ω [Here 𝐚( 𝐫
) 𝐛( 𝐫) means that 𝐚
^∗( 𝐫) ·𝐛( 𝐫)
=0.], 𝐯_k is an arbitrary vector field and α_k is an
arbitrary scalar field. The solution γ_k given in equation (<ref>) can be easily verified by
plugging it in equation (<ref>).
A possible choice for 𝐩_k is 𝐩_k
=ℛ𝐏_k^∗, being ℛ the 90
^∘ rotation operator in the counterclockwise direction. We notice that
ℛ𝐏_k^∗( 𝐫) =𝐏
_k^∗( 𝐫) ×𝐞_3 where 𝐞_3 is the unit vector along the x_3 direction.
Finally, we highlight that by means of the explicit solution of equation (<ref>) one can easily check if
_k is bounded or continuous. Specifically, we have that if _k and 𝐏_k are continuous (piecewise continuous) and
|_k|/|𝐏
_k| is bounded, then _k is continuous (piecewise continuous).
We conclude this Section with a remark about the scalar case.
When _k∥𝐏_k, i.e.
𝐏_k( 𝐫) =ε_0χ_k(
𝐫) _k( 𝐫) being
χ_k a scalar field, Eq. (<ref>) returns a scalar susceptibility tensor (homogeneous material):
_k=1/ξχ_kℐ,
where ℐ is the unit dyad. Indeed, Eq. (<ref>) follows from (<ref>) by choosing 𝐩_k( 𝐫) =𝐏_k^∗( 𝐫
) ×_3, 𝐯( 𝐫
) =𝐄_P^∗×_3, α
_k( 𝐫) =χ_k^∗( 𝐫)
/χ_k( 𝐫), and by observing that 𝐮𝐮
^∗+( 𝐮^∗×_3)
( 𝐮×_3) gives the (2D)
unit dyad ℐ when 𝐮 is an arbitrary unit vector. In this case, the prescribed mode is a material independent mode <cit.>.
Two isofrequential modes.
Let ω_1=ω_2≠ω_j for j>2, and ( ω_1,ξ_1,𝐏_1) and (
ω_2,ξ_2,𝐏_2) be the prescribed resonances modes. Let the solution be expressed as
_1 ( 𝐫)= ∑_l,m=1^2Γ_lm( 𝐫) 𝐔_l( 𝐫) 𝐏_m^∗( 𝐫).
where Γ_lm( 𝐫) ∈ℂ and
𝐔_l = ε_0ℰ (ω_1) 𝐏_l/ξ_l, l=1,2.
To find the unknown coefficients Γ_lm, we observe that by imposing Eq. (<ref>) on the two prescribed resonance modes we have:
𝐔_r ( 𝐫)= γ_1 ( 𝐫) ·𝐏_r ( 𝐫) for a.e. 𝐫∈Ω, and r=1,2.
Then, by left multiplying this expression by 𝐔^∗_s( 𝐫), we have
𝐔_s^∗·𝐔_t=∑_l,m=1^2(
𝐔_s^∗·𝐔_l) Γ_lm(
𝐏_m^∗·𝐏_t) in Ω, s,t=1,2,
which, in matrix form, gives
𝐆_U( 𝐫) = 𝐆_U( 𝐫) Γ( 𝐫) 𝐆_P( 𝐫),
where ( G_U) _st=𝐔_s^∗·𝐔_t,
( G_P) _ik=𝐏_i^∗·𝐏_k and
Γ is the matrix made by the unknown coefficients Γ_lm.
When both 𝐆_U and 𝐆_P are invertible at location 𝐫, the solution of (<ref>) exists, is unique and is given by
Γ( 𝐫) = 𝐆_P^-1( 𝐫).
In the remaining cases, i.e. 𝐆_P and/or 𝐆_U non invertible, the solution may not exist or be unique.
It is worth noting that matrices 𝐆_U and 𝐆_P are Gram matrices and, therefore, 𝐆_U=𝐆_U^†, 𝐆
_U≥ 0, 𝐆_P=𝐆_P^† and
𝐆_P≥ 0.
Moreover, the inverse of (<ref>) is (when it
exists)
χ=∑_l,m=1^2_ml^D𝐏_m𝐔_l^∗,
where ^D=( 𝐆_U^I
𝐆_P) ^-1.
§.§ Parameterization of the frequency response
Once the inverse of the susceptibility tensor is found at each each prescribed angular frequency
ω_k, we need to reconstruct the dispersion relation ( 𝐫,ω), which has to satisfy the causality throught the Kramers-Kronig conditions and the Hermitian symmetry, namely ( 𝐫,-ω)=^* ( 𝐫,ω). To this purpose, we parameterize the dispersion relation, as follows
( 𝐫,ω) =∑_m=1^M𝐚
_m( 𝐫) φ_m( ω)
where M is the number of terms, each expansion function φ_m is causal and Hermitian and each tensor field 𝐚_m is real. The φ_ms depend on the actual realization of the artificial material. A possible choice consists in assuming each expansion function φ_m of the
Lorentz-Drude type:
φ_m( ω) =ω_p,m^2/( ω_0,m
^2-ω^2) +jωβ_m,
where causality requires β_m>0.
Tensors fields 𝐚_ms can be found by point matching, for instance. Within this approach, we enforce the following constraints ∀ k =1, …, N
∑_m=1^M𝐚_m (𝐫) Re{φ_m( ω
_i)} = Re{γ_k^-1 (𝐫) }
,
∑_m=1^M𝐚_m (𝐫) Im{φ_m( ω
_i)} = Im{γ_m^-1 (𝐫) }
,
where Re{·} and Im{·} are the real and imaginary parts of their argument, respectively. Moreover, from (<ref>) and (<ref>), it follows that M=2 N to have existence and uniqueness of the solutions in terms of the unknown tensor fields 𝐚_ms.
We remark that parameters ω_p,m, ω_0,m and β_m depend on the actual realization of the artificial material. For instance, ω_0,m does not need to be equal to the resonant (angular) frequency ω_m prescribed for the Synthesis of the Modes. In the remaining of the paper we select parameters ω_p,m, ω_0,m and β_m, to avoid the appearance of any resonance due to the expansion functions, at the resonant frequencies prescribed for the Synthesis of the Modes.
§ TUNABILITY AND ESSENTIAL MODES
The tunability of the resonance refers to the possibility of changing the properties of a material in a controlled manner. The Synthesis of Modes entails tunability in a natural manner via the material eigenvalues ξ_k.
Indeed, after (<ref>), we have that a material with dielectric permittivity given by χ / ξ_k, being χ the result of the synthesis of modes, resonates at the angular frequency given by ω_k. In other terms, we can control the frequency behaviour of a material (value of the frequency resonances and spatial distribution of the related mode), by simply scaling χ by a proper factor.
From another perspective, the proposed approach to the synthesis of the modes allows to get the resonance frequencies and related spatial modes as a function of an individual parameter: a scaling factor in front of the synthesized χ.
This feature open the door to a systematic design of reconfigurable materials.
The concept of essential modes refers to the maximum number of modes that can be arbitrarily prescribed at a given angular frequency ω_k. Equation (<ref>) provide the values of the Γ_lm giving the sought inverse of the dielectric susceptivity tensor in (<ref>). This equation shed light on a special and not obvious physical feature of the modes: two modes are capable of defining uniquely the material property of the scatterer, at the prescribed angular frequency. In other words, γ(·,ω_k) is in a one-to-one correspondence with two of its modes at ω_k, at ω_k. From another perspective, only two modes can be assigned in a completely independent manner or, equivalently, all the modes depend upon two arbitrarily selected modes, at a prescribed angular frequency.
We term two arbitrary modes in a one-to-one correspondence with χ(·,ω_k) as essential modes.
It is worth noting that he number of essential modes is two in a 2D problem and three in a 3D problem.
§ APPLICATION OF THE THEORY OF SYNTHESIS OF MODES
In this Section, we show the effectiveness of the resonance synthesis method by means of three application examples. We demonstrate (i) the capability of the method to synthesise several modes, each one having prescribed polarization density distribution at prescribed frequencies, (ii) the tunability of resonant response, by a proper scaling of the dielectric susceptibility tensor and (iii) the concept of essential modes. In the first two examples, the reference geometry is an indefinite cylinder with square L× L cross-section with L=10cm under the illumination. In the third example the geometry consist of coated spherical gold nanoparticle.
The numerical model for solving the electromagnetic problem is derived from Ref. <cit.>. The parameters of the Lorentz-Drude expansion functions φ_k, introduced in Eq. (<ref>), are given in Table <ref>. The plot of each individual expansion function is shown in Figure <ref>. The positions of the peaks of the expansion function are uniformly spaced over the bandwidth of interest. We assume ω_p,k=ω_0,k and β_k=0.1ω_0,k. With this latter choice, each expansion function is localized in a neighborhood of its peak position, but does not present a sharp resonance that could hide those arising from the Synthesis of Modes. The amplitude and the shape of the expansion function are briefly discussed in Appendix <ref>.
Synthesis of the modes. In this first application, we prescribe the modes at the three angular frequencies shown in Table <ref>. Specifically, at the angular frequency ω_1 we prescribe two modes: the first one has a polarization density field 𝐏_0, whose shape resembles the number “0" and it is associated with the eigenvalue ξ_A=1; the second mode has a polarization density field 𝐏_1, whose shape resembles the number “1" and it is associated with the eigenvalue ξ_B=2. At the angular frequency ω_2, we prescribe the modes 𝐏_1 and 𝐏_2, where 𝐏_2 has a shape which resembles number 2. Modes 𝐏_1 and 𝐏_2 are associated with eigenvalues ξ_A=1 and ξ_B=2, respectively. Finally, at the angular frequency ω_3 we prescribe modes 𝐏_2 and 𝐏_0, associated with eigenvalues ξ_A=1 and ξ_B=2, respectively. Tables <ref> and <ref> summarize these choices.
The synthesis is carried out in two steps: i) we evaluate γ_i ( 𝐫) at the three prescribed frequencies; ii) we interpolate the corresponding dielectric susceptibility as in Eq. (<ref>), by solving (<ref>) and (<ref>).
In the first step, the theory for the synthesis of two isofrequential modes
is applied at each individual angular frequency using equation (<ref>): (i) for (ω_1, ξ_A, 𝐏_0) and (ω_1, ξ_B, 𝐏_1) at ω_1, (ii) for (ω_2, ξ_A, 𝐏_1) and (ω_2, ξ_B, 𝐏_2) at ω_2 and (iii) for (ω_3, ξ_A, 𝐏_2) and (ω_3, ξ_B, 𝐏_0) at ω_3.
Figures <ref>, <ref>, and <ref> show the real and imaginary part of every element of the relative dielectric permittivity tensor ε_R,k=χ_k+1, at ω_1, ω_2, and ω_3, respectively.
To validate the proposed method, we performed two tests, where the dielectric susceptibility profile is either χ^𝙰 ( 𝐫, ω ) = χ ( 𝐫, ω ) / ξ^𝙰 or χ^𝙱 ( 𝐫, ω ) = χ ( 𝐫, ω ) / ξ^𝙱, where χ ( 𝐫, ω ) is the outcome of the synthesis of modes.
The first test was a direct test and it consisted in i) computing the modes at the three frequencies and in ii) comparing them with the prescribed polarization density field. This test was passed successfully.
As second test, we evaluate the induced polarization density fields at the three frequencies ω_1, ω_2, and ω_3, when the cylinder is excited by a linearly polarized plane wave, propagating along the horizontal axis. These polarization fields are showed in Fig. <ref> (e-c) assuming a susceptibility tensor χ^𝙰(𝐫,ω) and in Fig. <ref> (d-f) for χ^𝙱.
The induced polarization density fields is very close to the prescribed modes. In quantitative terms, Table <ref>, shows the 2-norm of the relative difference between the actual 𝐏 and its projection along the subspaces generated by the prescribed modes, at each specific angular frequency:
ρ_k^i = ‖𝐏_i( ·,ω_k) -Π^i_k𝐏_i( ·, ω_k) ‖/‖𝐏_i( ·, ω_k ) ‖
with k=1,2,3 and i=𝙰,𝙱. In (<ref>), 𝐏_𝙰( ·,ω_k) and 𝐏_B ( ·,ω_k) are the polarization vectors at ω_k and for material 𝙰 and 𝙱, Π^𝙰_k and Π^𝙱_k are the projector into the linear space for the modes at the k-th angular frequency ω_k and for material 𝙰 and 𝙱. The detail about projectors Π^𝙰_ks and Π^𝙱_ks is given in Table <ref>.
We stress that
𝐏_i ( ·, ω_k) is the polarization vector for the
physical system under the prescribed illumination at ω_k.
This example clearly illustrates the concept of tunability of the resonant response: by just uniformly halving the value of the susceptibility distribution (passing from χ^𝙰 to χ^𝙱) the resonance modes in correspondence of the peaks change from the ordered sequence 0, 1, 2to 1, 2, 0.
Tunability. In this second application we determine the dielectric susceptibility by synthesizing at the frequency ω_1 the degenerate modes 𝐏_ and 𝐏_∨, whose polarization density field distribution resembles the characters and ∨, respectively; and at ω_2 the degenerate modes 𝐏_- and 𝐏_|, whose prescribed field distribution resembles the characters - and |, respectively. To validate the performed synthesis, we excite the infinite cylinder with a plane wave polarized along (𝐞_1+𝐞_2)/√(2). We show the real and imaginary part of the induced polarization field distributions at ω_1 in Figures <ref>(c), (d), and in Figures <ref>(g), (h) at ω_2. It is immediately apparent that at ω_1 the induced polarization field is a linear combination of the two prescribed degenerated modes 𝐏_ and 𝐏_∨, while at ω_1 the induced polarization field is a linear combination of 𝐏_- and 𝐏_|. From the quantitative perspective, the 2-norm relative difference ρ between the actual 𝐏 and its projection along the subspaces generated by the prescribed degenerated modes, is equal at 2.9908 × 10^-2 at ω_1 and 3.5310 × 10^-2 at ω_2. In this case Π_1 projects onto {𝐏_, 𝐏_∨}, whereas Π_2 projects onto {𝐏_-, 𝐏_| }.
Essential modes.
This final application case demonstrates a key feature of the Theory of the Synthesis of Modes, i.e. the concept of Essential Modes.
Specifically, given a scatterer operated at a prescribed angular frequency ω_1 and described by the dielectric susceptivity tensor χ(·,ω_1), we compute two resonance modes (ω_1, ξ_A,𝐏_A) and (ω_1, ξ_B,𝐏_B) and, then, we apply our Theory of the Synthesis to these modes. Since the tensor of the dielectric permittivity is in an one-to-one correspondence with two arbitrary modes, as discussed in a previous Section, we expect that the tensor χ_s(·,ω_1) of the dielectric permittivity synthesized by means of (ω_1, ξ_A,𝐏_A) and (ω_1, ξ_B,𝐏_B) via (<ref>), is equal to χ(·,ω_1).
The scatterer of this example consists of a coated (thickness 100 nm) circular (radius 200 nm) gold nanorod operated at f=500 THz (ω_1=π× 10^15 rad/s, free-space wavelength of 600 nm). The relative dielectric permittivity of the gold nanoparticle is 9.44-j 1.51, whereas that of the coating is 4.
Figures <ref> and <ref> show the real and imaginary parts for the selected modes 𝐏_A and 𝐏_B. The synthesized dielectric permittivity tensor is almost equal to that of the prescribed scatterer. As a figure of merit we evaluated the maximum relative error over the scatterer domain Ω:
e=max_𝐫∈Ω||χ(𝐫,ω_1)-χ_s(𝐫,ω_1)||_2/||χ(𝐫,ω_1)||_2,
which, in this case, is equal to 3.3 × 10^-11. In (<ref>) χ is the prescribed tensor of the dielectric susceptibility, whereas χ_s is the tensor of the synthesized dielectric susceptibility.
§ CONCLUSIONS
In this work we introduced a theoretical framework to find the permittivity profile of a dielectric object to synthesize at will its resonant modes. Specifically, we are able to control the spatial distribution of the polarization density field and the resonance frequency of a set of modes. The equations for the synthesis are straightforward and in an explicit form, making them suitable for specific customization. Moreover, we can prescribe the modes at many different frequencies.
The only limit, arising from the underlying physics, consists in the possibility of assigning at most two modes to each individual frequency and eigenvalue (up to three modes in a 3D setting). Indeed, from the theory of the synthesis of modes arises naturally that, at a prescribed angular frequency, the dielectric susceptivity tensor is in one-to-one correspondence with two of its modes, that we termed as essential modes.
We also demonstrated the concept of tunability: the proposed approach enables the design of the permittivity of a dielectric object that not only allows the synthesis at will of its resonant modes, but also allows to changes the resonant modes of the dielectric object in a controlled manner, by multiplying the designed permittivity by a proper multiplicative factor.
We also demonstrated the concept of tunability: our approach enables the design of the permittivity of a dielectric object, that not only allows the synthesis at will its scattering resonances, but also allows when such permittivity is multiplied by a proper multiplicative factor, it changes its resonant behaviour in a controlled manner. This is relevant from the practical point of view because this operation (multiplication by a constant) appears to be a simple operation.
With this theoretical framework, future development will be aimed to design a real world material approximating the synthesized dielectric susceptibility. Metamaterials are the natural candidates to this purpose.
The method introduced can be transplanted to different linear physical systems, where the constitutive relationship is linear and local, including thermal and mechanical systems.
§ METHODS
All the numerical calculations have been carried out by using the numerical method of <cit.>. All the value of the parameters used for generating numerical results have been included into the article.
§ DATA AVAILABILITY
All the data supporting the conclusions of this study are included in the
article. Source data are provided with this paper.
§ CODE AVAILABILITY
The computer code and algorithm that support the findings of this
study are available from the corresponding author on request.
§ GREEN FUNCTION
The component of the Green function for the illumination are
G_11( 𝐫) =-ζ_0/4r^3[
krx_2^2H_0( kr) +( x_1^2-x_2^2)
H_1( kr) ]
G_12( 𝐫) =-ζ_0/4r^3x_1
x_2[ 2H_1( kr) -krH_0( kr) ]
G_21( 𝐫) =G_12( 𝐫)
G_22( 𝐫) =-ζ_0/4r^3[
krx_1^2H_0( kr) +( x_2^2-x_1^2)
H_1( kr) ] ,
being ζ_0 the characteristic impedance of vacuum, k=ω/c_0 the
wavenumber, and c_0 the speed of light in vacuum.
§ LORENTZ-DRUDE EXPANSION FUNCTION
The (normalized) amplitude of the elementary Lorentz-Drude expansion function is:
| φ (ω) |/( ω_p / ω_0)^2 =1/√([ 1 - ( ω/ω_0)^2]^2 +( ω/ω_0)^2 ( β/ω_0)^2).
Its maximum value is
| φ (ω) |_max/( ω_p / ω_0)^2 =1/β/ω_0√(1 + 3/4( β/ω_0)).
and it is achieved at
ω/ω_0 = √(1+1/2( β/ω_0)^2)
The plot of (<ref>) for different β / ω_0 ratios is showed in Figure <ref>.
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|
http://arxiv.org/abs/2307.04526v2 | 20230710124959 | Self Expanding Neural Networks | [
"Rupert Mitchell",
"Martin Mundt",
"Kristian Kersting"
] | cs.LG | [
"cs.LG",
"I.2.6"
] |
Automatic Debiased Machine Learning for Covariate Shifts
Michael Newey and Whitney K Newey Research was sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. This research was supported by NSF Grant 1757140
August 12, 2023
====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
The results of
training a neural network are heavily dependent on the architecture chosen;
and even a modification of only the size of the network,
however small,
typically involves restarting
the training process.
In contrast to this,
we begin training with a small architecture,
only increase its capacity as necessary
for the problem,
and avoid
interfering with
previous optimization
while doing so.
We thereby introduce
a natural gradient based approach
which intuitively expands both the width and depth of a neural network
when this is likely to substantially reduce the hypothetical converged training loss.
We prove an upper bound on the “rate” at which neurons are added,
and a computationally cheap lower bound on the expansion score.
We illustrate the benefits of such Self-Expanding Neural Networks in both classification and regression problems,
including those where the appropriate architecture size is substantially uncertain a priori.
§ INTRODUCTION
Correctly tailoring a model's capacity to an arbitrary task is extremely challenging, especially when the latter is not yet well studied.
This challenge can be side stepped by choosing an architecture which is so large that a poor solution is nevertheless unlikely to occur <cit.>, e.g. due to the double-descent phenomenon. However, since it is hard to predict what size would be large enough this will often in practice entail using a massively overparameterized network <cit.> <cit.> <cit.>.
Surely it is possible to detect that the existing capacity of the network is insufficient and add more neurons when and where they are needed?
In fact, biological neural networks are grown by adding new neurons to the existing network through the process of neurogenesis.
The popular review <cit.> discusses the relatively recent discovery that this process is still active in the adult mammalian brain
<cit.>,
and <cit.> <cit.> identify it as a key ability underpinning lifelong learning.
Thus inspired,
we propose an analogous process for adding both neurons and layers to an artificial neural network during training,
based on a local notion of “sufficient capacity” derived from first principles in close relation to the natural gradient <cit.> <cit.>.
Any method for artificial neurogenesis
must answer three questions to avoid the problem of locally insufficient capacity <cit.>.
It must determine when the current capacity is insufficient and that neuron(s) must therefore be added.
It must identify where these neurons should be introduced.
Finally, it must choose what initialization is appropriate for these neurons.
These questions, if they are addressed at all in the literature, are normally addressed piecemeal or in ad-hoc ways. For example, very few methods address the question of what <cit.> <cit.>.
is answered either by assuming predetermined schedules <cit.> <cit.>,
or by waiting for the training loss to converge <cit.> <cit.>,
neither of which are informative about where.
“Whenever you parry, hit, spring, ..., you must cut the enemy in the same movement.”
[
Miyamoto Musashi, The Book of Five Rings (circa 1645)
]
Our metaphorical enemy is not a loss which is momentarily poor, or even one which is converging to a poor value:
it is a deficiency in our parameterization such that the optimizer cannot make progress.
We argue that by inspecting the degrees of freedom of the optimizer in function space,
one may not only strike faster in answer to when, but answer where and what in the same stroke.
From a mathematical perspective, these degrees of freedom available to the optimizer
are given by the image of the parameter space under the Jacobian,
and the derivative with respect to the loss in function space will not in general lie in this subspace.
It is however possible to project this derivative onto that subspace,
and the natural gradient, ^-1,
is exactly the change in parameters which changes the function according to this projection.
In order to measure the size of that projection for a given parameterization,
we introduce the natural expansion score η = ^T ^-1.
Specifically, the capacity of a neural network is locally insufficient when this score is small for the current parameterization. We therefore add neurons when this substantially increases η, where they will maximally increase η, and choose what initialization to use for the new parameters according to how it increases η. To summarize, our contributions are:
* We introduce the natural expansion score which measures the increase in rate of loss reduction under natural gradient descent when width or depth is added to a neural network.
* We show how such additions may be made during training
without altering the function represented by the network.
Our neurogenesis inspired Self-Expanding Neural Networks (SENN) thus avoid interfering with previous optimization or requiring restarts of training.
* We prove that the number of neurons added simultaneously in SENN is bounded. We further introduce a computationally efficient approximation as a provable lower bound to increases in natural expansion score resulting from additions.
* We demonstrate SENN's effectiveness for regression and classification.
In the remainder of this paper, we proceed as follows:
In section <ref> we summarize existing growth methods,
in section <ref> we then describe SENN,
and in section <ref> we illustrate its operation in practice.
§ RELATED METHODS FOR GROWING NEURAL NETWORKS
The problem of adding nodes to neural networks during training has been under consideration for over 30 years (e.g. Dynamic Node Creation (DyNC) <cit.>),
but remains substantially unsolved.
There does not seem to exist a unified answer to , and ,
as we summarize in table <ref>.
Most methods cannot add depth and sideline at least one of these questions.
Inspired by neurogenesis like SENN, <cit.> examine the case of representational learning with stacked autoencoders,
where they exploit local reconstruction error to determine and to add neurons.
Due to their more general setting,
DyNC,
Progressive NNs (PrNNs) <cit.> and Dynamically Expandable NNs (DENNs) <cit.>
use simple training loss convergence or even task boundaries to answer , but must then fall back on ad-hoc preset decisions for .
(However, DENNs use subsequent pruning to mitigate the excess capacity introduced by the preset.)
All four methods freeze old neurons or continue training from their present values,
but randomly initialize new neurons in answer to .
While ActiveNAS <cit.> can add both width and depth,
it does so by completely restarting training with a fresh initialization of the whole network after every modification.
It then waits for convergence,
and uses preset answers to similar to the previous methods.
The final cluster of three methods all aim to improve on random initialization as an answer to what.
Splitting Steepest Descent (SSD) <cit.> and Firefly <cit.> make small changes to the existing function and answer by optimizing the consequent loss reduction.
The former answers by waiting for convergence and examining the loss, whereas the latter simply adds more capacity every N epochs.
Gradmax <cit.> is the closest to SENN in spirit,
but is based on vanilla rather than natural gradient descent. More importantly, potential extensions of the method to the and questions are mentioned briefly and their investigation deferred to future work.
All three of these latter methods are only able to avoid redundancy of added neurons with existing neurons to the extent that the network is already converged. Of these three, only GradMax completely avoids changing the overall function.
In contrast, SENN provides a monolithic answer to all three questions via the natural expansion score.
§ SELF-EXPANDING NEURAL NETWORKS
To provide a cohesive answer to , , with Self-Expanding Neural Networks,
we start with the definition of the natural expansion score as the foundation:
The natural expansion score η = ^T ^-1 is given by the inner product of the natural gradient ^-1 with the gradient .
With this definition
we will describe we add capacity without interfering with the existing optimized parameters
in section <ref>.
We then in section <ref>
give an intuitive account of what our score η measures,
and why we use this to decide to add capacity.
Section <ref> gives a more mathematically precise account of the meaning of η,
and what this says about initializations should be used for new capacity.
Section <ref> extends the argument of <ref> to deciding new capacity should be added and whether it should be depth or width,
allowing us to put the ingredients of SENN together and summarize this combination.
Finally, sections <ref> and <ref> cover the practical questions of convergence guarantees and computational efficiency respectively.
add conditions for Wout equals and sigma p equals
§.§ How to add: expanding without changing the overall function
In order to explain how to add without changing the overall function,
we will consider the illustration in figure <ref>.
This shows a perceptron with two hidden layers, each with three neurons.
The number of neurons a hidden layer may be increased by introducing a new copy of the activation function σ_p
and connecting it to the neurons of the preceding layer with some linear transform _p.
As shown on the left of the figure, we connect the new neuron to the subsequent layer (in this case the output )
with a linear transform initialized to zero.
In doing so, we guarantee that we will not perturb the function specified by the existing parameters.
Although _p will initially receive zeroed gradients since the output transform is zero,
this latter transform will immediately receive non-zero gradients and thereby become non-zero.
The new neuron may thus be used in future optimization.
In addition to width expansion, we now consider inserting an entirely new layer,
as shown on the right of figure <ref>.
In essence, a particular linear transform, _2 in the figure, is replaced with a single layer perceptron.
To this end, we assume our nonlinearity σ_p to be parameterised, and there to exist a choice of those parameters such that σ_p = is the identity.
If we require the initial linear transform _p of the inserted perceptron to be invertible (but otherwise arbitrary),
then we may choose the output linear transform of the perceptron to be the matrix product _2 _p.
With these choices made, the inserted perceptron is equivalent to the linear transform _2 it replaces,
and the overall parameterized function once again remains unchanged.
We thus have the first ingredient of SENN:
SENN Ingredient 1: How to add more capacity without changing the overall function.
We add proposed neurons p to layer i
by concatenation along the the ith hidden dimension
(0⊎_i+1) ∘ (σ_p ⊎σ_i) ∘ (_p ⊎_i) = _i+1∘σ_i ∘_i,
and initialize the output weights of p to zero.
We insert a new layer q by replacing some linear transform _i
with the composition (_i _q) ∘ (σ_q = ) ∘_q,
where _q is invertible and σ_q is initialized to be the identity.
We must therefore choose a suitable parameterized activation function.
Rational activation functions satisfy the above conditions and were shown to obtain good real world performance <cit.>.
We use the simplified parameterization
σ_ (x) = α x + (β + γ x)/(1+x^2),
where = {α, β, γ} are the three parameters of σ,
and setting = { 1, 0, 0 } results in the identity function, as required.
Since this parameter count is small, we do not share the activation function weights within our layers.
§.§ When to add: deciding whether more capacity is useful
Having decided how to add,
perhaps the most natural way to evaluate the utility of making
some change to the parameterization is to ask what immediate effect this has on the total loss.
However, we cannot do this as we have assumed the overall function to remain unaltered.
We must therefore consider additional information such as the gradients of the function.
Specifically, one can favor adding neurons which maximally increase the euclidean norm of the gradients ||||_2.
As found in <cit.> this norm functions well for selecting which neurons to add when the network is close to convergence since
it is a direct measure of the rate at which gradient descent will decrease the loss.
Unfortunately, comparing the gradient norms ||||_2^2 and ||'||_2^2 for the current parameterization and some new expanded parameterization '
is insufficient to determine whether or not more capacity is needed in the first place.
This is primarily because it does not account for redundancy in the parameterization:
if there is some neuron a such that the gradients of the linear weights in the next layer “listening” to it have some large norm ||_a||_2,
then we could introduce an exact copy of this neuron a' for which the corresponding norm would also be ||_a'||_2 = ||_a||_2.
Since the squared euclidean norm is additive across parameters,
we could unboundedly increase ||||_2^2 just by adding very many copies of this one neuron a.
[
More generally, the same problem would occur when considering a new neuron c whose activations were some linear combination of those of some existing neurons a and b.
]
In SENN, we avoid this problem with the following simple notion of redundancy.
We are using our parameters to express a point in function space.
At some point in optimization we are therefore also using them to express a small change in function space.
There is some direction that our optimizer “wants” to move in (i.e. the direction in function space which most quickly reduces the loss).
We can define new parameters as being useful in a way which is non-redundant with the old parameters to the extent that they allow the optimizer to better express the direction in function space it “wants” to move in.
Our natural expansion score η = ^T ^-1 captures this combined sense of usefulness and non-redundancy in a way which will be made more mathematically precise in the next section.
This description of its function is sufficient, however, to justify our answer to when:
SENN Ingredient 2: When to add more capacity.
A new neuron or layer will be helpful and non-redundant if it provides a fractional increase in η = ^T greater than some threshold τ.
we find a potential new neuron or layer for which this is true, we add it.
We defer specific choices for τ to section <ref>,
at which point we may draw on the derivation of η.
§.§ What to add: determining the initial value of new neurons
We are assuming fisher information metric on output is euclidean here. Maybe mention in footnote?
The reader may at this point be expecting us to tackle the question of additional capacity is most useful,
but this would put the cart before the horse.
Additional capacity is useful to the extent that it can be initialized in a way which is useful,
which we now consider.
To simplify mathematical notation in this section,
we consider the output to be concatenated over the entire training dataset.
While the gradient of the loss with respect to the output _ tells us how the loss changes for arbitrary changes in ,
the only changes in we can actually achieve with some parameterization Θ are given by Jacobian product for some small parameter change ∈Θ.
Let _Θ be the orthogonal projection onto this space of directions in output space.
The vector _Θ_ is then the portion of _ which lies in the space of achievable output changes,
and its squared norm ||_Θ_||_2^2 is a scalar measure of how much of _ this portion is.
The vector _Θ_ is the image under the Jacobian
of some tangent vector in the parameter space.
By the definition of orthogonal projection, minimizes || - _||_2,
but if there are redundant directions in Θ then there may exist multiple such .
There is however a unique _* which minimizes ||||_2 among those which minimise || - _||_2.
The Moore-Penrose inverse, ^+, of is the unique matrix such that _* = ^+ _ for arbitrary _.
However, is a map from parameter space to total output space, which depends on dataset size N.
This dependency can be avoided by working with maps from the parameter space to itself,
such as the following
average over the dataset = 1/N^T,
known as the Fisher information matrix.
The natural gradient is then given by ,
where = 1/N^T _ is the gradient of the loss with respect to the parameters averaged over the training set,
and existence of = + ϵ is guaranteed by the addition of a small multiple ϵ of the identity.
In the limit of small ϵ this is exactly our _*.[
In fact, an alternative definition of the Moore-Penrose inverse is:
^+ := lim_ϵ→ 0(^T + ϵ)^-1^T
]
We are now able to rewrite the squared norm ||_Θ_||_2^2 in the familiar form of definition <ref>:
||_Θ_||_2^2 =
_*^T ^T _* =
^T ^-1^T ^-1 =
N ^T ^-1 =
N η .
Here, the factor of the dataset size N appears because the average gradient and our η are normalized according to the training set size.
With this formula, we have now derived η from first principles
and may use it to choose between specific initializations, yielding our third SENN ingredient:
SENN Ingredient 3: What Initialization to Use.
If ' ∈Θ' is an initialization of an expanded parameterization Θ' such that the overall function remains unchanged (see section <ref>), then the best such initialization _*' is given by
_' (η').
When we add new neurons or layers, we choose initialization to use by this method.
§.§ Where to add: completing the algorithm
Much as the euclidean norm ||||_2 measures the rate of loss reduction according to vanilla gradient descent,
our η measures the rate of loss reduction according to natural gradient descent.
This gives a uniform way of comparing the effect of new capacity no matter where it is added in the network or whether it takes the form of new neurons in an existing layer or a new layer.
In particular, one may compare the η values of the best initializations (see section <ref>) for each such variety of addition.
[
In general one can also adjust for the “size” of each addition in some relevant sense.
We found it sufficient to just penalize adding entire new layers versus single new neurons by some constant factor.
]
SENN Ingredient 4: Where to Add.
A choice of whether to add width or depth, and in the network the new neuron/layer will be added,
specifies a particular extension of the current parameter space Θ'.
We make those choices which correspond to the extension Θ_*' = _Θ'_' (η') for which the best initialization is possible.
Our newfound knowledge of η as a rate of loss reduction in hand,
we return to the question of specifying the expansion threshold τ,
which we deferred from section <ref> in our previous answer to when.
An increase from the current natural expansion score η_c to a new score η_p due to some proposed expansion p corresponds to an increase in the rate of loss reduction by natural gradient descent.
We define this increase to be “sufficient” when it corresponds to a relative increase η_p / η_c > τ
in loss reduction rate greater than the expansion threshold τ.
For example, with the intuitive choice τ=2,
each addition must at least double the rate of loss reduction.
Following the well known intuition that a network does not practically converge without setting the learning rate to zero,
it is generally considered to have converged once changes in loss become sufficiently small.
In analogy to monitoring plateaus in loss,
we further require the increase in loss reduction resulting from new capacity to surpass an absolute stopping criterion α. While we answer when, and cohesively with η during training,
we thus concur with all prior works on terminating training.
Overall, we may now summarize all ingredients of SENN on the basis of the natural expansion score:
SENN: Summary.
When we add width or depth we do so without changing the overall function.
We add new capacity when this produces a relative increase in score η_p / η_c > τ
larger than the expansion threshold τ.
We add new capacity where it would most increase η,
and choose what initialization to use in order to maximize η.
We ensure the addition process terminates by additionally comparing each Δη
to the absolute stopping criterion α,
and not adding capacity when η_p - η_c ≤α.
§.§ Bounds on convergence of expansion
Consider repeatedly running our addition algorithm for a network with initial expansion score η_0.
The expansion threshold τ guarantees that η_i > τη_i-1 after the i-th addition.
Since η = ^T ^-1
is the squared length of the projected gradient in output space ||P_Θ_||_2,
it is non-negative and bounded above by η≤λ = ||_||_2^2.
Since η_i grows exponentially with i
and is bounded above by λ
the maximum number of sequential additions i < N_s
increases logarithmically with λ.
Specifically, N_s < (lnλ - lnη_0)/lnτ.
This bound becomes large when η_0 is small,
but we also know that η_1 > α from the stopping criterion α.
The maximum number of additions N_s from repeatedly running the expansion algorithm is bounded:
N_s < 1 + (lnλ - lnα)/lnτ.
(Proof in supplementary material.)
For example, if τ = 2 and α / λ > 10^-3 then N_s < 1 + 3ln10/ln2 < 11.
Note that exponentially large ratios between α and λ produce only linearly large bounds on N_s.
We now consider the number of additions N_T made over the course of training with natural gradient descent.
Intuitively, λ is the total possible loss reduction and α is the minimum reduction which justifies expanding the network.
If every time we expand the network it only achieves this minimum reduction then we must expand a total of roughly N_T ≈λ / α times.
If the loss function has constant curvature equal to the fisher ,
then the total loss reduction possible with the current parameters is given by 1/2η
and we have N_T < λ / α exactly.
More generally,
we expect that when is an underestimate of the true curvature,
η will overestimate the usefulness of new neurons causing N_T to be larger,
and vice versa for an overestimate.
See supplementary for more in depth discussion.
§.§ Efficiently computing a lower bound on score increase
Recall that the natural expansion score η is given by the inner product of the gradient with the natural gradient ^-1.
Since working with the natural gradient can be challenging due to the matrix inverse ^-1,
we will make use of established approximation techniques.
Specifically, when we need the natural gradient for the whole network we will use the iterative conjugate gradient method, as suggested for the Hessian in <cit.>,
performing Fisher-vector multiplication cheaply via auto-differentiation.
check if there is a better citation which uses F not H
When we require the inverse Fisher _l^-1 for the linear transform in some layer l considered in isolation,
we approximate _l by the Kronecker product _l ≈_l = _l ⊗_l,
choose S or G
where _l is the second moment of the activations at the input of the linear transform,
and _l is given by the second moment of some distribution of gradients with respect to the output of the linear transform.
The relevant gradient distribution is determined by the choice of metric on the output space implicit in the exact definition of one is using,
which for us is the euclidean metric.
The advantage of this Kronecker factorization is that _l may be inverted by inverting _l and _l separately:
_l^-1 = _l^-1⊗_l^-1,
which is much cheaper.
If ∂ is the gradient with respect to the weights as a matrix,
then the natural gradient is given by ^-1∂^-1<cit.>.
The natural expansion score η is given by the inner product of the gradient with the natural gradient as vectors,
which in this matrix form becomes the elementwise inner product
η = ∑_i,j∂_ij (^-1∂^-1)_ij,
which can also be expressed as a trace: η = [∂^T ^-1∂^-1].
The trace formula for η is reminiscent of the definition of
the pearson correlation coefficient r^2 = xy^2 / (xxyy).
The gradient for is given by the expectation ∂ = ^T,
where is the input activation vector,
is the derivative of the loss with respect to the outputs,
and the expectation is over the dataset.
Let the residual gradient
_r = - ^T^-1
insert/handle layer indices here
be the part of the gradient not predicted by the current activations .
Then if _p is the activation vector of a set of proposed neurons,
and _p is their second moment,
then the “correlation coefficient” of the new activations with the residual gradients is a lower bound Δη' on the improvement Δη in natural expansion score
(proof in appendix via block LDU decomposition of joint activation covariance):
Δη' := [_p^-1_p _r^T_l^-1_r _p^T]
is a lower bound Δη' ≤Δη = η_p - η_c on the improvement in natural expansion score due to some proposed addition of neurons p to a layer l.
Intuitively, Δη' is the fraction of variance in residual gradients “explained” by the output of our new neuron(s).
This result holds for adding an arbitrary number of new neurons to an existing layer.
If a layer was inserted while retaining residual connections around it,
then the same result would hold if we treated the activations of the new layer as “new neurons” in the old layer to calculate Δη'.
Because our activation function can represent the identity,
we will automatically add these connections if in fact they are necessary,
so we in fact use this same method for evaluating our actual layer insertions.
The bound Δη' can be computed for an arbitrary proposal p of additional neurons
using only those intermediate activations and gradients which it would be necessary to cache
in order to calculate the gradient and (Kronecker factored approximate) natural gradient
via backpropagation.
Therefore, if we have an outer optimizer which computes and ,
then we may optimize arbitrarily many proposals p for arbitrarily many steps with an inner optimizer
without incurring any additional costs related to the evaluation of the existing network.
The costs of this inner optimizer instead scale with the size of the (very small) networks whose addition to the existing network is being considered.
§ EXPERIMENTS
We now apply Self-Expanding Neural Networks to regression and classification, to illustrate the behavior of the natural expansion score and demonstrate SENN's efficacy.
§.§ Width Addition in Least-Squares Regression
We first show that the evolution over training of the possible improvements Δη' in natural expansion score due to potential width expansions is meaningful.
In order to do so we consider the application of a single layer SENN to a one dimensional least squares regression task as shown in figure <ref>,
i.e. SENN with depth addition deliberately disabled.
The reason to have only one hidden layer is that this is effectively least squares regression with basis functions given by the neurons of that layer.
We can therefore plot the normalized score increase Δη' / η_c of the best neuron for each basis function location and length scale.
Where Δη' / η_c > 1 there exists an acceptable proposal.
Accepted/rejected proposed neurons are shown on this landscape in red/black at key points in training.
We see in the leftmost figure that the best such proposal is accepted because it achieves a large improvement in η,
and it corresponds to a basis function location close to datapoints with visibly large prediction error which we have been unable to reduce using the existing neurons.
The next figure to the right shows the same landscape after the new neuron is introduced,
and it can be seen that the Δη' / η_c values for neurons with similar locations to it have been dramatically reduced
since they would be redundant.
The second figure from the right shows the result of optimizing the new expanded parameters until the point at which the next neuron would be added.
It can be seen that the prediction errors in the region of the previously introduced neuron are now practically invisible,
and that the next neuron is to be introduced in a different region in which errors remain.
The rightmost figure shows the function approximation at the conclusion of training,
and it can be seen that the prediction errors are negligible and proposals with large relative increase in η are not to be found in the region considered.
The reader may note that there are some possible new neurons with small length scales which would surpass the expansion threshold which we do not find;
we could deliberately try optimizing initializations at this lengthscale to find these,
but this would likely result in overfitting.
Overall, SENN thus identifies regions of locally insufficient capacity in our parameterization,
targets these regions precisely with new added neurons,
and uses this selectively added capacity to achieve a good final fit.
§.§ Layer Addition in Classification
consider briefly mentioning width addition.
We now highlight SENN's depth expansion in the context of
classification.
Specifically, we consider two-dimensional inputs from the half-moons dataset <cit.>.
In figure <ref> we plot Δη' / η_c for the best layer addition proposals as a function of overall optimizer steps.
Visualizations of the learned decision boundary at initialization and just before layer additions are shown.
We can observe that Δη' / η_c increases approximately monotonically during three phases,
punctuated by large drops when layers are added.
In the initial phase the network has zero hidden layers (i.e. is linear),
and the simplicity of the decision boundary at the end of this phase reflects this.
Since the datapoints are not linearly separable,
the large Δη' / η_c value correctly indicates that the introduction of a hidden layer is necessary in order to further reduce loss.
The visible increase in decision boundary complexity and accuracy over the course of the second phase confirms this.
The beginning of the third phase marks the introduction of a second hidden layer and we wait until Δη' / η_c rises again,
indicating an exhaustion of this new capacity, before reexamining the decision boundary.
The increase in boundary complexity is less visible this time, but close inspection reveals that the boundary has become narrower and more rounded.
Conclusively, we have intentionally constructed a scenario where depth addition is necessary for a good fit to lie in the space of solutions,
and seen that SENN inserts new layers when this is necessary for global expressivity.
§.§ Dynamic Selection of Appropriate Architecture Size in Image Classification
Finally, we examine the ability of self-expanding neural networks to choose an appropriate size when classifying MNIST <cit.> images.
The leftmost plots of figure <ref> show SENN's total hidden size and validation accuracy during training on the full dataset as a function of total batches seen.
This use of mini-batching is not strictly necessary for MNIST but we use it to better reflect the realities of training modern neural networks.
Our SENN is initialized with a single hidden layer of size 10, and promptly adds a second hidden layer, also of size 10.
All five seeds considered then proceed to consistently add width to these layers at a moderate rate until a total hidden size of around 40 is reached,
at which point far fewer productive extensions of the network are found and addition slows dramatically.
It can be seen that this results in respectable validation performance (>97%) by the end of training with very modest hidden neuron counts (50-60).
It is of particular note that our method produces strong anytime performance:
we are able to continually expand size, and even insert layers, during training without any attendant drops in validation accuracy.
Indeed, our method exhibits mostly monotonic improvement up to stochasticity from batching,
a property not shared by methods which rely on reinitializing a new network, e.g. <cit.>. This makes SENN a perfect fit to prospective applications in e.g. active or continual learning, in the spirit of our original neurogenesis inspiration.
Having verified sensible performance of SENN on the full MNIST dataset,
we now examine the way in which they adapt their final converged size to the amount of information in the dataset.
To this end, we take class-balanced subsets of MNIST of varying sizes and train SENNs to convergence.
To maximize clarity in our examination of this relationship, we restrict the SENN to width addition.
The converged hidden sizes are shown together with the standard error across five seeds in the rightmost plots of figure <ref>.
The first of these shows log width against linear subset size for ease of comparison to the leftmost panel. It can be seen that the final width tails off rapidly with subset size.
The rightmost plot shows instead linear width against logarithmic subset size,
in which we can now distinguish three regimes.
For the smallest subsets, the initial hidden size of 10 is sufficient.
For subsets between 10% and 60% of the standard training set,
the final hidden size increases logarithmically,
but past that point further increases in subset size do not similarly increase the final network size.
We posit that this is due to substantial redundancy within the MNIST training set,
leaving further capacity growth unnecessary. Thus, SENN does not only provide desirable any time performance, but also tailors its size suitably to the available data.
§ CONCLUSION
We have introduced the natural expansion score η and shown how it may be used to cohesively answer the three key questions , and of growing neural networks.
We have demonstrated its ability to capture redundancy of new neurons with old and thereby make sensible expansion decisions
across time and tasks.
While we have focused on providing a thorough mathematical grounding of the natural expansion score in this work,
we acknowledge that the multilayer perceptrons on which it was demonstrated
differ in scale and complexity from many of the architectures in active use for deep learning in the modern big data regime.
Dually, however, prospects for further development are promising, as our theoretical results regarding η apply for arbitrary expansions of parameterized models,
and our method of expansion would extend naturally to, for example, convolutional neural networks or normalizing flows
where layers may be initialized invertibly.
This work was supported by the project “safeFBDC - Financial Big Data Cluster” (FKZ: 01MK21002K), funded by the German Federal Ministry for Economics Affairs and Energy as part of the GAIA-x initiative, and the Hessian research priority programme LOEWE within the project “WhiteBox”. It benefited from the Hessian Ministry of Higher Education, Research, Science and the Arts (HMWK; projects “The Third Wave of AI” and “The Adaptive Mind”).
named
§ PROOFS
§.§ Theorem 1: Bounded rate of addition
In this section we prove theorem 1 of the main body.
We will assume ≻ 0 to be positive definite, with the following straightforward consequence
The natural expansion score is non-negative η = ^T ≥ 0.
If ≻ 0, then ≻ 0,
and ^T ≥ 0 for all .
Considering the effect of the expansion threshold τ we obtain the following bound:
Let η have initial value η_0 and be bounded above by λ > η.
If threshold τ guarantees that η_i > τη_i-1 for the i-th addition,
then the maximum number of successive additions N_s is bounded by
N_s < lnλ - lnη_0/lnτ.
Due to the threshold τ, η grows at least exponentially: η_i > τ^i η_0.
But η is bounded: λ≥η_i > τ^i η_0.
Since ln is monotonic, we may take logarithms:
lnλ > i lnτ + lnη_0.
and rearrange to get i < lnλ - lnη_0/lnτ for all additions i.
This true for every i-th addition which is accepted, and so in particular also true for the last N_s-th addition.
Considering also the effect of the stopping criterion α we obtain theorem 1:
If the stopping criterion α guarantees that η_i > η_i-1,
then the maximum number of successive additions N_s is either 0, or bounded by
N_s < 1 + lnλ - lnα/lnτ.
Either N_s = 0, or there is a first addition with natural expansion score η_1 for which
η_1 - η_0 > α.
From lemma <ref> we then have η_1 > α.
We may then substitute α into lemma <ref> in place of η_0 to obtain
a bound on further additions, yielding
N_s < 1 + lnλ - lnα/lnτ.
This theorem is important because it guarantees that SENN will add a limited number of neurons or layers before continuing training.
Intuitively, this is because it rapidly becomes the case that any new neuron is either not relevant to rapidly decreasing the loss, or is redundant with some already extant neuron.
§.§ Theorem 2: Lower bound on increase in natural expansion score
We now prove theorem 2 of the main body, concerning a lower bound on the increase in natural expansion score η due to the addition of new proposed neuron(s) to a layer.
Let the joint activations = [ _c; _p ] of the current and proposed neurons have second moment
^T = = [ _c _cp; _pc _p ].
We will assume the Fisher matrix for the layer to which neurons are to be added to factorize as = ⊗, where ≻ 0 is positive definite.
We first derive a convenient form of a known result discussed in, for example, <cit.>,
related to the joint covariance of multivariate Gaussian distributions.
Let _p = _p - _pc_c _cp be the Schur complement of _c in .
Let also = [ _c; _p ] be an arbitrary vector,
and be the linear operator defined by = _p - _pc_c _c,
i.e. the residual part of _p not predicted by _c.
Then, ^T = _c^T _c _c +
()^T _p.
The following may be obtained by performing a block LDU decomposition:
=
[ _c _cp; _pc _p ] =
[ _c 0; _pc_c _p ][ _c 0; 0 _p ][ _c _c _cp; 0 _p ]
which we may then use to decompose :
=
[ _c _cp; _pc _p ]^-1 =
[ _c -_c _cp; 0 _p ][ _c 0; 0 _p ][ _c 0; -_pc_c _p ]
The desired result then follows by substitution into ^T:
^T =
[ _c _p ][ _c -_c _cp; 0 _p ][ _c^-1 0; 0 _p^-1 ][ _c 0; -_pc_c _p ][ _c; _p ]
=
_c^T _c _c +
(_p - _pc_c _c)^T _p^-1 (_p - _pc_c _c)
Recall from section 3.6 that η may be expressed as a trace:
η = [^-1^T^-1^T]
where is the derivative of the loss with respect to the outputs (i.e. layer pre-activations) of the linear transform.
We can use lemma <ref> to write the increase in natural expansion score Δη as
Δη = [^-1^T^-1^T]
- [^-1_c_c^-1_c ^T]
= [^-1 ()^T_p^-1() ^T]
where we can take inside the expectations by linearity.
It is computationally convenient for us to be able to have an expression in terms of residual gradients instead of residual activations, so we note the following:
()^T = _r _p^T
where _r = - _c^T_c^-1_c is the residual gradient.
()^T = (_p - _p _c^T_c^-1_c)^T
= _p^T - _c^T_c^-1_c _p^T
= ( - _c_c^-1_c) _p^T
= _r _p^T
Finally, we establish the following relationship between _p^-1 and _p^-1:
_p^-1 - _p^-1= (_p - _pc_c _cp)^-1 - _p^-1≽ 0.
The matrix inverse _p^-1 can be expanded as the following power series
_p^-1 = (_p - _pc_c _cp)^-1 =
∑_n=0^∞_p (_pc_c _cp_p)^n
We observe that this is a sum of positive semi-definite matrices, and truncate the series at n=0 and rearrange:
_p^-1 - _p = ∑_n=1^∞_p (_pc_c _cp_p)^n ≽ 0
We may now prove theorem 2 from section <ref>.
Δη' is a lower bound on the increase in natural expansion score Δη due to the addition of some proposed neurons p:
Δη≥Δη' = [^-1_r _p^T_p^-1_p _r^T]
Substituting lemma <ref> into corollary <ref> we have
Δη = [^-1_r ^T_p^-1_r^T].
The difference between Δη and Δη' is given by
Δη - Δη' = [^-1_r ^T (_p^-1 - _p^-1) _r^T].
This is the squared norm of _r ^T as a vector according to the Kronecker product
^-1⊗ (_p^-1 - _p^-1).
The first factor is positive semi-definite by assumption, the second by lemma <ref>,
and the Kronecker product of positive semi-definite matrices is positive semi-definite.
Therefore Δη - Δη' ≥ 0 and so Δη≥Δη'.
The significance of this lower bound on Δη is that _r and ^-1 may be computed once,
and then used to optimize very many proposals with different activations _p.
That is, performing N steps of gradient descent to optimize proposed neurons p scales linearly in the evaluation cost of _p and _p^-1.
These linear costs are unaffected by the number of neurons currently in the layer being added to,
and unaffected by the total number of layers in the network.
§ THE CONSEQUENCES OF NON-FISHER CURVATURE FOR TOTAL NEURONS ADDED
In section 3.5 we discussed the total number of neurons added during training, and in particular the extent to which we could provide bounds on this.
As noted there, in the case where the Fisher is constant over training and exactly equal to the hessian,
the dynamics of training are very simple.
The loss L has its global minimum at the point reached by a step of exactly ^-1,
and it can be seen by integration that the reduction in loss due to such a step is exactly Δ L = 1/2^T ^-1 = 1/2η.
The stopping criterion α corresponds to the requirement that parameter expansions should enable a further reduction in loss of at least 1/2α.
Since η≤λ is bounded by λ, the maximum possible reduction in loss is Δ L_max = 1/2λ.
If we pessimistically assume that every parameter expansion enables the minimal loss reduction of only 1/2α,
then the total number of added neurons N_T is still bounded by N_T < λ/α.
The case where the true hessian of the loss is some constant multiple of the Fisher = κ which is itself constant,
is almost as simple.
The parameters evolve along the same trajectory, only they move a factor of κ faster than they would if =.
This also results in a rescaling η = κη_B of natural expansion scores relative to the baseline value η_B in the case where was accurate.
While this has no effect on the behaviour of the expansion threshold τ,
the inflated η values mean that the effective value of α is reduced by a factor of κ
and so the total number of added neurons N_T is now only bounded by N_T < κλ/α.
We will now try to describe the effect of more general failures of to represent the true curvature .
Local expansion behaviour, i.e. without further parameter optimization, is bounded by lemma <ref> of appendix <ref>.
Assuming the baseline case of =, we may substitute λ = 2Δ L_max.
If we assume small step sizes, the rate of loss reduction L = -η is given by the natural expansion score by definition,
regardless of .
If at all times t during training the rate of reduction of expansion score -η(t) < -η_B(t) is lower than the baseline scenario,
then η will at all times be greater than expected.
Since the rate of loss reduction L = η is given by η, L will decrease faster than expected and the remaining maximum possible loss reduction Δ L_max will be at all times less than expected.
It can be seen from lemma <ref> that discrepancies in these directions relative to baseline will result in fewer additions being made.
We now only need to establish conditions under which the actual rate of reduction in η is lower than the expected rate.
The rate of change during optimization (indicated by overdot) of the various components of η can be described as follows:
= -^-1
= = -^-1
^T ^-1 = -^T ^-1^-1
η = -^T ^-1 - ^T ^-1 - ^T ^-1^-1
= -^T ^-1( 2 + ) ^-1
Since in the base case _B = and _B = 0, we have that
if + 1/2≼
then -η≤ -η_B.
Putting the above results together, we have that if at all times during training + 1/2≼,
then the bound on total additions N_T < λ/α should hold.
Incorporating the previous result regarding = κ,
it also appears that if at all times + 1/2≼κ, then N_T < κλ/α.
Assuming positive definite and the loss surface smooth (i.e. and finite),
then there will exist some finite κ for which the condition holds and so N_T will be bounded.
§ HYPERPARAMETERS AND IMPLEMENTATION DETAILS
All experiments were run on a single Nvidia A100 or V100 GPU, using no longer than one day each.
Our implementation uses the JAX <cit.> autodifferentiation and Flax <cit.> neural network libraries.
The full source code used to run the experiments is provided in the supplementary material, and will be made publicly available on publication of this work.
In all experiments we optimize our parameters via natural gradient descent with a learning rate of 0.1 and Tikhonov damping of magnitude 0.1.
In the image classification experiments we use batches of size 1024 and a weight decay of rate 0.001.
We initialize our dense layers with the default initialization of Flax (LeCun Normal) <cit.>,
and use a unit normal initialization for the parameters of our rational functions.
For the visualization experiments we use τ=2, for the image classification experiments we use τ=1.007 and τ=1.03
for the whole dataset and variable subset experiments respectively.
Larger thresholds τ result in longer training times but more conservative network sizes and higher accuracy of η estimates due to being a closer approximation to the curvature near convergence on the existing parameters.
Any extra costs are negligible for the visualization experiments, so we use the intuitive value of 2,
but we choose τ values for the image classification experiments in light of this natural trade-off.
We use α=0.0025 for all experiments apart from the whole dataset image classification, for which we use α=0.25.
Here the latter choice compensates for larger noise in Δη' introduced by use of a validation batch, as will be discussed shortly.
We adjust the expansion score increases for layer additions by a constant factor of 2 in the visualization experiments and 60 in the image classification experiments.
These values are selected to be within an order of magnitude of the actual layer sizes expected in classification of a toy dataset versus images, and so of the number of new neurons a new layer represents.
We calculate the natural gradient via the conjugate gradient method with a maximum iteration count of 100 when optimizing the existing parameters.
When optimizing the initializations of proposed neurons or layers we use the Kronecker factored approximation of the Fisher matrix for the relevant layer
based on derivatives of the predictions of the network as in <cit.>.
We compute Δη' based on this and normalize it with respect to the output gradient magnitudes of the particular task.
When comparing Δη' / η_c to τ we use the η_c value given by for the layer in question.
When considering adding layers, we ensure new layers are invertible by adding a regularization term of 0.01(ln)^2 when optimizing the initialization of their linear transform ,
and by setting the minimal singular values of to be at least 0.001 times its average singular value before adding the layer to the network.
In our visualization experiments we do not use batching, so we consider adding depth and width every 30 steps,
and add at most one layer per 90 steps.
In the image classification experiments we use batching and so consider adding width and depth every 10 epochs,
adding at most one layer each time.
We use the same scheme for initializing proposed new neurons or layers as for initializing the starting network.
In our whole dataset image classification experiment we then optimize proposal initializations to maximize Δη' via 300 steps of vanilla gradient descent
on a fixed batch of 1024 images.
We consider 10000 neuron proposals and 100 layer proposals per location, and use a learning rate of 0.3,
reducing this by a factor of 3 as necessary to maintain monotonic improvement in Δη' for each proposal.
We take the best proposal on this batch of size 1024 for each depth and width addition location,
and reevaluate its Δη' on a fixed validation batch of size 1024 when deciding whether and where to add.
The variable degree of overfitting of the best proposal results in some noise in Δη' at each location which we compensate for by choosing a relatively large α.
For our other experiments we optimize proposal initializations using 3000 steps of the Metropolis Adjusted Langevin Algorithm (MALA) <cit.>,
using a unit gaussian prior on initializations during these steps.
We use a temperature T of 10 and an initial step size of 0.3, and adjust by a factor of 3 every 10 steps if necessary to maintain an acceptance rate of around 0.6.
We consider 100 width proposals and 100 layer proposals for each location,
and obtain 100 final MALA samples i for each location width could be added and each location depth could be added.
We then construct a categorical distribution over each set of 100 samples via (1/TΔη_i'),
and use the corresponding expectation of Δη' when deciding when and where to add capacity and whether it should be depth or width.
We draw initializations for new capacity from this categorical distribution,
except in the initial least squares regression experiment, where we use _i Δη_i' over the 100 samples i to make figure 2 more intuitive.
|
http://arxiv.org/abs/2307.03988v1 | 20230708143306 | PCG-based Static Underground Garage Scenario Generation | [
"Wenjin Li",
"Kai Li"
] | cs.AI | [
"cs.AI",
"cs.RO"
] |
Journal of Class Files, Vol.
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals
PCG-based Static Underground Garage Scenario Generation
Wenjin Li, Kai Li
Wenjin Li, Kai Li are with the Department of Computer Science and Technology, Southern University of Science and Technology, Shenzhen, 518055, China
August 12, 2023
============================================================================================================================================================================
Autonomous driving technology has five levels, from L0 to L5.
Currently, only the L2 level (partial automation) can be achieved, and there is a long way to go before reaching the final level of L5 (full automation).
The key to crossing these levels lies in training the autonomous driving model.
However, relying solely on real-world road data to train the model is far from enough and consumes a great deal of resources.
Although there are already examples of training autonomous driving models through simulators that simulate real-world scenarios, these scenarios require complete manual construction.
Directly converting 3D scenes from road network formats will lack a large amount of detail and cannot be used as training sets.
Underground parking garage static scenario simulation is regarded as a procedural content generation (PCG) problem.
This paper will use the Sarsa algorithm to solve procedural content generation on underground garage structures.
Automated driving, underground garage planning, reinforcement learning, procedural content generation, Sarsa
§ INTRODUCTION
According to a recent technical report by the National Highway Traffic Safety Administration (NHTSA), 94% of road accidents are caused by human errors <cit.>. Against this backdrop, Automated Driving Systems (ADSs) are being developed with the promise of preventing accidents, reducing emissions, transporting the mobility-impaired, and reducing driving-related stress <cit.>.
Autonomous driving simulation is an important part of ADSs.
However, simulation lacks interactive and changeable scenarios <cit.>. Researchers are still using authentic human-made ways to build one scenario for huge training.
Procedural Content Generation for Games (PCG-G) is the application of computers to generate game content, distinguish interesting instances among the ones generated, and select entertaining instances on behalf of the players <cit.>.
In our project, we consider the underground garage as the game content that should be generated.
The problem can normally be divided into three parts. The first part is to create the digit grid map for each type of floor, as a PCG task. The second part is to convert each type of floor to the design diagram.
The last part is to simulate the whole 3D scenario map depending on the design diagram.
To simplify the simulation, we combine the last two parts as one part.
In reinforcement learning <cit.>, an agent seeks an optimal control policy for a sequential decision-making problem.
We regard the first part as a sequential decision-making problem.
Markov decision processes (MDPs) are effective models for solving sequential decision-making problems <cit.> in uncertain environments.
The agent's policy can be represented as a mapping from each state it may encounter to a probability distribution over the available actions <cit.>.
Generalized policy iteration (GPI) was demonstrated as a class of iterative algorithms for solving MDPs in <cit.>.
It contains policy iteration (PI) and value iteration (VI) as special cases and has both advantages of PI and VI.
Temperal-difference <cit.> is the specific implementation of GPI <cit.>.
TD methods are guaranteed to converge in the limit to the optimal action-value function, from which an optimal policy can be easily derived.
A classic TD method is Sarsa <cit.>.
The on-policy algorithm, in which policy evaluation and policy improvement are identical, has important advantages.
In particular, it has stronger convergence guarantees when combined with function approximation, since off-policy approaches can diverge in that case.
In this paper, we use the Sarsa algorithm to create a digit grid map.
Simulation is an important step during the conversion <cit.>.
We consider the simulator can generate test scenarios automatically, including static buildings, dynamic traffic flow, and real-time calculated lighting and weather.
This paper aims to solve the static scene generation problem.
§ RELATED WORK
Abdullah <cit.> compared the space utilization efficiency of diagonal, parallel, and perpendicular parking methods and concluded that perpendicular parking methods have the highest number of spaces, using a university as a specific example.
Sawangchote <cit.> developed a heuristic algorithm for the layout of parking spaces in small-scale garages based on the space utilization of different parking methods;
Xu Hanzhe <cit.> carries out a parking space layout design based on a greedy algorithm to study the influence of irregular contours and obstacles on the layout of parking spaces and get the layout plan with the most number of parking spaces.
Julian Togelius <cit.> finds that the result of the composition of a bunch of different algorithms is better than the result of any single algorithm and He used answer set programming to do procedure content generation.
Huimin Wang <cit.> has previously proposed a model-based reinforcement learning algorithm to implement the path planning problem. The path planning problem has similar features when it applies to the specialized PCG problem. We consider that generation on a garage can use the method of path planning on agent moving. Besides, Arunpreet Sandhu <cit.> comes up with the WFC algorithm to generate similar images.
Akatsu <cit.> provides an idea for evaluating underground garage structures by feeding a series of indicators obtained from a realistic traffic survey into a modeled underground garage structure to obtain a series of evaluation results.
§ METHODOLOGY
§.§ Overall
We consider dividing the underground garage construction into two main parts, PCG task and simulation. Notations using throughout this report are as follows:
Since the most important thing in static underground garage scenario generation problems is the planning of parking stalls. For parking space planning problem, it is essentially an optimization problem of object placement, the objects to be placed will have the following distinction:
* static object: object's position will not change after confirming the position
* dynamic object: objects can wait for further optimization after confirming the position of static objects
Now we only need to consider the dynamic object distribution, in order to better describe the entire underground garage object planning situation, here we rasterize the underground garage by using three matrices S_i,j, R_i,j, C_i,j to describe the state of an underground garage.
In this paper, we will use reinforcement learning to plan the distribution of dynamic objects, by combining the distribution with the distribution of static objects to obtain the S_i,j as the result of parking space planning, and finally combine the R_i,j and C_i,j as the plane structure of the static underground garage to pass into the Unity3D engine for 3D modeling to finally generate the static underground garage scenario.
We provide the following requirements for a reliable garage:
* Reality: The generated basement structure needs to adapt to real-world standards (such as national standards and regulations)
* Feasibility: Ensure that at least one route to any exit and entrance can be found for each parking space arranged in the basement structure
* Randomness: The structure and contour of the basement are randomly generated, and the solution generated each time will change according to the change of the random process
* Bijection: Each generated basement structure has a unique corresponding random process, and this random process must correspond to a unique basement structure
* Customizability: The structure of the basement can be self-defined
§.§ Static objects generation
First, we give a definition of structure matrix 𝒮(i,j):
𝒮(i,j)={
0 , parking space or free space
-1 , obstacle
1 , lane
2 , entrance
3 , exit
.
At the beginning of getting this matrix, we should confirm the location of those static objects, which can be divided into three steps: contour generation, entrance and exit generation, and obstacle generation.
First, we need to generate the contour of the underground garage. Divide a w× h rectangle into w× h blocks and each block has a width and height of 1. We consider generate n groups of 2n points in this rectangle and use the line of two points of each group as the diagonal of the rectangle to generate a rectangle and then after expand all rectangles to its corresponding squares, We will treat the concatenation of all rectangles as a generated underground garage contour. The following algorithm shows the generation of underground garage contour.
After contour generation, we can get all squares in the floor plan, which mean we get ζ and ψ and then assign values to all those squares in ζ and ψ:
𝒮(ζ) = 0
𝒮(ψ) = -1
Secondly, we need to determine the position of the entrance and exit. After contour generation, in order to generate a reliable position of entrance and exit, we give a definition of ξ and η. A frontier square needs to satisfy the following conditions:
𝒮(ξ) = 0
∑_i=1^8𝒮(ρ_ξ) < 0
An inner square needs to satisfy the following conditions:
𝒮(η) = 0
∑_i=1^8𝒮(ρ_η) = 0
Since entrances and exits can only be generated in ξ and cannot be generated on the corners of ξ, in this condition, we only generate entrance and exit on those squares satisfy the following condition:
ϵ∈ξ
∑_i=1^8𝒮(ρ_ϵ) = -3
M(ϵ_i,ϵ_j) ≥σ_1
Thirdly, we need to consider the position of obstacles in this underground garage. We only generate obstacles on those squares satisfying the following conditions:
o ∈η
M(o_i,o_j) ≥σ_2
§.§ Reinforcement Learning
Reinforcement learning (RL) is a basis to solve our PCG problem. In this paper, we first focus on finite Markov decision processes (finite MDPs).
A finite Markov decision process can be
represented as a 4-tuple M = {S, A, P, R}, where S is a
finite set of states; A is a finite set of actions; P : S× R × S × A → [0, 1] is the probability transition function; and R : S × A →ℛ is the reward function. In this paper, we denote the probability of the transition from state s to another state s' when taking action a by P(s', r|s, a) and the immediate reward received after the transition by r_s^a <cit.>.
A policy is defined as a mapping, π: S× A→ [0,1]. In this paper, we use π(s) to represent the action a in state s under the policy π. To measure the quality of a policy, action-value function, q_π(s, a) is used to estimate the expected long-term cumulative reward of taking action a in state s under a policy π. It is formally defined as:
q_π(s,a)=𝔼_π[∑_k=0^∞γ^kR_t+k+1| S_t=s, A_t=a]
where γ is a discount factor, R_t is the reward at time-step t, and E_π is the expectation with respect to the policy π.
The goal is to find an optimal policy π_* which maximizes the expectation of long-time discounted cumulative reward from any starting state s∈ S:
π_*=*argmax_πE_π [∑_t=0^∞γ^t R_t|s_0=s]
In this paper, we format PCG as an optimization problem <cit.>, which is represented as a 2-tuple (M, E ), where M is finite MDPs which can generate one 2D integer array and E is an evaluation function which evaluates the quality of array. We have one agent with policy π. It will tack action in state s and send a message to the environment.
The environment receives the message and changes the state to the next state and sends rewards to the agent.
Finally, the agent and environment produce a finite Markov decision array:
S_0,A_0, R_1, S_1, A_1, R_2, S_2, A_2, R_3,…, S_T-1, A_T-1, R_T
where T is the termination time. Evaluation function E is calculated from M
E=∑_t=1^T-1 R_t
R_T is always a negative value and it is not included in E. In other words, we come back to the previous unfailed state to compute E.
Generalized policy iteration (GPI) contains two processes, policy evaluation (E) and policy improvement (I):
π_0E→ q_π_0I→π_1E→ q_π_1I→π_2E→…I→π_*E→ q_*
where q_π_i is action value function under π at episode i. The process is terminated when q and π converges to q_* and π_*. For Sarsa algorithm, policy evaluation and policy improvement are carried out simultaneously in each episode.
The agent and environment in MDP are clear. Our design is divided into two sections.
In the first section, we design the MDP for our PCG task. In the other section, we design the environment penalty based on the principle of parking lot design.
§.§ Sarsa
We use the Sarasa algorithm to solve the PCG task. First, we define the parameters of MDPs. We consider a car in a 2D place as an agent to perform a colouring task, which colours the undefined square to a lane spuare. Agent's state at timestamp t is defined as the multiple dimensional vectors:
S_t=(D, M, A_t-1)
Where D is a 4-dimensional vector that each element point to the distance between the free space, border, or obstacle and agent in the direction, M is a 25-dimensional vector that symbols to the perception range of the agent. It satisfies that all points have a Manhattan distance of less than 2 from the agent.
The agent takes action from the action set
A={UP, DOWN, LEFT, RIGHT, STAY}
The goal is to colour the road as much as possible until it comes back to the start and takes action STAY, leading to a terminate state. Agent receives rewards depending on the increment of the number of parking spaces. The agent also receives a penalty for some wrong actions.
To evaluate one policy π, we predict one Markov decision array containing S, A, R for each episode. We update q(S_t, A_t) during the prediction, following the function:
q(S_t, A_t) = q(S_t, A_t) + α× (R_t+1 + γ× q(S_t+1, A_t+1)-q(S_t, A_t))
where α and γ are parameters, with 0≤α, γ≤ 1.
We use greedy method to improve one policy:
π(s)=*argmax_a q(s,a)
where π(s) is the greedy action under policy π. We consider using ϵ-greedy to take action, where the agent has ϵ chance of taking greedy action with maximum value otherwise taking action equivalently. The probability of taking greedy action π(s) in state s is:
p(s, π(s)) = (1-ϵ)+ϵ/|A|
§.§ Penalty design
The principle of parking lot design has been proposed for optimizing parking area space.
* Use rectangular areas where possible
* Make the long sides of the parking areas parallel
* Design so that parking stalls are located along the lot's perimeter
* Use traffic lanes that serve two rows of stalls
<ref> conforms the above principle, where green square refers to lane square, orange square refers to parking square or free square, and white square refers to entrance or exit. Contrary to <ref>, <ref> has many problems: no cycle, existing non-rectangular and non-parallel areas, and many lanes serving only one row of the stall.
The agent can not only receive a reward after the action but also a certain penalty we defined. The reasonable penalty guides agents to do actions they want. Based on the design principle, we propose several penalties below:
* Turn-back penalty when the agent takes the opposite action from the last action.
* Interval penalty based on the interval of the same actions.
* Wheeling penalty at an improper position with a certain direction.
* Step penalty for each timestamp to prevent agents from cycling consistently.
§.§ Convert matrix to simulated underground garage
After generating structure matrix 𝒮(i,j), we need to convert this matrix to a simulated underground garage. Here we first atomize the elements of the matrix, we define the below equation:
n = ∑_i=1^4𝒮(θ_η)
and for any square η, if:
𝒮(η) = 1
we define η as:
η={
Crossroads , n = 4
T-Junctions , n = 3
Straight road , n ≤ 2
.
and if:
𝒮(η) = 0
we define η as different types in Figure 2:
η={
Type1 , n ≥ 3 or across n = 2
Type2 , adjacent n = 2,
Type3 , n = 1
Type4 , n = 0
.
Then, we only need to model each type of square η in the simulator and use scripts to construct the simulated underground garage.
§.§ Construction of underground garage structure
We know that autonomous vehicles typically use multiple types of sensors to collect and process environmental information to support the vehicle's decision-making and control systems <cit.>.
The parking garage structure we generate is intended to provide training scenarios for autonomous vehicles, and the information collected during autonomous vehicle training comes from the simulated scenes, such as the lighting of light sources, the materials of various object surfaces, and information on the different light reflections of objects in the scene, and so on <cit.>. If we can better simulate the various objects in these scenes, the amount of information contained in the overall static parking garage scene will be greater, and it will better provide training data for autonomous vehicles, achieving better training effects.
The construction details of a static underground parking garage mainly include object surface texture mapping, such as:
* Lane marking texture mapping
* Wall texture mapping
* Floor texture mapping
* Lighting texture mapping
As well as collision bodies in the underground parking garage, such as:
* Column mesh collision body
* Speed bump collision body
* Parking barrier
And here we give the detailed procedure of underground garage generation in Unity3D:
* The structure matrix 𝒮_(i,j) previously generated by using reinforcement learning is used as the generated underground structure, and the R_(i,j) and C_(i,j), which define the length and width of each plot of land in reality, are passed as input into Unity3D engine.
* In the Unity engine, each different state of the land is first modeled, and then the entire underground plane is automatically generated based on the arrangement of elements in the specific structure matrix.
* After generating the plane, three-dimensional information such as walls, pillars, ceilings, obstacles, etc. are further generated based on the outline of the underground structure.
* According to the generated structure, more detailed descriptions are made, such as light tubes, ventilation ducts, and other underground details.
* According to the demand, some objects that may appear underground, such as parked vehicles and no parking signs, are randomly generated.
§ EXPERIMENTAL SETUP
§.§ Evaluation
After generating the underground garage structure, we need to evaluate it, but there is no unified and credible standard for the evaluation function. So we proposed the following three dimensions to describe the value of the underground garage structure by combining the evaluation system of several papers:
* the number of the parking spot
* the average parking time
* the number of unused squares
So the evaluation function is like:
y^' = k_1 * N_S + k_2 * T_S + k_3 * U_S
To obtain the proportion of weights accounted for by each of these three criteria, here we assume that there exists a corresponding evaluation function for a certain underground garage structure, and the value distribution of all solutions for that structure is roughly Gaussian distributed.
Based on this, we can know that if we have enough sampling points and judge the value size relationship of the structure in the sampling points, we can correspond these sampling points to the Gaussian distribution curve one by one, and then make the estimated value order of the sampling points the same as before by adjusting the weights of our evaluation function, so that we get an evaluation function with a certain degree of confidence, and when more and more points are sampled, the final evaluation function will be more credible.
Here, we sampled a series of more representative experimental results and derived the above values for the three coefficients:
y^' = N_S + (-5) * T_S + (-1) * U_S
We conducted a 5000-episode cycle test for Sarsa algorithm with one garage contour. For each episode, we save the matrix and evaluation on it to the dictionary. In the end, we select top 200 matrix with high evaluation function value.
§.§ Simulation of Underground Garage
The main hardware devices used in the simulation to generate the underground garage scenario are: CPU: Intel(R) Core(TM) i7-10750H CPU @ 2.60GHz, GPU: NVIDIA GeForce GTX 1650 and the software are: Unity3D 2021.3.5f1c1, Visual Studio 2022
§ RESULTS
§.§ Sarsa Result
<ref> indicate that the agent easily achieves the local limit at episode 400. Then it straight down to a small value. It maintains a trend of first converging to the limit and then sharply decreasing. It will keep searching for a solution if the test doesn't stop.
However, we observed that as the number of episodes increases, there are instances where the agent obtains lower payoffs. This can be attributed to the ϵ-greedy strategy, which sometimes leads the agent directly to the termination state. To increase the converge rate, We make the ϵ decrease slowly. We also refresh the value of ϵ if the matrix keeps at 100 consecutive episodes.
<ref> shows the matrix with the highest evaluation value during the test. It is slightly inferior to <ref> and <ref> manually constructed.
§.§ Simulated Underground Garage
<ref> shows the underground garage model simulated by modelling the structure matrix generated by the above reinforcement learning algorithm for 3000 iterations as input.
<ref> shows the underground garage model simulated by modelling the structure matrix generated by the above reinforcement learning algorithm for 3000 iterations as input.
§ DISCUSSION
For the evaluation function, there is no unified credible evaluation function, and the coefficient given in this paper is only a fitting operation for the real value curve. At the same time, since the structure of an underground garage with different contours has an impact on the three evaluation indexes we selected, the value of the coefficients for different contours may also be inconsistent, which may require more sampling and training through neural networks to come up with the coefficients for each underground garage contour later <cit.>.
However, happily, we were able to correctly evaluate the generated underground garage parking space structure according to the evaluation function obtained from the sampling on the 7*9 square contour, as it can be seen that Fig. 5 and Fig. 6 are the manually designed structures considered to be of higher value according to the cognitive design, and Fig. 1 to Fig. 4 are the top four structures of value filtered according to the evaluation function from the results of the algorithm generating 5000 episodes, and it can be seen that the filtered structures, although not perfect, can meet several of the most basic requirements in designing an underground garage parking space, and are indeed a little more valuable than the manually designed structures.
§ CONCLUSIONS
Sarsa, an on-policy TD algorithm, performs well in this paper. It can generate reliable graphs eventually. However, the state set is so large that it can not converge into one solution that reaches the highest repayment.
This study demonstrates the feasibility of using reinforcement learning to programmatically generate underground garage grid maps. We have yet to reach a target that can generate a reliable underground garage based on some contour. PCG of underground garage design has a long way to go.
In terms of simulation, we are currently able to construct the corresponding 3D underground parking garage and the generated garage has certain details: real-time lighting, ventilation ducts, column network structure, etc.. The current garage details such as various pipe layouts are not yet practical and various scene elements can be further rendered to achieve a more realistic effect. This will allow us to further enhance the accuracy and reliability of the generated underground garage maps. These findings provide valuable insights for the development of intelligent underground garage planning and design tools.
In the future, we will extend this work with other AI technologies, such as classification <cit.>, knowledge graphs <cit.>, deep learning <cit.>.
IEEEtran
|
http://arxiv.org/abs/2307.04367v1 | 20230710064801 | Explanation Needs in App Reviews: Taxonomy and Automated Detection | [
"Max Unterbusch",
"Mersedeh Sadeghi",
"Jannik Fischbach",
"Martin Obaidi",
"Andreas Vogelsang"
] | cs.SE | [
"cs.SE"
] |
Explanation Needs in App Reviews: Taxonomy and Automated Detection
Max Unterbusch
University of Cologne
[email protected]
Mersedeh Sadeghi
University of Cologne
[email protected]
Jannik Fischbach
Netlight Consulting GmbH | fortiss GmbH
[email protected]
Martin Obaidi
Leibniz University Hannover, Software Engineering Group
[email protected]
Andreas Vogelsang
University of Cologne
[email protected]
August 12, 2023
============================================================================================================================================================================================================================================================================================================================================================================================================================
Explainability, i.e. the ability of a system to explain its behavior to users, has become an important quality of software-intensive systems. Recent work has focused on methods for generating explanations for various algorithmic paradigms (e.g., machine learning, self-adaptive systems).
There is relatively little work on what situations and types of behavior should be explained. There is also a lack of support for eliciting explainability requirements.
In this work, we explore the need for explanation expressed by users in app reviews. We manually coded a set of 1,730 app reviews from 8 apps and derived a taxonomy of Explanation Needs.
We also explore several approaches to automatically identify Explanation Needs in app reviews. Our best classifier identifies Explanation Needs in 486 unseen reviews of 4 different apps with a weighted F-score of 86%.
Our work contributes to a better understanding of users' Explanation Needs. Automated tools can help engineers focus on these needs and ultimately elicit valid Explanation Needs.
Explainability, Requirements, NLP
§ INTRODUCTION
Software systems are becoming more intelligent and ubiquitous than ever before, increasing the criticality of their impact on humans. Driven by modern artificial intelligence, it is becoming increasingly difficult for an external user, but also for the developers of these systems, to understand their inner workings and thus their decisions and actions. The ability to provide explanations—a natural ability of humans—is therefore considered an important capability of software systems. As such, explainability is now accepted as a critical quality attribute <cit.> and represents an emerging topic in the field of RE <cit.>.
Researchers have explored the foundations of explainability from different angles. There are several approaches to generating explanations for different algorithmic paradigms.
However, there has been relatively little focus in the literature on what users actually need explanations for <cit.>.
This lack of knowledge limits our ability to effectively elicit explainability requirements and apply existing explanation generation methods.
Thus, the first problem we address in this paper is as follows: We lack knowledge about what users need explanations for.
App reviews have been overlooked as a potential source of Explanation Needs. Pagano and Maalej <cit.> found that app reviews contain valuable RE-related information because they represent rich and readily available textual data that provides insights into thousands of user experiences.
Unlike interview or survey data, app reviews are collected “in the field” under natural circumstances. Users are motivated enough to publish their opinions about an app; they are not forced or paid to do so. This underlines the importance that users place on their concerns. In addition, users are not asked about any specific aspect. The review messages are open to any feedback the users want to give to the app vendors or developers.
We set out to understand users' need for explanation, which we refer to as Explanation Need. Our focus is to characterize the occurrence of Explanation Needs in app reviews and to investigate the types of Explanation Needs that users express. We conducted a qualitative analysis of 1,730 English app reviews of 8 different apps. As a result, we propose a taxonomy of Explanation Needs in app reviews to help developers and researchers distinguish between different types. One of the key benefits of the taxonomy is that it enables researchers and engineers to extract explainability requirements in a systematic and rigorous manner. By categorizing users' Explanation Needs from their perspective into distinct categories, the taxonomy highlights areas where a system may lack transparency or fail to meet users' expectations. This, in turn, provides valuable insight into the types of explanations that are most needed.
Our qualitative analysis shows that Explanation Needs in app reviews are valuable and contain rich information, but are relatively sparse. Explanation Needs have only appeared in about 5% of the app reviews studied. However, manually analyzing app reviews can be challenging due to the sheer volume of reviews and the varying levels of detail and insight they provide.
Tool support to filter the reviews for relevant content would be valuable to allow development and stakeholders to efficiently exploit this source of information <cit.>. We identify this as the second problem addressed in this paper: We lack tool support to automatically identify Explanation Needs in app reviews. To support the use of app reviews, we investigated several classifiers (rule-based, traditional ML, and transformer approaches) to automatically detect Explanation Needs in app reviews. We evaluated and compared the classifiers in a 10-fold cross-validation on an extended set of 5,078 manually labeled app reviews. In addition, we evaluated our baseline rule-based approach and our best-performing classifier on an additional set of 486 unseen and unmodified reviews of 4 new apps to test how well the approaches generalize and perform in a realistic setting.
Our best-performing classifier—a fine-tuned BERT model—achieved a weighted F-score of 93% in a 10-fold cross-validation and a weighted F-score of 86% when evaluated on unseen data. We make the following contributions:
* We provide a taxonomy of Explanation Needs derived from a large set of app reviews.
* We provide a performance analysis of several classifier approaches to detect Explanation Needs automatically in app reviews.
* We publish a set of 5,564 app reviews that we manually labeled according to our proposed taxonomy.
* To strengthen transparency and facilitate replication, we make our code, dataset, and trained models publicly available.[10.5281/zenodo.7740411 .]
§ TERMINOLOGY AND RELATED WORK
§.§ Explainability and User Needs in Explanations
Explainability has gained significant attention from various research fields, including Human-Computer Interaction, Cyber-Physical Systems, and Psychology <cit.>. Since 2019, when it was proposed as a non-functional requirement <cit.>, it has become a trending topic within the SE and RE communities <cit.>. Research has shown that explainability can enhance trustworthiness, transparency, accountability, fairness, ethics, and other quality aspects by overcoming the black box nature of software systems <cit.>. Chazette et al. developed a concise definition of explainability that meets the requirements of SE and RE communities <cit.>: A system S is explainable with respect to an aspect X of S relative to an addressee A in context C if and only if there is an entity E (the explainer) who, by giving a corpus of information I (the explanation of X), enables A to understand X of S in C. The explainer entity does not have to be the system itself. Achieving explainability depends on specific variables: the system's aspect, the addressee, and the context. Accordingly, Kohl <cit.> and Chazette <cit.> emphasize the significance of identifying users' specific needs for explanations and providing customized explanations correspondingly. Indeed, in cases where users do not require explanations, ensuring explainability may not be necessary <cit.>.
Studying app reviews for explanation need identification is a relatively under-researched area. Consequently, a taxonomy of Explanation Needs can aid in advancing knowledge and eliciting requirements for developing explainable systems. Constructing taxonomies provides numerous benefits, including supporting the communication of complex concepts, revealing relationships between entities, and uncovering knowledge gaps. In a similar approach for a different domain, Sadeghi et al. <cit.> developed a taxonomy of reasons for Explanation Needs. They primarily distinguish between four categories of situations requiring explanations: Training, Interaction, Debugging, and Validation, yet the authors focused on Interaction. For Interaction, the taxonomy further breaks hierarchically down into disobedience, failure, and context-aware behavior. That work considered the system, the user, and the environment in their taxonomy; in contrast, our focus will be on the user only.
§.§ App Store Mining and Classifying App Reviews
Pagano et al. <cit.> conducted a
comprehensive analysis of app stores to determine their usefulness for
requirements engineering. They collected over a million app reviews and found that feedback messages can facilitate communication between users and developers. However, they discovered that a significant amount of the feedback collected was of poor quality and lacked informative value. They argue that although app stores can facilitate user-centered RE through the use of user feedback, it is essential to employ appropriate tools and techniques to filter and pre-process relevant contributions. In response to the need for tool support in app store mining, the RE community developed various solutions to extract valuable insights from app store reviews. Guzman and Maalej <cit.> proposed a method to filter features mentioned by users and extract corresponding sentiments, allowing for a detailed analysis of user experience with individual app features. Chen et al. presented a tool that filters app reviews, groups and ranks them, and provides visualizations of the insights <cit.>.
Particularly relevant to this paper are contributions that classify app reviews according to predefined labels, such as problem reports, inquiries, and user experience, or non-functional requirements such as reliability, usability, and portability. To achieve this classification, researchers typically use traditional ML and DL methods for classifying app reviews into various categories <cit.>. Active Learning strategies have also been experimented with, which can help reduce human labor and improve classification accuracy in certain scenarios <cit.>.
Recently, BERT achieved state-of-the-art performance classifying English app reviews into feature requests, problem reports, and irrelevant <cit.>. In this paper, we compare a simple rule-based approach as a baseline, different ML-based approaches, and a DL-based approach using the BERT-Base model <cit.> for detecting Explanation Needs in reviews automatically.
§ CHARACTERIZATION OF EXPLANATION NEEDS
We define an Explanation Need as a knowledge gap that a user intends to close and present our findings on such needs in app reviews in this section. To consider a review as an Explanation Needs, the user must explicitly raise a question or express a need for an explanation. Rhetorical questions ([sic] “What the hell?”) do not qualify as Explanation Needs as they are not intended to elicit an answer. Direct requests ([sic] “Please could you please check it?”) are also excluded since they do not indicate a specific gap in knowledge.
It is important to note that we distinguish between Explanation Needs and Explainability Need, a non-functional requirement identified for software systems. On the other hand, Explanation Needs are needs perceived by users. Following the formatting of Chazette et al.'s definition of explainability <cit.>, we formally define Explanation Needs as:
An addressee A has incomplete knowledge about an aspect X of system S in context C and requests a corpus of information I provided by an entity E that allows A to understand X of S in C.
§.§ Study Design
In the endeavor to identify users' Explanation Needs, this research aims to explore the potential of app reviews as a source of information. By analyzing the rich textual data of reviews, we seek to uncover the types of explanations that users are looking for. To guide our investigation, we formulated the following research questions (RQ):
RQ1: What types of Explanation Needs have been expressed in app reviews?
RQ2: How prevalent are Explanation Needs and their types in app reviews?
Answering RQ1 is crucial for identifying common issues faced by users and prioritizing areas for improvement in app development. It aims to identify and understand users' Explanation Needs in app reviews, guiding the development of more transparent and user-friendly software systems. To answer our research question, we undertake a qualitative analysis to develop a taxonomy for Explanation Needs in app reviews. The provision of conception classification and taxonomy is generally valuable since it provides a standardized framework and facilitates a common ground to communicate and research in emerging fields of knowledge <cit.>. As depicted in Figure <ref>, the qualitative analysis toward addressing RQ1 involved three phases: (1) Dataset Selection, (2) Analysis and Preliminary Taxonomy Extraction (3) Verification and Taxonomy Finalization.
Phase 1.
In the first phase, we selected the datasets for our analysis.
The original dataset used in our study was assembled by Brunotte <cit.>. Although a more recent version of the dataset exists with a larger number of reviews, we focused our analysis on a subset of 1,730 reviews provided to us directly by the authors. It allowed us to conduct our analysis more targeted and manageable. In the remainder of this paper, we refer to this dataset as . comprise app reviews from eight distinct apps available on the Apple App Store and Google Play. The domains represented in span several categories, including health and wellness, finance, technology, and lifestyle, making it well-suited for exploring the nature of user feedback and Explanation Needs in mobile app reviews. Table <ref> provides an overview of this dataset.
Phase 2.
Using the dataset as our basis, we extracted the preliminary taxonomy of Explanation Needs. A single coder initially analyzed all 1,730 app reviews based on the definition of Explanation Needs outlined in <ref>. The coder then filtered out 1,600 reviews that did not express any Explanation Need, and the remaining 130 cases were labeled as Explanation Need on a tentative basis. While there was a possibility that some of these cases could be excluded by the other coders in subsequent phases, these 130 cases still provided a foundation for further analysis in terms of categorization and taxonomy extraction. Following the template by Saldaña <cit.>, the coder also developed a codebook to maintain, organize, and share the codes with the other authors.
The initial coding resulted in an early version of the taxonomy, which was subject to further refinement through extensive discussions and revisions by the authors involved in the study. Hence, as this phase's output, a preliminary taxonomy was generated, which classified different types of Explanation Needs and established boundaries between them. Nevertheless, at this point, the codebook yet had rather generic and fuzzy definitions of the categories or loose criteria for differentiating them. Therefore, we proceed to the next phase to further verify the applicability of the taxonomy and codebook.=-1
Phase 3.
In the final phase, we aimed to verify and refine the preliminary taxonomy by involving two other coders. We sampled 130 app reviews tentatively identified as Explanation Needs by the first coder, plus a random selection of 70 reviews that were not labeled as such. The resulting dataset was shuffled and divided equally between the coders, with each responsible for categorizing their respective half as Explanation Need or not. For the reviews categorized as Explanation Need, the coders then had to check if they could be classified under one of the leaf nodes of the preliminary taxonomy. The goal was to ensure the preliminary taxonomy and codebook's completeness and accuracy and identify any deficiencies.
The coders then engaged in several rounds of discussions and classification. During the first iteration, the coders compared the labels assigned by the initial coder to the new labels the additional coders gave. From the 130 cases identified by the initial coder as Explanation Need, 48 cases were excluded by either of the new coders. So we were left with 82 app reviews that the new coders also tentatively labeled as Explanation Need, with each case being assigned a specific type of explanation. During the second iteration, all the coders went through these 82 reviews to further discuss and evaluate each case.
Moreover, at this point, coders attempted to prune and/or extend the taxonomy categorization to produce the final taxonomy and to consolidate their descriptions and boundaries recorded in the codebook. Throughout the last iteration, 5 additional app reviews that did not meet the requirements and specifications of the final taxonomy were excluded, resulting in a total of 77 cases labeled as Explanation Need.
§.§ Results: A Taxonomy of Explanation Needs
As shown in Figure <ref>, the taxonomy has a hierarchical structure and consists of two levels. We refer to the lowest level elements, namely , , , , and , as categories of Explanation Needs. To make the categories more tangible, we included a non-exhaustive list of aspects for each category. These aspects are more concrete groupings of related and typical Explanation Needs that we could observe in the data. However, they are not part of the taxonomy in a narrow sense.
Given the Explanation Need <ref>, a key distinction we make in the first level of our taxonomy is whether such a need for some explanation is an issue's primary or secondary concern. More precisely, if the user perceives their lack of knowledge as the only issue, then the Explanation Need becomes a Primary Concern, whereas if they see other substantial problems aside from their knowledge gap, it becomes a Secondary Concern. In the latter case, an underlying problem exists, typically a deficiency, which substitutes the Explanation Need as the primary concern. Therefore, offering an explanation may increase the overall understanding of the situation, but an explanation alone cannot solve the underlying problem.
As depicted in Figure <ref>, the Explanation Needs belonging to the , , and categories represent a primary concern. In general, is when users are unfamiliar with the system or particular features, either because they are new to it or the system's features have been changed. We found the following aspects to characterize best:
* Instruction. Users seek instructions for achieving specific goals, such as how to use a system, feature, or settings option. This aspect requires that the users clearly intend what they aim to do. Instruction aspect excludes reviews if there is an identifiable deficiency, such as an error or failure (see aspect Fix). Example [sic]: “How do you edit from this app???”.
* Features Offered. Users seek information about specific or general systems' features or functionality. Therefore, users are unaware of what the system can exactly do. Example [sic]: “... is there anyway to sort this out ...?”.
* Effect-Of. Users want to obtain information on the potential outcomes of specific actions. The users know how to perform such an action but are not sure what the impact will be. Example [sic]: “If I invest in dividend paying stocks, will the dividends be added to my portfolio?”.
The next category is , including aspects that arise in the ordinary operation of a user familiar with the system. These aspects assume expected behavior, not accounting for deficiencies such as errors or failures. The category was found to encompass the following aspects:
* Algorithm. Users struggle to comprehend why a system generates a particular output, wanting to know the factors that influenced the computation. The output is unique to each user, therefore the programmed logic that is the same for all users is not included in this aspect (see aspect Design Decisions). Example [sic]: “In the last 3 months my credit went up a total of 10 points and then dropped down 7 points December 2. This doesn't make sense.”
* Design Decision. Users wonder why things are a particular way (status quo) or not a certain way (counterfactual). It is not an output of the system that might be individual to each user, but the programmed logic, which the developers have agreed on. Hence, in contrast to the Algorithm aspect, the Design Decisions are the same for multiple (if not all) users. Example [sic]: “why does the app force portrait mode?”
* Signification. Users seek clarification on definitions, visual elements (such as symbols, colors, and highlighting), information visualizations, or related issues in order to understand the system's intended meaning. Example [sic]: “I like this app, but when there may be something in red I just don't understand. Does it means something is wrong?”
The last category in the primary concerns is category. It represents general Explanation Needs that are not necessarily provoked during the interaction with a system. Further, aspects to be explained may be shaped by overarching business goals or specific project or process requirements <cit.>. Here we determined the following aspects:
* Mission. Users seek clarification on the system's purpose, utility, and vision, with a particular focus on specific features and the system as a whole. Example [sic]: “Why do we need to access this app to get the information we used to get by phone from the doctor?”
* Purchase & Subscription. Users inquire about purchase or subscription matters, such as feature exclusivity in premium. This aspect only applies when there are multiple product lines with varying purchase or subscription plans. Example [sic]: “Do I have to pay for it on all devices?”
* Privacy. Users express privacy concerns regarding data collection, processing, and forwarding practices, as well as legal privacy rights and app permissions (e.g., GPS activation). If the inquiry is not focused on privacy but rather on the aspects that affect software decisions, it falls under the Algorithm aspect. Example [sic]: “Not sure why you need date of birth to register a navigation app, very suspicious as far as I'm concerned.”
Moving to the secondary concerns, we have and categories. Accordingly, here the Explanation Need is only the secondary concern of users, and there is a substantial underlying problem (at least in the user's perception) that is their primary concern. Overall, the aspects are somewhat reproachful and the primary concern typically is a subjective deficiency from the user's point of view.
* Change. Users seek explanations for changes to a system, including modifications to the user interface or workflow. This aspect is more critical than genuine. However, it does not necessarily involve the need for re-learning the system, which is covered by the Instruction aspect. Example [sic]: “It just keeps getting worse. Why do you do this?”
* Feature Gap. Users want to know why a feature is incomplete or missing. This aspect doesn't cover cases where a feature is not supported for an individual user's use case (see aspect Compatibility). Example [sic]: “Why would you have a database where you can only add and not edit or delete?”
* Compatibility. Users are confused by a feature(s) not being supported or compatible with their use case. So, they are prevented from using a set of features due to external conditions that are not part of the system. This aspect excludes errors or failures. Example [sic]: “Only big downfall is that USA account holders for some reason ... cannot use the boost feature. No clue why and no one has given answers to why it doesn't work.”
Finally, the category describes a situation with an undeniable objective deficiency such as an error or failure <cit.> in the system. It differs with , where the primary concern is a subjective deficiency in a user's eyes. We found the following aspects to be typical for :
* Fix. Users ask about fixes or workarounds to solve errors/ failures or ask whether errors/failures are known to the developers. Example [sic]: “Anyone experiencing the same or know what to do about it?”.
* Cause. Users ask for the underlying faults that cause errors, failures, or obviously erroneous outputs. They are interested in knowing the cause of the errors/failures to potentially attempt to fix them themselves. On the contrary, they do not ask for any support (see aspect Fix). Example [sic]: “Is it a loading problem or a glitch??”.
* Confusing Message. Users feel misled by rare messages (such as uninformative or incongruous alerts) and assess the messages as incomplete, inaccurate, or erroneous. The messages can potentially be faulty explanations. Example [sic]: “I constantly get warnings that I don't have enough shares to sell and I cannot find any solutions”.
§.§ Discussion of Results
Through a rigorous study of app reviews, we have developed the Explanation Needs taxonomy, which addresses RQ1 and provides a valuable resource for researchers and developers seeking to understand the concerns and requirements of end-users. By categorizing user needs in the taxonomy, we can better recognize and address various requirements in a more systematic manner, ultimately improving the quality, transparency, and user-friendliness of the application. The proposed taxonomy serves as an enabler, allowing for a more effective approach to addressing user needs and fostering a deeper understanding of the end-user experience. As such, the Explanation Needs taxonomy has significant implications for app development and can contribute to the development of more explainable systems that better meet the needs of users.
With the Explanation Needs taxonomy, we were able to tackle the RQ2, which aimed to gain a more statistical view of the types of Explanation Needs expressed in app reviews. So we applied the taxonomy to multiple sets of data, composed of 5,564 reviews in total. Table <ref> provides an overview of all the datasets used in this paper. As discussed in Section <ref>, the taxonomy extraction was based on the and the final labeling was achieved through several rounds of cross-checking to ensure the validity and reliability of our findings.
However, to gain deeper insights into the types of information and Explanation Needs in the app reviews and to further assess the coverage and applicability of our taxonomy, we also labeled the reviews of our extended datasets, which we create for classifier implementation and validations (see Section <ref> for more details). The labeling process of the rest of the data (i.e., the app reviews 9 to 22 in Table <ref>) was carried out after consolidating the taxonomy and codebook, the latter of which provides complete information on inclusion and exclusion criteria, as well as typical and atypical examples. Following this, a single coder categorized the app reviews in and that had already been labeled as Explanation Needs (see Section <ref> for more details).
Besides the description of the apps, source and number of reviews, Table <ref> provides a breakdown of the distribution of different types of Explanation Needs per app. It shows the number of occurrences of each type of Explanation Needs for each app, as well as the total number and percentage of Explanation Needs across all apps. By examining this table, we can answer the RQ2 by identifying the areas where users require the most explanations. This analysis can help shed light on the nature and extent of Explanation Needs in app reviews.
For example, it shows that the majority of cases fall under the Primary Concerns category, accounting for 52.3% of all app reviews. This implies that users' primary issue with the app is their lack of understanding and knowledge, without any substantial problems aside from it. This finding highlights the importance of addressing users' primary concerns and providing sufficient explanations to enhance their overall understanding of the app's functionality. Furthermore, the category is the most frequent type within the Primary Concerns and accounts for 20.7% of the total number of Explanation Needs across all apps. This means that a significant proportion of user feedback in app reviews is related to ordinary interaction with the system. As users engage with the app, they may encounter unexpected behaviours, have questions about design decisions, or need clarification on the meaning of certain visual elements or notions. Accordingly, it is not surprising to have a relatively high number of types since these issues could arise regardless of the app's specific functionality, and, therefore, could be relevant to a wide range of users. Additionally, the category may be particularly salient to users, as it directly affects their experience using the app, and they may be more likely to leave reviews on these types of issues. Similarly, the category stands out with the second-highest percentage of Explanation Needs in the primary concern, accounting for 18.6% of all Explanation Needs, indicates that users frequently encounter difficulties in understanding how to use certain features or functionalities of the app. This finding highlights the importance of providing concise instructions or tutorials to help users learn how to use the app effectively.
Overall, the high percentage of and indicates that the app's user interface or design could be improved. Our results hence may suggest that the application design and development should primarily focus on the usability of the apps by making them more intuitive and user-friendly.
Another interesting observation is that the category, which is classified as a secondary concern, has the highest percentage of Explanation Needs at 32.3%. This could be attributed to its subjective nature, as the primary concern of this category is a perceived deficiency from the user's point of view, which may be difficult to address directly. Additionally, this deficiency is not necessarily related to a specific bug or technical issue, but rather a mismatch between the user's expectations and the app's performance or features. This finding suggests that users are more likely to express their discontentment and frustration in reviews. Last but not least, our qualitative analysis also reveals an important insight. We found that although app reviews provide a wealth of information about users' Explanation Needs, the proportion of reviews that contain such information is relatively low, at only 5.1%. This indicates a need for more efficient and automated techniques to extract useful content from reviews. Therefore, our study has motivated us to pursue our second contribution, which is described in more detail in Section <ref>. By developing machine learning-based approaches to extract Explanation Needs from reviews, we hope to improve the efficiency and effectiveness of analyzing large volumes of user feedback.
§.§ Threats to Validity
A potential threat to internal validity is the use of quantitative coding, which can be interpretive and subjective. This means that our analysis may be influenced by our own biases or assumptions, which could affect the accuracy of our findings. Poor English and typos in some reviews can also lead to inaccurate conclusions, but we made a conscious effort to evaluate unintelligible reviews. In addition, a threat to external validity could be survivorship bias, as our results may not be representative of those with low technological literacy, as they may be less likely to write and publish app reviews in the first place. Also, the we used in our taxonomy extraction is relatively small, with only a few cases of Explanation Needs observed (4.6% as shown in Table <ref>). Accordingly, it might limit the generalizability of our taxonomy categories. However, to mitigate the potential threat of a small sample, we conducted a thorough and saturated coding process and verified the validity of our taxonomy categories on an extended dataset.
§ AUTOMATIC DETECTION OF EXPLANATION NEEDS
§.§ Corpora Creation
To determine the best method for detecting Explanation Needs in a structured way, we follow the recommendations by Dell’Anna et al. <cit.>. They stress that the results of a simple cross-validated experiment do not allow to draw definite conclusions about the performance of a classifier in an operational context. In other words, we cannot necessarily infer from such an experiment whether the classifier is able to generalize and is thereby suitable for use on unseen data in practice. Hence, we evaluate our approaches on two datasets:
CrossVal-DS. We use this dataset to train and compare all models applying 10-fold cross-validation. The main purpose of is to compare the performance of different NLP classifiers and to select the best-performing method. It includes all reviews of created in Section <ref>. However, this dataset with 77 Explanation Needs is not sufficient for training an NLP classifier. Accordingly, we extend the dataset with further reviews and manually label them with respect to the tags “explanation need” and “no explanation need”. We make use of a dataset collected by Maalej et al. <cit.> that has already been utilized in the RE community to classify app reviews into problem reports, inquiries, and irrelevant ones <cit.>. Additionally, we collect further app reviews from 9 popular apps, using custom Python web scraping tools for the Apple App Store[<https://pypi.org/project/app-store-scraper/>] and Google Play Store[<https://pypi.org/project/google-play-scraper/>]. For each of the apps, we scraped as many reviews as possible and then drew a random sample of 100 reviews to include an equal-sized subset of the reviews per app. A detailed overview of is provided in Table <ref>. In total, comprises 5,078 reviews of which 261 contain Explanation Needs (5.14%).=-1
General-DS. To investigate the generalizability of the best-performing classifier, we apply it to a set of unseen reviews that are not associated with any of the apps contained in . Specifically, we scrape and annotate reviews about the four randomly selected apps called WeChat, Memrise, Duolingo, and GitHub (see Table <ref>). The main purpose of is to report the performance of our best classifier in a realistic setting. In total, comprises 486 reviews of which 24 contain Explanation Needs (4.94%).
§.§ Annotation Validity
To verify the reliability of our annotations, we calculated the inter-annotator agreement in terms of Cohen's Kappa <cit.>. We involved a total of four annotators in the creation of and and assessed the inter-rater reliability on the basis of 485 reviews that each have been labeled by two out of the four annotators.
In case of a high imbalance of ratings, Cohen's Kappa is low and indicates poor inter-rater reliability even if there is a high agreement between the raters (Kappa paradox <cit.>). Thus, Cohen's Kappa is not meaningful in such scenarios. Consequently, Cohen's Kappa should always be reported together with the percentage of agreement and other paradox-resistant measures (e.g., Gwet's AC1 measure <cit.>). We calculated all measures (see Table <ref>) using the cloud-based version of AgreeStat[<https://www.agreestat.com/>]. Cohen's Kappa and Gwet's AC1 can both be interpreted using the taxonomy developed by Landis and Koch <cit.>: values ≤ 0 as indicating no agreement and 0.01–0.20 as none to slight, 0.21–0.40 as fair, 0.41–0.60 as moderate, 0.61–0.80 as substantial, and 0.81–1.00 as almost perfect agreement. Table <ref> demonstrates that the inter-rater agreement of our annotation process is reliable as we achieve an average percentage of agreement of 95%. Despite a high agreement of over 90%, Cohen's Kappa yields a relatively low value, which paradoxically suggests only moderate agreement. A more meaningful assessment is provided by Gwet's AC1 as it did not fail in the case of prevalence and remains close to the percentage of agreement. The achieved Gwet's AC1 of 0.945 indicates a nearly perfect agreement. Therefore, we assess and as reliable and suitable for the implementation and evaluation of our Explanation Need detection approach.
§.§ Methods
We define the detection of Explanation Needs as a binary classification problem, in which we are given a certain review 𝒳 and we are required to produce a nominal label y ∈𝒴 = {explanation need, no explanation need}. Since app store reviews are written in natural language, we build our classifier based on different methods established for NLP.
Rule-based Approach.
Instead of using a random classifier as the baseline approach, we involve simple regex expressions for the detection of Explanation Needs. We iterate through all reviews in the test set and check if a question mark or the word “why” is contained. We hypothesize that both expressions might be a feasible indicator for the presence of an Explanation Need. Following this assumption, we classify a review as an Explanation Need if it contains at least one of the two expressions and vice versa.
Machine Learning-based Approach.
We investigate the use of supervised ML models that learn to predict Explanation Needs based on a labeled dataset. Specifically, we employ established binary classification algorithms: NB, SVM, RF, DT, LR, AB, and KNN. To determine the best hyperparameters for each binary classifier, we apply Grid Search, which fits the model on every possible combination of hyperparameters and selects the most performant. We use two different methods as word embeddings: BoW and TF-IDF. In Table <ref> we report the classification results of each algorithm as well as the best combination of hyperparameters.
Deep Learning-based Approach.
With the rise of DL, more and more researchers are using DL models for NLP tasks. In this context, the BERT model <cit.> is prominent and has already been used for question answering and named entity recognition. BERT is pre-trained on large corpora and can therefore easily be fine-tuned for any downstream task without the need for much training data (Transfer Learning). In our paper, we make use of the fine-tuning mechanism of BERT and investigate to which extent it can be used for the detection of Explanation Needs. First, we tokenize each app store review. BERT requires input sequences with a fixed length (maximum 512 tokens). Therefore, for reviews that are shorter than this fixed length, PAD are inserted to adjust all reviews to the same length. Other tokens, such as the CLS, are also inserted in order to provide further information on the review to the model. CLS is the first token in the sequence and represents the whole review (i.e., it is the pooled output of all tokens of a review). For our classification task, we mainly use this token because it stores the information of the whole review. We feed the pooled information into a single-layer feedforward neural network that uses a softmax layer, which calculates the probability that a review contains an Explanation Need or not.
§.§ Evaluation Procedure
is strongly imbalanced as only 261 are positive samples. To avoid the class imbalance problem, we apply Random Under Sampling. We randomly select reviews from the majority class and exclude them from the dataset until a balanced distribution is achieved. Our final dataset consists of 522 reviews of which 261 contain an Explanation Need and the other 261 do not. We follow the idea of cross-validation and divide the dataset into a training, validation, and test set. We opt for 10-fold cross-validation as a number of studies have shown that a model that has been trained this way demonstrates low bias and variance <cit.>. Please note that undersampling stands in conflict with our goal to understand how well our classifier generalizes and performs in a realistic setting. Hence, we do not undersample allowing us to report our final results on a realistically distributed test corpus.
We use standard metrics for evaluating our approaches, such as Precision, Recall, and a weighted F-measure. Since a single run of a k-fold cross-validation may result in a noisy estimate of model performance, we repeat the cross-validation procedure five times and average the scores from all repetitions.
Since our classifier is supposed to assist development teams by detecting relevant Explanation Needs in reviews automatically, we favor Recall over Precision. A high Recall corresponds to a greater degree of automation of Explanation Need detection because it is easier for users to discard FP than to manually detect FN. Consequently, we seek high Recall to minimize the risk of missed Explanation Needs and acceptable Precision to ensure that the development teams are not overwhelmed by FP. To attain a accumulated, single metric from Precision and Recall, the simple F-Measure (F1) is frequently used in binary classification tasks. It is defined as the harmonic mean between Precision and Recall, and thus assigns equal importance to both metrics. To account for our preference for Recall over Precision, it is imperative to make adjustments to the way in which the two metrics are weighted. We evaluate our approaches based on a weighted F-Measure:
F_β = (1+β^2) ·Precision ·Recall/(β^2 ·Precision) + Recall
where β is the ratio to which Recall is more important than Precision <cit.>. Berry <cit.> defines β as follows:
β= time_a ·λ/time_v
where time_a is the average time that a human would need to assess an artifact manually (i.e., the time spent by a human determining whether a particular review is an Explanation Need or not), and time_v is the average time that a human would need to verify whether a positive detection by a tool is actually a True Positive (i.e., the time spent by a human neglecting a FP detection of an Explanation Need). Further, λ is the inverse of the share of relevant artifacts within all artifacts. In other words, λ is the average number of artifacts that an analyzer would need to investigate in order to find a single relevant artifact. In our case, λ is calculated as follows:
λ= (285/5564)^-1 ≈19.52
because we identified a total of 285 Explanation Needs in our dataset of 5,564 reviews. Thus on average, one out of 19.52 app reviews contains an Explanation Need. Since the time required to vet a single answer of our classifier is no more than the time required to manually check if an app review contains an Explanation Need, the weight ratio β is equal to λ. Hence, we define β as 19.52.
§.§ Experimental Results
In the following, we describe the results of our experiments. First, we compare the performance of different NLP classifiers on . Second, we investigate the generalizability of the best-performing method on .
Selection of Best-Performing Method
Table <ref> reveals that our shallow rule-based approach shows a strong performance in detecting Explanation Needs. It achieves a high F_19.52 score for both classes and is able to demarcate between reviews that contain Explanation Needs and those that do not. In comparison, all ML-based approaches exhibit a significantly poorer performance. For example, DT trained on TF-IDF embeddings achieves a Macro-F_19.52 score of 58% (deterioration of 35% compared to the baseline approach). The best performance in this category is achieved by RF trained on BoW embeddings with a Macro-F_19.52 score of 76%. Our experiment shows that the choice of sentence embedding has no significant effect on the performance of the ML-based approaches. Most of the approaches achieve a Macro-F_19.52 score of about 70% regardless of the applied sentence embedding. Our fine-tuned BERT model, on the other hand, shows a considerably stronger performance and achieves a Macro-F_19.52 score of 93%. Interestingly, despite its rich language understanding, the BERT model fails to outperform our simple rule-based approach. In fact, both approaches achieve the same Macro-F_19.52 score and posses consequently the same predictive power. Our experiments thus show that both approaches are suitable for identifying Explanation Needs in . To investigate the generalizability of the rule-based approach and the BERT model, we apply both approaches to a larger set of unseen reviews written for other apps contained in .=-1
Generalizibility of Best-Performing Method
When applied to unseen data, both approaches show a clear performance drop in the detection of Explanation Needs (see Table <ref>). While both approaches continue to show very high F_19.52 scores for the “no explanation need” class, the F_19.52 score for the “explanation need” class has decreased significantly. The largest performance drop is evident in the rule-based approach, which only shows an F_19.52 score of 67% in detecting explanation needs across all reviews of all four apps. Similarly, the trained BERT model fails to match the very good F_19.52 score of 94% that it could achieve when applied to the balanced training set. Instead, it achieves a score of 79% on the unseen data, which corresponds to a decrease of 15%. Overall, the BERT model outperformed the rule-based approach and achieved a significantly better Macro-F_19.52 score of 86%.
The higher Macro-F_19.52 score is mainly attributable to the fact that the BERT model shows a significantly better Recall with regard to the Explanation Need class. In other words, the BERT model identified more Explanation Needs in the reviews than the rule-based system. Our experiment demonstrates that this performance deviation does not depend on a specific app about which the respective reviews were written. In fact, when applied to the reviews about WeChat, Duolingo and Github, the BERT model exhibits better performance. In the case of the reviews about Memrise, it achieves the same Recall as the rule-based approach. Both the rule-based approach and the BERT model show the most significant performance loss with regard to Precision and generate a great number of FP. Using both approaches, two of three reviews that are supposed to contain an Explanation Need are FPs, causing high filtering costs for practitioners.
§.§ Discussion of Results
Our experiments show that the rule-based approach achieves the same performance as the BERT model when evaluated on , but performs worse when applied to unseen data. The rule-based approach fails to recognize more than 30% of the Explanation Needs and seems to generalize less effective than the BERT approach. When analyzing the data in , we see that the detection of Explanation Needs cannot be broken down to the presence of questions and question words. Explanation Needs do not necessarily contain question marks or question words. In many cases, questions are formulated but question marks are not included: Would you please keep us updated on what's going on. I have several texts and don't know how to keep them. Don't want to lose it. The BERT model understands the semantics of sentences better and dependents less on the sentence's syntax. The rule-based approach could be extended by adding more interrogatives (e.g., how) and interrogative verbs (e.g., don't understand) to enhance the Recall of the approach, however, this may lead to an unreasonable increase in FPs. The resulting filtering effort would diminish the use of the approach in practice.
From a critical point of view, our best classifier does not perform flawlessly. It does not identify all Explanation Needs in and predicts a number of FPs. We argue that the recall value needs to be improved above 90% to qualify the approach for practical use. Otherwise, the practitioners would have to go through the reviews manually to detect false negatives, which is time-consuming given the high number of reviews and the fact that Explanation Needs rarely occur. The achieved precision value of 37% is not optimal, but in our view still justifiable. It is much easier for the practitioner to neglect two false positives from 3 reviews predicted as Explanation Needs than to go through 20 reviews manually to discover a single Explanation Needs.
Our classifier marks a first step toward automatic Explanation Need detection. Further studies should focus on optimizing the classifier in terms of recall. We hypothesize that the extension of the training set and the use of further language models might be beneficial. So far, we have only focused on the BERT-Base model <cit.>, although other studies <cit.> show that alternative models such as RoBERTa can achieve even better performance. To assist practitioners in filtering FPs, it may also be useful to have the classifier mark the specific clause in each review that has caused the review to be categorised as Explanation Needs <cit.>. This will help practitioners to understand the inner workings of the classifier and also increase its acceptance.
§.§ Threats to Validity
A threat to internal validity are the annotations themselves as an annotation task is subjective to a certain degree. To minimize the bias of the annotators, we performed two mitigation actions: First, we conducted a workshop prior to the annotation process to ensure a common understanding of Explanation Needs. Second, we assessed the inter-rater agreement by using multiple metrics (Gwet's AC1 etc.).
Despite our efforts to make the labeling process as transparent and systematic as possible, there may still be some variability in the resulting gold standard, e.g., misinterpretation of the users' intention, blurred boundaries between the categories, too broad or too narrow judgement, or human mistakes. Using the adjusted F_β-score as an evaluation metric poses a threat to construct validity. We used an adjusted β value of 19.52, which was calculated based on the frequency of Explanation Need occurrences in app reviews. This value is in the order of β values calculated for other “needle in the haystack” tasks <cit.>. However, it is possible that the value may deviate when calculated based on another dataset. Our results have shown that generalization of our tested classifiers is fairly moderate when applied to unseen, dissimilar test data. This may indicate that more data is needed to train a classifier that generalizes better.
Lastly, app reviews are not the only relevant source of user feedback <cit.>.
§ CONCLUSION
This work is a further step towards user-centered explainability engineering.
It contributes to a better understanding of users' Explanation Needs and lays the foundation for future research and development in this area. The proposed taxonomy of Explanation Needs provides a rigorous approach for extracting explainability requirements from app reviews, ensuring that they meet users' expectations. In addition, our approach represents the first step towards automatic explanation need detection and reduces the manual effort required by engineers and researchers to identify Explanation Needs in reviews. To facilitate practical use of the approach, it needs to be optimized for recall so that practitioners can efficiently focus on eliciting valid Explanation Needs. Finally, our published set of manually labeled app reviews will enable researchers in the field to improve their own models and approaches for detecting Explanation Needs.
§ ACKNOWLEDGEMENTS
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No.: 470146331, project softXplain (2022-2025).
IEEEtran
|
http://arxiv.org/abs/2307.05972v1 | 20230712073824 | Self-Distilled Quantization: Achieving High Compression Rates in Transformer-Based Language Models | [
"James O' Neill",
"Sourav Dutta"
] | cs.CL | [
"cs.CL",
"cs.LG"
] |
Contrastive Learning for Conversion Rate Prediction
Yanlong Du
August 12, 2023
===================================================
We investigate the effects of post-training quantization and quantization-aware training on the generalization of Transformer language models. We present a new method called self-distilled quantization (SDQ) that minimizes accumulative quantization errors and outperforms baselines. We apply SDQ to multilingual models XLM-R_Base and InfoXLM_Base and demonstrate that both models can be reduced from 32-bit floating point weights to 8-bit integer weights while maintaining a high level of performance on the XGLUE benchmark. Our results also highlight the challenges of quantizing multilingual models, which must generalize to languages they were not fine-tuned on.
§ INTRODUCTION
A main aim of neural network quantization is to reduce the size and computational demands of a model while maintaining its performance. There are two main approaches: quantization-aware training (QAT) <cit.> and post-training quantization (PTQ) <cit.>. Both of these approaches have limitations in terms of dealing with accumulative quantization errors that are propogated within the layers of a neural network during the forward pass <cit.>. To address this issue, we propose a method called Self-Distilled Quantization (SDQ) that combines self-attention and output distillation with quantization to compress large language models. SDQ involves injecting quantization noise into the student network during training and distilling knowledge from a fine-tuned teacher network from both its final output and outputs of intermediate self-attention layers. By distilling knowledge of the self-attention layers, as depicted in <ref>, we further reduce the compounding effect of quantization errors in the network. We use SDQ for self-attention models and demonstrate its effectiveness in compressing multilingual models XLM-R_Base and InfoXLM_Base, achieving high compression rates while maintaining performance on the XGLUE benchmark. Lastly, we identify that quantization error is largest at the output of self-attention modules.
We find that SDQ sets state-of-the-art (SoTA) results and when used with current QAT methods it improves over the original QAT method without distillation.
We focus on quantizing cross-lingual Transformer models, namely XLM-RoBERTa <cit.> and InfoXLM <cit.>. To our knowledge, this is the first study that focuses on quantizing cross-lingual language models and in turn how quantization effects generalization on more than one language. We now move to a background on quantization for neural networks.
§ RELATED WORK
Combining quantization and distillation has been previously explored by <cit.>, who used three different schemes to combine low bit precision and knowledge distillation (KD) using a 4-bit ResNet network. <cit.> used a distillation loss with respect to a quantized teacher network to train a student network, and also proposed differentiable quantization, which optimizes the location of quantization points through SGD. <cit.> used iterative quantization, supervised by a teacher network, to retrain an FP-32 model with low precision convolution weights (binary, ternary, and 4 bits). <cit.> used QAT and fine-tuning to mitigate the regularization effect of KD on quantized models. Q8BERT <cit.> and fully Quantized Transformer <cit.> applied QAT with the Straight-Through Estimator to approximate non-differentiable quantization in INT-8 format. We now describe the methodology of SDQ.
§ METHODOLOGY
We begin by defining a dataset 𝒟: = {(X_i, y_i)}^D_i=1 with samples s_i = (X_i, y⃗_i), where each X_i := (x⃗_1, …, x⃗_N) and x⃗_i ∈ℝ^d is the i-th vector. For structured prediction y_i ∈{0, 1}^N × d_y and for single and pairwise sentence classification, y_i ∈{0, 1}^d_y, where d_y is the number of classes.
Let y⃗^S=f_θ(X_i) be the output prediction (y^S∈ℝ^d_y) from the student f_θ(·) with pretrained parameters θ:= {W_l, b⃗_l}_l=1^L for L layers and the outputs of self-attention blocks are denoted as A⃗_l.
The loss function for standard classification fine-tuning is defined as the cross-entropy loss ℓ_CE(y⃗^S, y⃗).
Self-Distilled Quantization
For self-distilled quantization, we also require a fine-tuned teacher network f_Θ, that has been tuned from the pretrained state f_θ, to retrieve the soft teacher labels y^T := f_Θ(x⃗), where y^T∈ℝ^C and ∑_c^C y^T_c=1. The soft label y⃗^T can be more informative than the one-hot targets y⃗ used for standard classification as they implicitly approximate pairwise class similarities through logit probabilities.
The Kullbeck-Leibler divergence (KLD) ℓ_KLD is then used with the main task cross-entropy loss ℓ_CE to express ℓ_SDQ_KLD as shown in <ref>,
ℓ_SDQ_KLD= ℓ_CE(y⃗^S, y⃗)+ατ^2 D_KLD(y⃗^S, y⃗^T)
where D_KLD(y⃗^S,y⃗^T) = ℍ( y⃗^T) - y⃗^Tlog(y⃗^S), ℍ(y⃗^T)=y⃗^Tlog(y⃗^T) is the entropy of the teacher distribution and τ is the softmax temperature. Following <cit.>, the weighted sum of the cross-entropy loss and the KLD loss ℓ_SDQ_KLD=ℓ_CE(y⃗^S, y⃗) +ατ^2 D_KLD(y⃗^S, y⃗^T) is used as our main SDQ-based KD loss baseline, where α∈ [0, 1]. However, D_KLD only distils the knowledge from the soft targets of the teacher but does not directly reduce accumulative quantization errors of the outputs of successive self-attention layers. This brings us to our proposed attention-based SDQ loss ℓ_SDQ_Att-KLD shown in <ref>,
ℓ_SDQ_Att-KLD=ℓ_CE(y⃗^S, y⃗) +ατ^2 D_KLD(y⃗^S, y⃗^T)
+β1/LH∑_l=1^L∑_h=1^Hℓ_Attention(A^S_lh,A^T_lh)
where α and β are regularization terms and ℓ_Attention computes the loss between the student and teacher outputs of each self-attention block in L layers and H attention heads per layer. We also consider two baselines, ℓ_SDQ_Att which is the same as <ref> without ατ^2 D_KLD(y⃗^S, y⃗^T) and ℓ_SDQ_Hid which applies the Mean Squared Error (MSE) loss between the hidden state outputs instead of the attention outputs. The gradient of D_KLD(·,·) is expressed as ∂ D_KLD(y⃗^S_i, y⃗^T_i)/∂y⃗^S_i = τ(y⃗^S_i/τ - y⃗^T_i/τ) and as τ→∞, the gradient is approximately 1/(d_yy⃗^S_i - y⃗^T_i). Similarly, the gradient of the MSE loss on a single self-attention output in layer l and head h is 1/n_lh(a⃗^S_j - a⃗^T_j) for a single sample input x. Hence, we see the connection between derivatives between the KLD loss and the MSE loss when combining them in a single objective. We now move to desribing how SDQ is used in two QAT methods.
We can then expressed the gradient step as,
W_(t+1) := W_t - η𝔼_
(X,y)∼𝒟(τ(y⃗^S/τ - y⃗^T/τ)
+ (a^S-a^T)) - ∇_W_ty⃗^S
Note that scaled dot-product is used (normalization by √(d_k)) to avoid vanishing gradients of the Softmax, which may occur when d_k is large.
The parameters for the j-th attention head K^j, V^j ∈ℝ^d × l, U^j ∈ℝ^o for j = 1,…, n_a where n_a is the number of attention heads. Then we summarize the formulation of multi-headed self-attention as Equation (<ref>),
Z^j = Softmax(QK^j/√(d_k)(V^j)^⊤Q^T)QU^j
Z̃ = Concat(Z^1, …Z^n_a)
Z = Feedforward(LayerNorm(Z̃ + Q))
where Z^j∈ℝ^n × d_a and Z̃∈ℝ^n × d_an_a, with d_a being the dimensionality of the self-attention output.
Iterative Product Distilled Quantization
We first consider using SDQ with iPQ <cit.>. This is achieved by quantizing m subvectors for each k columns of W where a codebook for each k subvectors is learned to map each subvector to its nearest neighbor in the learned codebook C ∈ℝ^k× d where k is the number of codewords.
The codebook is updated by minimizing ||W - W̃||_2^2 = ∑_i^d || W_[:, i] - ϕ(w_[:, i])||^2_2 where ϕ(·) is the quantization function. This objective can be efficiently minimized with the k-means algorithm and the codewords of each layers are updated with SGD by averaging the gradients of each assigned block of weights. This is done iteratively from the bottom layers to the top layers throughout training where the upper layers are finetuned while the lower layers are progressively being quantized <cit.>. When using iPQ with SDQ, omitting the KLD loss and cross-entropy loss, the objective is ℓ_SDQ_iPQ =∑_l=1^L-F[||W_l - W̃_l||^2_2 + β/L-F∑_i^d(A^S_l,i - A^T_l,i)^2] where F is the number of finetuned layers (non-quantized) at that point in training. Hence, SDQ progressively quantizes the layers throughout training when used with iPQ.
Block-Wise Distilled Quantization Noise
For the majority of our QAT-based experiments we use Quant-Noise <cit.>. Quant-Noise is a SoTA QAT method that applies (fake) block-wise quantization noise at random to each weight matrix. Concretely, blocks of weights b_kl in W_l are chosen at random at a rate p and quantization noise is added to the chosen blocks.
For describing the use of SDQ-based QuantNoise in Transformers, we can define A^S = Softmax(W̃_QW̃_K/√(d_k)W̃^⊤_VW̃^⊤_Q)W̃_QW̃_U where W̃ represents (fake) quantized weights and is given as W̃ = ϕ_INT-8(W) = s(round(W/s + b) - b) where s and b are scalars learned throughout training and represent the scaling factor and offset respectively. We can than pass A^S and A^T to <ref> to compute the loss.
Algorithm <ref> illustrates the proposed framework of SDQ with QuantNoise (SDQ_Att-KLD) in PyTorch pseudocode used for sentence classification. In this code student_model contains forward hooks that apply QuantNoise.
§ EMPIRICAL RESULTS
We begin by referring the reader to the supplementary material for the experimental setup in <ref> and <ref>.
Before discussing the main results on XGLUE, we first analyse the mean absolute quantization error and the Frobenius norm of the elementwise difference in self-attention blocks between an INT-8 dynamically quantized InfoXLM_Base and an unquantized FP-32 InfoXLM_Base in <ref>. We see in <ref> that the output layer contains the largest mean absolute error across each layer and highest error variance. In contrast, query, key, value (QKV) parameters have much smaller error. However, since most of the parameters are found in the QKV layers, the sum of the quantization error is larger, as seen in <ref>. This motivates us to focus on the output of the self-attention block when minimizing quantization errors with our proposed loss in <ref> as the mean error is higher near the output as it accumulates errors from previous layers in the block.
This is also reflected in the parameter distribution of each layer type across all layers in
<ref>, where the x-axis is the mean absolute quantization error and the y-axis is the layer indices. We see the quantization noise is more apparent on the output layer as the Gaussian distrbutions are non-smooth and have clear jitter effect.
XNLI Per Language Results
<ref> shows the baselines and our SDQ methods applied to XLM-R_Base and InfoXLM_Base. Here, both models are only trained on the English language and hence the remaining languages in the evaluation set test the zero-shot performance after INT8 quantization (apart from the first 3 rows that show FP-32 fine-tuned results). On average, we find that best student networks results are found when distilling using QNAT_Att-KLD SDQ with the outputs of an FP-32 teacher for InfoXLM_Base at 73.8% test accuracy points, where the original FP-32 InfoXLM_Base achieves 74.6%. Additionally we see that QNAT_Att-KLD improves over QNAT_KLD distillation, indicating that attention output distillation improves the generalization of the INT-8 student model. We also found that largest performance drops correspond to languages that less pretraining data and morphologically rich (Swahili, Urdu, Arabic), while performance in English for the best INT-8 XLM-R_Base (84.4%) is within 0.2% of the original network (84.6%) and the best InfoXLM_Base that uses QNAT_Att-KLD is on par with the FP-32 results.
We also find that languages which have less training data for pretraining and are morphologically rich (Swahili, Urdu and Arabic) suffer the largest drops in performance. From this we posit that FP-32 resolution allows for capacity that maintains information about more complex sentences.
In contrast, the standard supervised learning evaluation on English achieves comparable results to the original FP-32 results across all baselines and our proposed SDQ (the best INT-8 XLM-R_Base (84.4%) is within 0.2% of the original network (84.6%) and the best InfoXLM_Base that uses QNAT_Att-KLD is on par with the FP-32 results).
Quantization Results on XGLUE.
We show the per task test performance and the understanding score (i.e average score) on XGLUE for quantization baselines and our proposed SDQ approaches in Table <ref> (for brevity we denote InfoXLM_Base as I and XLM-R_Base). Our proposed QNAT_Att-KLD achieves the best average (Avg.) score and per task performance for all tasks, using a fine-tuned InfoXLM_Base (XNLI, NC, NER and QAM) and a fine-tuned InfoXLM_Base trained with QuantNoise and dynamically quantized post-training (PAWSX, POS, QAM, QADSM and WPR). We also find that QNAT_Att-KLD improves over QNAT_KLD, highlighting that the attention loss is improving quantized model performance.
Performance versus Compression Rate
<ref> shows how the performance changes for four approaches, including two of our proposed objectives (QNAT_KLD and QNAT_Att-KLD), when training InfoXLM_Base. As before, PTQ_dynamic is a dynamically quantization fine-tuned InfoXLM_Base and QNAT-PTQ_dynamic is the same as PTQ_dynamic except fine-tuned also using QuantNoise.
Unlike our previous results, here we apply fake quantization at inference to achieve compression lower than INT-8 and be comparable to previous work <cit.>. We see that performance is generally well maintained up until 8 bits. However, performance significantly degrades for all quantization methods for 4 and 2 bit weights. We find that QNAT_Att-KLD maintains higher performance when compared to the baselines and directly quantizing with no QAT (PTQ_dynamic) leads to the poorest results, also reflected in <ref> results with real dynamic quantization at inference time.
Ablation with Current QAT Methods
<ref> shows the results from XGLUE tasks where the first two columns describe how the student and teacher networks are trained and “Standard” refers to standard FP-32 fine-tuning. This includes iPQ <cit.> with scalar quantization (iPQ_Scalar), iPQ that uses expectation maximization to create the codebook during training (iPQ_EM) and previous results of QuantNoise (QNAT) as a reference point. In this setup, we only apply the attention loss, ℓ_Attention, to the layers that are quantized during iPQ.
In all cases, adding that SDQ distillation of the classification output and the self-attention outputs improves the average performance.
§.§ Summary of Results.
From our experiments we find that minimizing the MSE between the output of quantized self-attention modules and unquantized outputs (i.e distillation) leads to consistently better performance of quantized models. Moreover, most of the quantization errors after dynamic quantization are largely attributed to the query, key and value (QKV) parameters, but this is only due to the discrepancy between the number of parameters used for these layers with remaining fully-connected output layers within the self-attention block. We also find quantization error per layer type is largest on the output linear layer with a significantly larger mean error and variance when compared to QKV outputs. This is due to the accumulation of quantization errors within the self-attention block itself. Hence, this motivated us to focus on the output of the attention block to apply the distillation loss, to achieve a desired trade-off between the total number of losses applied during training (only 1 per self-attention block) and its direct effect on generalization performance of quantized models.
§ CONCLUSION
In this paper we proposed an attention-based distillation that minimizes accumulative quantization errors in fine-tuned masked language models. We identified that most of the quantization errors accumulate at the output of self-attention blocks and the parameter distribution of the output layer is effected more by quantization noise.
The proposed distillation loss outperforms baseline distillation without the attention loss and the resulting INT-8 models are within 1 understanding score points on the XGLUE benchmark with real quantization post-training. Moreover, fine-tuning the teacher network with quantization-aware training can further improve student network performance on some of the tasks.
Further compression can be achieved up to 4-bit and 2-bit weights but performance steeply degrades as the network capacity is drastically reduced coupled with the models having to generalize to multiple languages it was not trained on.
§ LIMITATIONS
Dataset and Experimental Limitations.
The datasets and tasks we focus on are from the XGLUE benchmark <cit.>. The structured prediction tasks, namely Named Entity Recognition (NER) and Part of Speech (PoS) Tagging, both have a limited number of training samples at 15k and 25.4k samples respectively. This is due to the difficulty in annotating on the token level, however it can still be viewed as a limitation when compared to the remaining sentence-level tasks the majority of tasks have at least 100k samples.
Methodological Limitations.
Below are a list of the main methodological limitations we perceive of our work:
* Our method requires a teacher model that is already trained on the downstream task which can then be used to perform knowledge distillation. This is limiting when there are constraints on the computing resources required to produce the quantized model.
* We have focused on the problem of reducing accumulative qunatization errors which become more apparent the deeper a network is. However, this problem is intuitvely lessened when the model is shallow (e.g 3-4 layers) but perhaps wider. Hence the results may be less significant if the model is shallower than what we have experimented in this work.
* By introducing the distillation loss we require an additional regualrization term β to be optimally set, relative to the main distillation loss α. This can be viewed as a potential limitation has it introduced an additional hyperparameter to be searched to obtain best results on a given task.
* Lastly, since intermediate layer outputs of the teacher network are required for self-attention distillation, we have to perform two forward passes during training. Since standard KLD distillation only requires the output logits, it is common to store the training data teacher logits, eliminating the need to perform two forward passes at training data. However, this is not an option with self-atttention outputs as the storage required offline scales with the number of self-attention heads, number of layers and the size of the training data.
§ ETHICS STATEMENT
Here we briefly discuss some ethical concerns of using such compressed models in the real world, specifically the two techniques used in this work, quantization and knowledge distillation. <cit.> have found that compressed models can amplify existing algorithmic bias and perform very poorly on a subset of samples while the average out-of-sample accuracy is maintained close to the uncompressed model. This general finding for pruning and quantization may be also extrapolated to our work (including distillation), hence it is important to recognize that our work, much like the remaining literature on compression, may have ethical concerns with regards to algorithmic bias and how that effects downstream tasks. However, smaller models are more cost-efficient and thus become more widely available to the general public. To summarize, it is important to analyse any aforementioned bias amplification for subsets of samples for downstream tasks compressed models are used for.
acl_natbib
§ SUPPLEMENTARY MATERIAL
§.§ Self-Attention in Transformers
Consider a dataset D = {(X_i, y_i)}_i=1^m for D ∈𝒟 and a sample s:= (X, y) where the sentence X:= (x_1, … x_n) with n being the number of words x ∈ X.
We can represent a word as an input embedding x⃗_w ∈ℝ^d, which has a corresponding target vector y⃗.
In the pre-trained transformer models we use, X_i is represented by 3 types of embeddings; word embeddings (X_w ∈ℝ^n × d), segment embeddings (X_s ∈ℝ^n × d) and position embeddings (X_p ∈ℝ^n × d), where d is the dimensionality of each embedding matrix.
The self-attention block in a transformer mainly consists of three sets of parameters: the query parameters Q∈ℝ^d × l, the key parameters K∈ℝ^d × l and the value parameters V∈ℝ^d × o. For 12 attention heads (as in XLM-R_Base and InfoXLM_Base), we express the forward pass as follows:
X = X_w + X_s + X_p
Z := ⊕_i=1^12softmax(XQ_(i)K^T_(i)X^T)XV_(i)
ℤ = Feedforward(LayerNorm(Z + X))
ℤ = Feedforward(LayerNorm(Z + X))
The last hidden representations of both directions are then concatenated ℤ' := ℤ⊕ℤ' and projected using a final linear layer W∈ℝ^d followed by a sigmoid function σ(·) to produce a probability estimate ŷ, as shown in (<ref>). Words from (step-3) that are used for filtering the sentences are masked using a token to ensure the model does not simply learn to correctly classify some samples based on the association of these tokens with counterfacts.
A linear layer is then fine-tuned on top of the hidden state, h⃗_X, emitted corresponding to the token. This fine-tunable linear layer is then used to predict whether the sentence is counterfactual or not, as shown in <ref>, where ⊂ D is a mini-batch and ℒ_ce is the cross-entropy loss.
ℒ_ce := 1/||∑_(X,y) ∈y⃗log( σ (h⃗_X,·W) )
Configurations We use XLM-R_Base and InfoXLM_Base, which uses 12 Transformer blocks, 12 self-attention heads with a hidden size of 768.
The default size of 512 is used for the sentence length and the sentence representation is taken as the final hidden state of the first [CLS] token.
§.§ Experimental Setup and Hardware Details
Below describes the experimental details, including model, hyperparameter and quantization details.
We choose modestly sized cross-lingual language models as the basis of our experiments, namely XLM-R_Base <cit.> and InfoXLM_Base <cit.>, both approximately 1.1GB in memory and these pretrained models are retrieved from the https://huggingface.co/modelshuggingface model hub.
We choose both XLM-R_Base and InfoXLM_Base because they are relatively small Transformers and are required to generalized to languages other than the language used for fine-tuning. Hence, we begin from a point that model are already relatively difficult to compress and are further motivated by the findings that larger overparameterized networks suffer less from PTQ to 8-bit integer format and lower <cit.>.
For both XLM-R_Base and InfoXLM_Base the hyper-parameters are set as follows: 768 hidden units, 12 heads, GELU activation, a dropout rate of 0.1, 512 max input length, 12 layers in encoder. The Adam Optimizer with a linear warm-up <cit.> and set the learning rate to 2e-5 for most tasks. For all sentence classification tasks the batch size is set to 32 and we fine-tune with 10 epochs. For POS Tagging and NER, we fine-tune with 20 epochs and set the learning rate to 2e-5. We select the model with the best average results on the development sets of all languages. For SDQ-based models, we report the best performing model for α∈ [0.1, 0.2, 0.5, 0.8] and β∈ [10, 100, 200, 500]. All experiments are carried out on Tesla V100-SXM2 32 Gigabyte GPUs <cit.> with no constraint on GPU hours used on these machines. In all reported results, we report the best (max) result from 8-16 different runs when searching for α and β depending on each particular task.
§ INTENDED USE OF EXISTING ARTIFACTS
The artifacts we use in this work are the datasets from the XGLUE benchmark that are used for evaluation and the pretrained models themselves.
§.§ Model Configuration and Hyperparameter Settings
XLM-R_Base and InfoXLM_Base uses 12 Transformer blocks, 12 self-attention heads with a hidden size of 768. The default size of 512 is used for the sentence length and the sentence representation is taken as the final hidden state of the first [CLS] token. A fine-tuned linear layer W is used on top of both models, which is fed to through a softmax function σ as p(c|h) = σ(Wh⃗) where c is used to calibrate the class probability estimate and we maximize the log-probability of correctly predicting the ground truth label.
<ref> shows the pretrained model configurations that were already predefined before our experiments. The number of (Num.) hidden groups here are the number of groups for the hidden layers where parameters in the same group are shared. The intermediate size is the dimensionality of the feed-forward layers of the the Transformer encoder. The `Max Position Embeddings' is the maximum sequence length that the model can deal with.
We now detail the hyperparameter settings for transformer models and the baselines. We note that all hyperparameter settings were performed using a manual search over development data.
§.§.§ Transformer Model Hyperparameters
We did not change the original hyperparameter settings that were used for the original pretraining of each transformer model. The hyperparameter settings for these pretrained models can be found in the class arguments python documentation in each configuration python file in the <https://github.com/huggingface/transformers/blob/master/src/transformers/> e.g configuration_.py.
For fine-tuning transformer models, we manually tested different combinations of a subset of hyperparameters including the learning rates {50^-4, 10^-5, 50^-5}, batch sizes {16, 32, 128}, warmup proportion { 0, 0.1} and ϵ which is a hyperparameter in the adaptive momentum (adam) optimizer.
Please refer to the huggingface documentation at <https://github.com/huggingface/transformers> for further details on each specific model e.g at <https://github.com/huggingface/transformers/blob/master/src/transformers/modeling_roberta.py>, and also for the details of the architecture that is used for sentence classification and token classification.
|
http://arxiv.org/abs/2307.10216v1 | 20230714141631 | A modified Ehlers model for the description of inelastic behavior of porous structures | [
"Martin Abendroth",
"Alexander Malik",
"Bjoern Kiefer"
] | cond-mat.soft | [
"cond-mat.soft"
] |
ConTrack: Contextual Transformer for Device Tracking in X-ray
Marc Demoustier, Yue Zhang, Venkatesh Narasimha Murthy(), Florin C. Ghesu, and Dorin Comaniciu
August 12, 2023
==================================================================================================
This paper describes a modification of Ehlers' model for the inelastic behavior of granular media. The modified model can be applied for describing the inelastic behavior of porous media. The key feature is a subtle change of the yield potential, which allows the correct orientation of the triangular-shaped yield surface cross sections depending on the hydrostatic stress state. The model is incorporated into a general framework for isotropic plasticity. An elastic predictor/corrector algorithm is employed to solve the constitutive equations. The necessary derivatives for a Newton update are also given in detail. The model is calibrated using stress, and strain data obtained from finite element simulations of a generic highly porous open-cell Wheire-Phelan foam.
§ INTRODUCTION
Porous structures or foams appear in many technical applications as well as in nature. Especially for technical applications it is desirable to have computational models for such structures in order to predict their strength, elastic and inelastic behavior. The mechanical properties of porous structures strongly depend on their underlying meso and micro-scale, whereas homogenization schemes as shown in Fig. <ref> are applied.
The mechanical behavior on the micro-scale is defined by the properties of the bulk material a foam is made of. Often, the behavior on the micro-scale is known or can be described using established constitutive models. The behavior on the macro scale depends significantly on the topology of the mesostructure, the size and shape of the struts, and their connections <cit.>.
Ashby <cit.> investigated the general physical properties of foams and lattices depending on their geometrical structure and topology. He divided between bending and stretching-dominated structures. Critical states of foam structures, such as failure or onset of inelastic deformation can be described by limit surfaces in stress space. A lot of research has been done to describe such surfaces, whereas the majority of publications focus on the transition between elastic and plastic domains (yield surfaces) <cit.> or on surfaces characterizing the onset of material failure (failure surfaces). A comprehensive overview of phenomenological yield and failure surfaces is given in <cit.>.
Since foams or other porous structures can fail or yield for pure compression as well as for pure tension, their yield or failure surfaces are usually closed and convex surfaces in stress space <cit.>. Since it is difficult and expensive to perform experiments with triaxial stress states or stress states with large hydrostatic components, the experimental data available are limited with respect to the sampled stress space. Jung and Diebels <cit.> reviewed contributions with respect to experimental and modeling approaches regarding open cell foams.
One way to generate sufficient data for complete yield and failure surfaces is through direct numerical simulation (DNS) of foam structures. Such techniques have been used in a number of publications e.g. <cit.>. Depending on the underlying mesostructure, it is observed that yield and failure surfaces of foam structures can depend on the three invariants I_1, J_2, and J_3 of the stress tensor. Especially, the dependence on the third invariant J_3 is remarkable. The deviatoric cross sections of the yield surfaces are not necessarily of circular shape but vary from rounded triangular to rounded hexagonal shapes. The triangular shapes can have opposite orientations, depending on the sign of the hydrostatic stress <cit.>. The microstructures used in the models for DNSs are often representative volume elements (RVEs) of regular foam structures, like the Kelvin or Wheire-Phelan cell, which also can show a certain anisotropy. But, even yield surfaces for isotropic foam structures <cit.> show non-circular deviatoric cross sections.
Constitutive models for foams have been developed by Deshpandy and Fleck <cit.>, their model depends only on the first and second invariant of the stress tensor, which results in circular deviatoric cross sections. Öchsner <cit.> gives a comprehensive introduction to elastic-plastic mechanics of foams, whereas it is also mentioned that the yield surfaces of foams in general can depend on the three stress invariants I_1, J_2 and J_3. Anisotropic failure and yield criteria have been developed by Tsai and Wu <cit.>, Barlat <cit.>, Bilkhu <cit.> and Nusholtz <cit.>. The constitutive behavior of foams becomes more complex if the shape of the yield surface is assumed to depend on internal variables describing the deviatoric and hydrostatic deformation state. Initial and subsequent yield surfaces have been investigated by Demiray <cit.> and Storm <cit.>. A data-driven model using neural networks has been developed by Settgast et al. <cit.>. Here, the yield surface is approximated by a regression-type neural network. The training data for the network are generated by DNSs of RVEs of foam structures.
Interestingly, failure and yield surfaces of foams have similar features as those for granular media like sand, rock, or other geomechanic materials.
A very flexible yield or failure criterion for such materials has been proposed by Bigoni and Piccolroaz <cit.>. It allows various adjustments of the shapes of both hydrostatic and deviatoric yield surface cross sections. As a single surface criterion it cannot provide the necessary shape flip of the deviatoric cross-section along the hydrostatic axis. Bolchoun, Kolupeav and Altenbach <cit.> provide a comprehensive collection of yield and failure criteria for geomaterials. Especially, their so-called geometric mechanical model (GMM) can describe most of the features of interest. Jung and Diebels proposed among others the Ehlers model <cit.> as a potential candidate for yield surfaces of open cell foams. In the following, we will focus on the Ehlers yield surface, but present a modification of the model, which allows us to model the change of the orientation of the triangular cross-section in hydrostatic tension and compression.
The paper is organized as follows: First, the original Ehlers model is recalled, followed by the description of its modification. The essential parameters of the model are discussed and certain restrictions are defined, which ensure the convexity of the yield surface. It follows a section, where a general framework for a constitutive model is presented. This is mainly based on a thermodynamically consistent frame work presented in the book of de Souza Neto <cit.>. To solve the constitutive equations a general return algorithm is presented, which can be implemented in finite element codes. To apply the modified model to foam structures a parameter identification procedure is necessary. DNSs of a Wheire-Phelan foam are used to gather stress and strain data, which are used to identify parameters of the model and their dependence on the load history. To show the accuracy of the developed model predictions of the model are compared with DNSs of the corresponding foam structures. The article is closed with conclusions and an outlook for improvements and possible extensions of the model. Within an appendix, all necessary derivatives for the implementation of the model as well as for the parameter identification procedure are given.
§ MODELLING
§.§ Original Ehlers Model
The original form of the Ehlers yield surface <cit.> has the form
F = √(J_2 [1 + γJ_3/J_2^3/2]^m + 1/2α I_1^2 + δ^2 I_1^4) + β I_1 + ϵ I_1^2 - κ = 0 ,
which is formulated in the space of the three stress invariants
I_1 = ,
J_2 = 1/2 : ,
J_3 = () ,
where denotes the symmetric Cauchy stress tensor, and
= dev() = - 1/3 I_1
its deviator, with as the second-order unit tensor. In (<ref>) α, β, δ and ϵ are parameters describing the shape of the meridian cross section, γ and m parameters describing the shape the deviatoric cross-section of F. Parameter κ scales the size of the yield surface. If κ depends on the equivalent plastic strain , it also describes the isotropic hardening behavior.
The right term in the square brackets of (<ref>) is related to the Lode angle θ, as
J_3/J_2^3/2 = - 2/3√(3)sin(3θ) ,
or vice versa, the Lode angle may be expressed using
θ = 1/3sin^-1( - 3 √(3)/2J_3/J_2^3/2) .
One may note that Eq. (<ref>) delivers values for θ in the range of [-π/6,π/6], whereas for the argument in (<ref>) any scalar value for θ can be given.
§.§ Modified Ehlers Model
The modification done to the original Ehlers model realizes a smooth change of parameter γ depending on the hydrostatic stress state. In general, it is considered that all parameters can depend on internal variables, especially on , which allows to model shape changes of the yield surface during a deformation process. The modified version reads as
F = √(J_2 [1 + γ A C]^m + 1/2α I_1^2 + δ^2 I_1^4) + β I_1 + ϵ I_1^2 - κ = 0 ,
with the additional term
A = /√(3) = sin( tan^-1( I_1/√(6 J_2)) ) ,
whereas represents the normalized stress or stress direction
= / with =√( : ) .
For convenience the term
C = J_3/J_2^3/2 = 2/3√(3)sin(3θ)
is introduced, which simplifies the notation later on.
The term A changes smoothly in the range [-1,1] depending on the hydrostatic stress. The sign change of A realizes the flip of the triangular-shaped deviatoric cross-section of the yield surface with respect to the state of the hydrostatic stress, as it is observed for yield surfaces of foam structures <cit.>. F can be also understood as a yield and/or flow potential for a constitutive model.
The shape of the deviatoric cross section (I_1=const.) can be expressed as
F^dev = √(J_2 [1 + γ A C ]^m) .
If Eq. (<ref>) is solved for J_2, an expression for the shape in the hydrostatic plane can be derived, which reads as
F^hyd = √(( ϵ^2 - δ^2 ) I_1^4 + 2 β ϵ I_1^3 + ( β^2 - 1/2α - 2 ϵ κ) I_1^2 - 2 β κ I_1 + κ^2/[1 + γ A C ]^m)
Fig. <ref> shows the shape of the deviatoric cross-section of the yield surface projected onto the deviatoric plane in principal stress space for varied parameters γ and m. If γ 0 the deviatoric cross section shows a triangular shape, which becomes more pronounced if the absolute value of γ increases. The sign of γ A controls the orientation of the shape, having a sharp tip either at an angle θ of 30° or 90°. Parameter m controls the sharpness of the tips of the yield surfaces' cross-section.
Fig. <ref> shows the meridian cross-section of the yield surface from the modified model for varied values of parameter α. For smaller values of α the yield surface extends symmetrically in the hydrostatic stress direction forming an elongated ellipsoid. The dashed lines show the value of the term A=tr()/√(3) in Eq. (<ref>) depending on the hydrostatic stress state expressed by I_1. As more elongated the yield surface becomes, as less curved that dependence appears. An important feature of A is, that it a priori changes from -1 to 1 within the bounds of I_1 given by the shape of the yield surface.
In Fig. <ref> the influences of the parameters θ, α, β, δ, ϵ, and κ on the meridian cross-section shape of the yield surface are displayed. The default parameters are set as θ=0, α=1/2, β=0, γ=1, δ=0 MPa^-1, ϵ=0 MPa^-1, κ=1 MPa, and m=1.
For a negative value of θ the meridian cross section is slightly thinner for negative hydrostatic stresses. Vice versa, for positive θ values the cross section is thinner for positive hydrostatic stresses. The meridian cross-section is symmetric if θ=0. Decreasing values of α lead to an elongated yield surface along the hydrostatic axis. The parameter β shifts the cross-section along the hydrostatic axis. The parameters δ and ϵ, both change the curvature of the meridian cross section. If both parameters are zero the cross section has an elliptical form. Finally, parameter κ scales the whole yield surface in a self-similar manner. For the use as a yield potential, F is required to be strictly convex, which requires certain restrictions on the parameters <cit.>. The conditions for a convex deviatoric cross-section are
γ ≤√(27)/9 m-2 or m ≤√(27)/γ+2/9 and γ≤√(27)/2 .
A sufficient condition for the convexity in the hydrostatic plane is derived from Eq. (<ref>). Convexity in the hydrostatic plane is ensured, if the second derivative of (<ref>) with respect to I_1 is smaller or equal to zero.
F_hydI_1 ≤ 0
This condition is a priori fulfilled if the parameters α≥ 0, δ≥ 0, ϵ≥ 0 and κ≥ 0.
The complete yield surface can be projected into the principal stress space. Fig. <ref> shows the original and the modified yield surface for the same set of parameters α=10^-8, β=0, γ=2.273, δ=0.0031 MPa^-1, ϵ=0.0517 MPa^-1, κ=0.769 MPa, and m=0.389. These parameters have been identified for an artificial foam structure <cit.>.
§ A CONSTITUTIVE FRAMEWORK FOR THE INELASTIC BEHAVIOR OF POROUS STRUCTURES
§.§ Constitutive Equations
For the derivation of the constitutive equations within a thermodynamically consistent framework we follow the book of de Souza Neto <cit.>.
We consider a small deformation setting. The strain rate tensor is additively decomposed into an elastic and plastic part.
= +
A free energy function depending on the elastic strain tensor and a set of internal variables is defined, which is also split into an elastic and a plastic part.
ψ = ψ( ,)
= ψ^el( )+ψ^pl()
= ψ^el( - )+ψ^pl()
A set of internal variables is denoted by . The corresponding Clausius-Duheme inequality reads
( - ρ̅ψ^el) : + : - ·≥ 0 .
Eq. (<ref>) implies a general elastic law
= ρ̅ψ^el ,
and a general hardening thermodynamical force
= ρ̅ψ^pl .
The plastic dissipation function is derived from (<ref>) and reads
Υ^pl = : - ·≥ 0 .
The elastic contribution to the free energy (<ref>) is given by
ρ̅ψ^el = 1/2 : : ,
where denotes the isotropic forth order stiffness tensor
= 2G + 3K ,
with the shear and bulk moduli G and K, respectively. The fourth order projection tensors and are used for the composition of the stiffness tensor. Using the elastic modulus E and Poisson's ratio ν shear and bulk modulus can be expressed as
G = E/2(1+ν) and K = E/3(1-2ν) .
The general isotropic elastic law in rate form reads
= : ,
which is also known as Hooke's law.
The plastic contribution to the free energy (<ref>) is given by
ρ̅ψ^pl = F(,) ,
where F(,) is the yield potential.
The plastic flow rule and the generalized hardening law are given by
= (,)
= (,)
The tensor denotes the plastic flow direction, which is derived from a plastic flow potential G as
= G .
In case G=F the plastic flow is called associated. If the generalized hardening modulus is derived from the yield potential F
= -F ,
Eq. (<ref>) is called an associative hardening law.
The constitutive equations are completed by the loading and unloading (Karush-Kuhn-Tucker) conditions
F ≤ 0, ≥ 0, F = 0 .
§.§ General Return Algorithm
The general return algorithm considered here is adopted from de Souza Neto <cit.>. First, the general outline of the algorithm is given, followed by a detailed adaption to the specific problem considered in this paper. A time interval [t_n,t_n+1] is considered, where it is assumed that at t_n all quantities _n, _n, _n are known. For a deformation driven process, a strain increment Δ=_n+1-_n is given. To compute the values _n+1, _n+1, _n+1 at the end of the time increment we have to solve the following system of equations
_n+1 = _n + Δ - Δ(_n+1,_n+1)
_n+1 = _n + Δ(_n+1,_n+1)
for the unknowns _n+1, _n+1, and Δ, subject to the constraints (Karush-Kuhn-Tucker conditions)
Δ≥ 0, F(_n+1,_n+1) ≤ 0, Δ F(_n+1,_n+1) = 0 ,
where
_n+1 = ρ̅.ψ|_n+1, _n+1=ρ̅.ψ|_n+1 .
The increment Δ is called the incremental plastic multiplier. Once the solution _n+1 has been obtained, the plastic strain at t_n+1 can be calculated as
_n+1 = _n + Δ - Δ .
The KKT-conditions (<ref>) allow only two distinct mutually exclusive cases. If Δ=0 there is no plastic flow nor any evolution of internal variables within the interval [t_n,t_n+1], which means that the step is purely elastic. Then the solution is simply given by
_n+1 = _n + Δ ,
_n+1 = _n .
In case that the plastic multiplier Δ > 0 the solution for _n+1, _n+1, and Δ must satisfy the equations (<ref>) and (<ref>) in combination with the constraints (<ref>)_2 and (<ref>)_3, which results in the system of equations
_n+1 = _n + Δ - Δ(_n+1,_n+1)
_n+1 = _n + Δ(_n+1,_n+1)
F(_n+1,_n+1) = 0 .
To solve the system (<ref>) a fully implicit elastic predictor/corrector return mapping algorithm is employed. The elastic trial state is defined by the state variables at t_n and a given strain increment Δ.
_n+1^el trial = _n + Δ, _n+1^trial = _n
_n+1^trial = ρ̅.ψ|_n+1^trial, _n+1^trial=ρ̅.ψ|_n+1^trial
If F(_n+1^trial,_n+1^trial) ≤ 0 then the solution is the trial state ( · )_n+1 = ( · )_n+1^trial, corresponding to an elastic step with update relations (<ref>) and (<ref>).
If F(_n+1^trial,_n+1^trial) > 0 a plastic step is considered, and by putting the trial state from (<ref>) into Eqs. (<ref>) the following system of equations can be set up
{[ _n+1 - _n+1^el trial + Δ_n+1; _n+1 - _n+1^trial - Δ_n+1; Φ(_n+1,_n+1) ]}
=
{[ ; ; 0 ]} ,
which has to be solved for _n+1, _n+1, and Δ with
_n+1 = ρ̅.ψ|_n+1, _n+1=ρ̅.ψ|_n+1 .
Linearization of Eqs. (<ref>) yields
{[ d + Δ : d + Δ·d + dΔ; d - Δ·d - Δ·d - dΔ; F : d + F·d ]}
=
{[ d^el trial; ; 0 ]} ,
From the potential equations the differentials d and d can be derived (see <cit.> p. 239).
d = : d + E·d
d = F·d + G·d
The linear operators , E, F and G are defined as
= ρ̅ψ, E = ρ̅ψ, F = ρ̅ψ, G = ρ̅ψ .
Inversion of (<ref>) yields the exressions
d = : d + B·d
d = A·d + J·d .
Since the potential ψ is split additively into an elastic and a plastic part (<ref>) the tangent moduli E and F as well as A and B vanish and the Eqs. (<ref>) can be inverted into
d = : d
d = J·d ,
where =^-1 and J=G^-1.
Substituting (<ref>) and (<ref>) into (<ref>) yields the symbolic matrix representation
[ + Δ B + Δ ; A - Δ J - Δ -; F F 0 ][ d; d; dΔ ]
=
[ d^el trial; ; 0 ] .
Eq. (<ref>) represents the general case, where and can depend on and .
§.§ Application to the Modified Ehlers Model
In the special case considered here, the set of internal variables contains only the equivalent plastic strain ={} and we define that =.
The equivalent plastic strain is chosen as the norm of the plastic strain tensor since it reflects deviatoric as well as volumetric deformations.
=
If the associative flow and hardening laws are considered (see <cit.> p. 241) we have
= G = F , =F and =1 .
This implies that the partial derivatives
= and = .
Under the assumption that the material parameters are constants Eq. (<ref>) simplifies to
[ + Δ ; 0 ][ d; dΔ ]
=
[ d^el trial; 0 ] .
For the general case of non-associate plastic flow and that hardening and plastic flow direction depending on the equivalent plastic strain and that = we have
[ + Δ ; F F ][ d; dΔ ]
=
[ d^el trial; 0 ] .
To solve the equations (<ref>) or (<ref>) a Newton scheme is applied. A vector of residuals is defined as
[ ^pl_n+1; R^F_n+1 ] =
[ _n+1 - _n - Δ + Δ_n+1; F(_n+1,_n+1) ]
=
[ ; 0 ] ,
and the Newton update for (<ref>) or (<ref>) reads
[ d_n+1; dΔ_n+1 ] =
^-1_n+1[ -^pl_n+1; -R^F_n+1 ] ,
where is the left most matrix defined in (<ref>) or (<ref>).
The update of the variables and Δ is performed as
_n+1 = _n + d_n+1 ,
_n+1 =_n + d_n+1 .
The Newton iterations are performed until the residuals become
^pl_n+1 < tol and R^F_n+1 < tol .
The inverted matrix A can be written using submatrices
^-1 =
[ _11 _12; _21 _22 ] .
Using (<ref>), linearization of (<ref>) with respect to d and considering that
d_n+1 = -^-1 d_n+1
= - d_n+1 ,
one finds that
d_n+1 = _epd_n+1 with _ep=_11 ,
where _ep denotes the consistent tangent tensor.
§.§ Parameter Identification
§.§.§ Yield Surface
The parameters for the modified model are obtained by analyzing a representative volume element (RVE) of a generic foam model as it is displayed in Fig. <ref>. A finite element model of the RVE is generated. The bulk material is assumed to be elastic-plastic described by a classical von Mises model with linear hardening. To determine the effective stresses and strains a homogenization approach as described by Malik et al. <cit.> is used. The FE analysis delivers for n_l systematically varied load cases with n_i load increments the data set
𝒟^sim={^sim_(i,l)} ,
where ^sim_(i,l) denotes the homogenized stress tensor for load case l and a corresponding homogenized equivalent plastic strain . The i values for are equally spaced in the range [0 …] for every load case l. The n_l load cases are defined in such a way that the stresses ^sim_(i,1…,n_l) cover one of the six sections of the full yield surface for a corresponding equivalent plastic strain as shown on the right side in Fig. <ref>. One section is representative, due to the symmetries of the yield surface. Further details regarding the sampling strategy can be found in <cit.>. In other words, 𝒟_i,: represents a discretized yield surface for a given equivalent plastic strain. For each value of a mean square error
(_i) = 1/2 n_l𝒟^sim_i,: - 𝒟^ana_i,:(_i)^2
is defined. The stresses in the set 𝒟^ana are obtained as
^ana_(i,l)(_i)=x ^sim_(i,l) ,
and obey Eq. (<ref>) for a given parameter vector _i=[α_i,β_i,γ_i, δ_i,ϵ_i,κ_i,m_i]^T, which has the length n_p. The scaling factor x is determined numerically using a Newton iteration scheme
x_n+1 = x_n - F(x_n ^sim,_i)/xF(x_n ^sim,_i) with x_0=1 ,
until |F(x_n ^sim,_i)| ≤tol for each element in 𝒟^sim_i,:. The index i denotes the i^th yield surface for and n the index for the Newton iterations. The elements in 𝒟^sim_i,: depend on the current parameters in _i. To find an optimal parameter set _i^* for a yield surface with the constrained minimization problem
_i^* = _i[(_i)] subject to g_j(_i) ≤ 0 ; j ∈ [1,2]
has to be solved. The two inequality constraints g_j are defined by the Eqs. (<ref>) and (<ref>) and are necessary to ensure the convexity of yield surface. A sequential least squares programming (SLSQP) algorithm is used to solve the constrained minimization problem defined in Eq. (<ref>). The algorithm requires the computation of the Jacobian of the objective function (<ref>). This can be done numerically using a finite difference scheme or much more efficiently analytically using
_i = 1/n_l∑_l=1^n_l(^sim_(i,l) - ^ana_(i,l)) : ^ana_(i,l)_i ,
with
^ana_(i,l)_i =
^ana_(i,l)x * x_i ,
considering the implicit derivative
F_i + Fxx_i != 0
Fxx_i = -F_i
x_i = -[ Fx]^-1F_i ,
such that
^ana_(i,l)_i = - ^ana_(i,l)x * [ Fx]^-1F_i .
The (*) product in Eq. (<ref>) and (<ref>) denotes the outer product between a vector and a second order tensor, such that the result can be interpreted as a vector containing n_p second-order tensors. The derivatives in (<ref>) are given in the appendix.
§.§.§ Plastic Flow Direction
For the case of non-associated plastic flow, the direction of the plastic strain is derived from a potential
=G with G=√(J_2[1+γ_G A C]^m_G +12α_G I_1^2) ,
which is a reduced version of the yield function, but with specific parameters α_G, γ_G and m_G. Again, these parameters may be scalar functions depending on internal variables (e.g. ). The objective function to be minimized expresses the difference between the plastic flow directions determined by the analytical model and the numeric simulations using the FE model of the foam RVE.
_G(_Gi)=1n_l∑_n_l( 1 - ^ana_(i,l) : ^sim_(i,l)^ana_(i,l)^sim_(i,l))
The optimal parameters _Gi^* are found by solving
_Gi^* = _Gi[_G(_Gi)] .
The Jacobian of (<ref>) needed for the SLSQP minimization algorithm reads as
_Gi=-1n_l∑_n_l_Gi( ^ana_(i,l)^ana_(i,l)) .
The derivative of the normalized flow direction in the above equation is given in the appendix.
§ RESULTS, APPLICATIONS, AND DISCUSSION
The results given in this section are specific for the foam structure <cit.> shown in Fig. <ref>. The local bulk material is assumed to be an isotropic elastic-plastic material with an elastic modulus E_loc=10 GPa and a Poisson's ratio of ν_loc=0.3. The local hardening law is linear with an initial yield stress of σ_0=20 MPa and a linear hardening coefficient σ_1=10 MPa. The relative density of the foam structure is ρ̅=20% and the so-called strut shape factor is chosen as k=1.0 (see Abendroth et al. <cit.> for details).
This structure is analyzed using DNSs with systematically varied stress states, which are defined as:
= λ ,
= I/√(3)sinα + /√(2)cosα ,
= ∑_k=1^3 e_k _k with _K=_k^⊗_k^ ,
using
[ e_1; e_2; e_3 ] =
[ cosθ - sinθ/√(3); 2 sinθ/√(3); -cosθ - sinθ/√(3); ] .
In the above equations λ serves as a scaling factor, α∈ [-π/2,π/2] represents the angle between the deviatoric and hydrostatic stress direction. The eigenvalues e_k of are defined using the Lode angle θ∈ [-π/6,π/6], and the three eigentensors M_k are expressed by the eigendirections of the effective stress tensor _k^. Since it is assumed that the foam structure shows an isotropic elastic-plastic behavior, the eigendirections are chosen to be equivalent to the unit directions of Euclidian space. The angles α and θ are varied uniformly in 39 and 19 steps within the given ranges. The scaling factor λ is chosen such, that the equivalent effective strain takes 50 equidistant levels between 0.5 and 25%. In Fig. <ref> stress data are depicted exemplary for =0.1 and =0.2 as black dots superimposed to the corresponding yield surfaces.
Using that stress data for each level of equivalent effective strain a set of material parameters for the modified model is identified using the parameter identification procedure described in section <ref>. Since a small deformation setting and an isotropic local material model is considered, the effective yield surface is always point symmetric with respect to the origin, which implies that the parameter β=0. In the first attempt, all model parameters except β have been identified. The values of these parameters are plotted over the effective equivalent plastic strain on the left-hand side in Fig. <ref>.
Despite the fluctuations for α and ϵ it is observed that parameter δ is very close to zero and that γ and m, which are the parameters defining the shape of the deviatoric cross-section, do not change significantly for increasing . The overall mean square error () has its maximum of 1.956·10^-4 MPa^2 at the lowest equivalent plastic strain value. That means, that the average error is about 0.014 MPa, or less then 1% of typical stress values, which can be interpreted as an excellent agreement between the analytical model and generic experiments. In a second parameter identification approach the parameters β=0, δ=0, and m=0.8 remain fixed during optimization (see Fig. <ref> right). The remaining parameters show an almost identical behavior as in the first attempt and even the mean square error does not change significantly. A further reduction of the number of variable parameters leads to the results shown in Fig. <ref>.
On the left-hand side of Fig. <ref> the parameter values for γ, ϵ, and κ depending on are plotted, and it is observed that also γ does not change significantly. That, finally leads to the case where only the two parameters ϵ and κ are used to describe the change of the yield surface shape during a plastic deformation process with very high accuracy as shown on the right panel in Fig. <ref>.
The dependency of the remaining two parameters ϵ and κ on can be described analytically using
ϵ = ϵ̂() = ϵ_0 + ϵ_1 exp( -ϵ_2 ) ,
κ = κ̂() = κ_0 + κ_1 + κ_2 [ 1 - exp( -κ_3 ) ] + κ_4 [ 1 - exp( -κ_5 ) ] ,
with
ϵ_0=0.02333 MPa^-1, ϵ_1=0.02818 MPa^-1, ϵ_2=2.841, κ_0=0.7022 MPa, κ_1=0.6243 MPa, κ_2=0.0752 MPa, κ_3=150.1, κ_4=0.08512 MPa, κ_5=11.96. The corresponding fixed parameters are α=7.068·10^-4, β=0, γ=0.9981, δ=0, and m=0.8
The dependencies of three dimensionless parameters of the flow potential G on the equivalent plastic strain are depicted in Fig. <ref>. Also here, the parameters can be expressed by scalar functions.
α_G = α̂_̂Ĝ() = α_G0 + α_G1 + α_G2[ 1 - exp( -α_G3) ]
γ_G = γ̂_̂Ĝ() = γ_G0 + γ_G1 + γ_G2[ 1 - exp( -γ_G3) ]
m_G = m̂_̂Ĝ() = m_G0 + m_G1 + m_G2[ 1 - exp( -m_G3) ]
The corresponding parameters are α_G0=0.08002, α_G1=0.0, α_G2=0.02243, α_G3=8.065, γ_G0=0.5049, γ_G1=-0.7625, γ_G2=-0.2236, γ_G3=0.01241, m_G0=2.036, m_G1=0.1791, m_G2=-0.1561, and m_G3=4.886. The mean square error reduces with increasing from 0.01995 down to 0.005594.
The above-identified parameters have been used to predict the stress-strain response for various load cases. In Fig. <ref> the predictions of the model for uniaxial positive and negative load cases are compared with the results from direct numerical simulations of the Wheire-Phelan RVE shown in Fig. <ref>. Here, an excellent agreement between model prediction and data from RVE simulations is observed.
In Fig. <ref> left) A deviatoric loading case is investigated. For the given strain state it is observed that σ_11 reaches a peak at the end of the elastic region followed by a short stress drop before the usual strain hardening starts, which is also reflected by the material model. The further evolution of stresses is slightly more deviating than for the uniaxial load cases but still accurate enough for most engineering applications.
As a final comparison, a cyclic load with an increasing strain amplitude in each half cycle is investigated and visualized in Fig. <ref> right). Here, the stress-strain curve of the developed material model deviates increasingly from the data with each cycle. Since the model considers only isotropic hardening effects, possible kinematic hardening of the RVE is not reflected. Nevertheless, the model is able to approximate the general material behavior with reasonable accuracy.
§ SUMMARY, DISCUSSION, AND OUTLOOK
A new constitutive model for the elastic-plastic behavior of foams or other porous structures has been presented. The model is a modification of Ehlers' model <cit.> and can especially describe the change of the orientation of the triangular-shaped deviatoric yield surface cross-section, depending on the hydrostatic stress state. The model is formulated using a consistent thermodynamic framework by de Souza Neto <cit.>. A general return algorithm is used to solve the constitutive equations. A comprehensive appendix contains all necessary partial derivatives needed for the implementation of the model into finite element codes and the application of the parameter identification procedure.
The model parameters are identified using a constraint parameter identification procedure, where data from direct numerical simulations of a representative volume element of a generic foam model are used as a substitution for experimental data. The parameter identification procedure is not generally restricted to generic data, sufficient experimental data can also be used. Care must be taken for the choice of the parameters since their exist certain constraints to ensure the overall convexity of the yield surface.
The predictions of the model are compared with data generated by DNS of the generic foam model and show an excellent agreement. The authors believe that this model is capable to describe a wide range of foam structures. The structure of the thermodynamical framework allows certain extensions such as the application for finite deformations and the use of additional internal variables. Also, other failure criteria like the Bigoni-Piccolroaz criteria <cit.> or the GMM criteria described in Bolchoun et al. <cit.> could be implemented as yield surfaces or flow potentials using the same framework.
The current version of the model is implemented in a small deformation setting. It does not allow the simulation of foam compaction processes, where different elements of the mesostructure come into contact. Also, any anisotropic properties that can be observed in some real foams are not considered here. But, these deficiencies could be eliminated in future versions, on which the authors are currently working on.
plain
§ APPENDIX
This appendix presents derivatives of the yield function and/or the flow potential, which are necessary for the implementation of the model into a finite element code. Furthermore, the derivatives necessary for the parameter identification procedure are given.
§.§ Necessary derivatives for the implementation of the model
§.§.§ Derivatives of the yield function
We start with subsequent substitutions of terms in Eq. (<ref>).
A = /√(3)
B = J_3/J_2^3/2
C = A · B
D = (1 + γ C)^m
E = J_2 D
W = √(E+1/2α I_1^2 + δ^2 I_1^4)
Therewith, Eq. (<ref>) can be expressed as
F = W + β I_1 + ϵ I_1^2 - k .
The first derivatives of the stress invariants with respect to the symmetric Cauchy stress tensor reads
I_1= , J_2= , and J_3= - 2/3J_2 .
In Eqs. (<ref>)2,3 the symbol denotes the deviator of the symmetric Cauchy stress tensor.
= - 1/3 = :
The derivatives of the powers of the first invariants in (<ref>) and (<ref>) are given by
I_1^2=2I_1 and I_1^4=4I_1^3 .
Using (<ref>) and (<ref>) the derivatives of (<ref>) – (<ref>) with respect to become
A = ( ^2 - I_1 )/√(3)^3
B = 2 J_2 J_3 - 3 J_3 J_2/2 J_2^5/3
C = A B + B A
D = γ m C(1 + γ C)^m-1
E = J_2 D + D J_2
W = E + α I_1 + 4 δ^2 I_1^3 /2 W .
Having these derivatives one finally gets
F = W + β + 2 ϵ I_1 .
If a Newton algorithm is considered for solving the constitutive equations the second derivatives of F and G with respect to are required. Here, it is important to keep in mind that is symmetric. The resulting following derivatives are symmetric fourth-order tensors. Several forth order unit tensors are needed for the formulation of the second derivatives. The general forth order unit tensor and its transpose can be derived from an unsymmetric tensor .
= , =^T
The symmetric fourth-order unit tensor is therewith defined as
= ^sym
= 1/2(+^T)
= 1/2 (+)
and reads in index notation as
_ijkl = 1/2(δ_ikδ_jl + δ_ilδ_jk) .
The volumetric fourth-order tensor is defined as
= ^vol=1/3⊗
and reads in index notation
_ijkl = 1/3(δ_ijδ_kl) .
The deviatoric forth order unit tensor is finally defined as
=^dev=- .
With the fourth-order unit tensors at hand, the second derivatives for the stress invariants can be formulated as follows.
I_1 =
J_2 = =
J_3 = ( - 2/3J_2)= + - 2/3⊗
The term + in (<ref>) reads in index notation _imkl s_jm + s_im_mjkl. The following second derivatives are needed to formulate the second derivative of the yield potential.
A =
^3(2 ⊗
- ⊗
- I_1 )
- (^2
- I_1 ) ⊗ 3 /√(3)^6
B =
J_3/J_2^3/2
- 3/2J_3⊗J_2/J_2^5/2
- 3/2J_2⊗J_3/J_2^5/2
- 3/2 J_3 J_2/J_2^5/2
+ 15/4 J_3 J_2⊗J_2/J_2^7/2
C =
A⊗B
+ B A
+ B⊗A
+ A B
D =
γ m ( 1 + γ C )^m-2[
( 1 + γ C ) C
+ γ( m - 1 ) C⊗C]
E =
D⊗J_2
+ J_2 D
+ J_2⊗D
+ D J_2
W =
2 (f+g+h) (f”+g”+h”) - (f'+g'+h') ⊗(f'+g'+h')/4 ( f+g+h )^3/2
The symbols in (<ref>) are placeholders for the following terms.
f = E, f'=E, f”= E
g = 1/2α I_1^2, g'=α I_1 , g”=α⊗
h = δ^2 I_1^4, h'=4 δ^2 I_1^3 , h”=12 δ^2 I_1^2 ⊗
Finally, the second derivative of the yield potential with respect to the Cauchy stress tensor reads
F = W + 2 ϵ⊗ .
In case that the parameters α, β, γ, δ, ϵ, κ and m are functions of the derivative
F = Fαα
+ Fββ
+ Fγγ
+ Fδδ
+ Fϵϵ
+ Fκκ
+ Fmm
is needed, with
Fα = I_1^24 W ,
Fβ = I_1 ,
Fγ = C J_2 m ( 1 + γ C )^m-1/2 W ,
Fδ = I_1^4 δ/W ,
Fϵ = I_1^2 ,
Fκ = -1 ,
Fm = J_2 ( 1 + γ C )^m log( 1+γ C )/2 W .
In Eq. (<ref>) the mixed derivatives of F with respect to and are required. Here, we assume that all parameters depend on .
F = W + β + 2 ϵ I_1
W = g f' - g' f/2 g^2
The symbols in Eq. (<ref>) are placeholders for the following terms:
f = E + α I_1 + 4 δ^2 I_1^3
f' = E + α I_1 + 8 δδ I_1^3
g = W
g' = W
The mixed derivatives for E and D read as
E = J_2 D + DJ_2 ,
D = Dγγ + Dmm ,
with
Dγ = m C( 1 + γ C )^m-2( 1 + γ m C ) ,
Dm = γC( 1 + γ C )^m-1(m log( 1 + γ C ) + 1 ) .
In the above equations the derivatives of W, E, and D with respect to are required and given as follows:
W = E + 12α I_1^2 + 2 δδ2 W ,
E = J_2 D ,
D = Dγγ + Dmm ,
with
Dγ = m C ( 1 + γ C )^m-1 ,
Dm = ( 1 + γ C )^m log( 1 + γ C ) .
§.§.§ Derivatives of the flow potential
For a non-associative flow rule the data-driven approach sketched in section <ref> generates a flow potential in such a way that its derivatives with respect to the stress tensor, which is equivalent to the plastic flow direction, can be arbitrarily scaled. Therefore, it is considered that the flow direction is normalized using
= G^-1G .
The derivatives of the normalized flow direction with respect to the stress tensor and the equivalent plastic strain , which are necessary in Eq. (<ref>) are then defined as
= G⊗( G^-1) + G^-1G ,
= G( G^-1) + G^-1G ,
with
( G^-1) = G^-3G : G ,
( G^-1) = G^-3G : G .
The parameters α, β, γ, δ, ϵ, m, and κ can be expressed by scalar functions α̂(), β̂(), γ̂(), δ̂(), ϵ̂(), m̂(), and κ̂(). These functions and their corresponding derivatives with respect to are specific for the porous structure to be modeled and need to be defined individually.
§.§ Derivatives necessary for the parameter identification procedure
§.§.§ For the yield function
For the parameter identification procedure the necessary derivatives for the Jacobian (<ref>) are
F = Fα + Fβ + Fγ + Fδ + Fϵ + Fκ + Fm
and
Fx = Wx + βI_1x + ϵI_1^2x ,
with
Wx = Ex + 12αI_1^2x + δ^2 I_1^4x/2 W ,
Ex = J_2x D ,
and
I_1x = I_1 , I_1^2x = 2 x I_1^2 , I_1^4x = 4 x^3 I_1^4 , J_2x = 2 x J_2 .
For the Eqs. (<ref>) – (<ref>) it is assumed that the usual stress argument is given by its scaled version x.
§.§.§ For the plastic flow direction
The derivative of the normalized plastic flow direction in (<ref>) is given as
_Gi( ) = _Gi^2 - * _Gi : ^3 ,
with
_Gi = G_Gi = [ Gα_Gi, Gγ_Gi, Gm_Gi]^T ,
whereas
Gα_Gi = I_1 G - Gα_Gi( E + α_Gi I_1 )2 G^2 ,
Gγ_Gi = Eγ_Gi G - Gα_Gi( E + α_Gi I_1 )2 G^2 ,
Gm_Gi = Em_Gi G - Gα_Gi( E + α_Gi I_1 )2 G^2 ,
and
Eγ_Gi = J_2 Dγ_Gi + Dγ_GiJ_2 ,
Em_Gi = J_2 Dm_Gi + Dm_GiJ_2 .
The derivatives Dγ_Gi and Dm_Gi are already defined in the Eqs. (<ref>) and (<ref>), whereas the parameters γ and m are to replaced by γ_Gi and m_Gi.
Therewith remain
Gα_Gi = I_1^24 G , Gγ_Gi = 12 GEγ_Gi , Gm_Gi = 12 GEm_Gi ,
and
Eγ_Gi = J_2 Dγ_Gi , Em_Gi = J_2 Dm_Gi ,
using Dγ_Gi and Dm_Gi as defined in the Eqs. (<ref>) and (<ref>).
|
http://arxiv.org/abs/2307.04219v1 | 20230709162247 | Large Satellite Constellations and Their Potential Impact on VGOS Operations | [
"Federico Di Vruno",
"Vincenza Tornatore"
] | astro-ph.IM | [
"astro-ph.IM"
] |
Derandomizing Codes for the Binary Adversarial Wiretap Channel of Type II
This work is supported in part by the U.S. National Science Foundation under grants CNS-2128448, CNS-2212565, CNS-2225577, EEC-1941529, ITE-2226447 and by the Office of Naval Research under grant ONR N000142112472.
Eric Ruzomberka1, Homa Nikbakht1, Christopher G. Brinton2, David J. Love2 and H. Vincent Poor1
1Princeton University 2Purdue University
August 12, 2023
==================================================================================================================================================================================================================================================================================================
Large LEO satellite constellations (or so-called Mega-constellations) will significantly change the view of the sky in some radio frequency bands. For VGOS telescopes it is important to understand the potential impact these constellations will have in their operations, what is the risk of its receivers going into non-linear behaviour and how much additional power would a telescope receive if observing in the same frequencies where satellites are transmitting. This work describes three of these new constellations (as they would look fully deployed) and summarizes the results of a particular study considering two VGOS telescopes (Onsala and Wettzell).
§ INTRODUCTION
The industrialization of spacecraft construction, and the lowering in costs of space launches has paved the way
for big plans in Low Earth Orbit (LEO). Large satellite constellations like Starlink phase 1(with 4400 satellites) and OneWeb phase 1 (with 648 satellites)
are already in the deployment phase, others like Project Kuiper (from Amazon) or Guowang (from China) are in their development phase and others with even larger numbers
are being filed into the International Telecommunication Union (ITU) system (see Table <ref>). With altitudes between 500 km and 1200 km, these new constellations will surround the planet almost homogeneously.
From a radio telescope point of view, the situation in the sky will change considerably. This change is already evident in the number of active satellites in LEO, from about 2000 in 2018, to more
than 5000 in 2022, and the trend suggests it may reach hundred of thousands in this decade <cit.>.
Until now, most of the satellites for internet communication
were located in the geostationary belt (at approximately 35780 km altitude), appearing fixed in the sky for a terrestrial observer <cit.>. The new LEO satellites will orbit the Earth with a period
of about 90 minutes and will be seen as hundreds to thousands of bright and fast-moving radio sources in the sky with downlinks in frequency bands from 10.7 GHz up to 76 GHz (see Section <ref>).
Contrary to the situation with terrestrial radio frequency interference (RFI), it is not possible to build radio telescopes far away from satellite transmissions [1],
the challenge is further increased due to the opposite pointing direction of radio telescopes and user downlink antenna beams.
The typical power flux density (PFD) of satellite constellations is in the order of -146 dBW/m^2 (<cit.>, <cit.>) in 4kHz or an equivalent to 62*10^6 Jy,
i.e. more than 7 orders of magnitude brighter than a typical VGOS source <cit.>. These strong signals will require a radio astronomy receiver to have a
large dynamic range to accommodate the RFI and still be able to detect faint cosmic sources in other frequency channels within the receiver band.
This is normally possible for modern radio astronomy receivers, but it can be different in some particular situations such as total power bolometric receivers or receivers with a low effective number of bits (ENB) <cit.>.
§ LARGE LEO CONSTELLATIONS
Radio astronomy has been dealing with satellite transmissions since the very first satellites were launched back in the 1960s. Implementing different strategies such as using analog receivers with large dynamic ranges, smart scheduling, and RFI flagging among others, radio telescopes have been more or less able to mitigate (or avoid) the effect of these strong radio transmissions towards Earth <cit.>. In conjunction with these strategies, spectrum management has also played a key role in dealing with the effects of satellites, several radio astronomy groups have worked at national, regional and international level for the protection of the radio astronomy service (RAS) frequency bands allocated by the International Telecommunication Union (ITU). Some with successful results, like the GLONASS example, and sometimes with battles that still ongoing 20 years after satellite deployment like in the IRIDIUM case <cit.>.
The exponential growth in the number of active satellites in Low Earth Orbit <cit.> could result in more than 2000 satellites above the local horizon at any moment in time. Radio telescopes are sensitive to any transmitter in line of sight through its main beam or antenna sidelobes.
§.§ Walker-Delta constellations
All these new constellations follow a "Walker Delta" type of distribution, composed of orbital shells at a certain altitude, each shell contains several orbital planes, with a certain inclination with respect to the Equator and distributed homogeneously in the 360 degrees of
right ascension. Each one of the constellation's planes contains N satellites, a representation of Starlink Phase 2 can be found in Figure <ref>.
A shell of a Walker-Delta constellation <cit.> is described by i = t/p/f where i is the inclination, t is the total number of satellites, p is the number of equally spaced planes, and f is the relative spacing between satellites in adjacent planes. This description makes it very simple to simulate any of these constellations with the purpose of studying its geometric distribution
in LEO and also its effect on radio telescopes. It is also possible to use existing Two-Line Elements (TLEs) to obtain the approximate position of existing
satellites in space, which can be useful to compare observations to simulation.
Figure <ref> shows a qualitative view of the sky from the Wettzell VGOS station (lat 49 degrees), with the position of different satellite constellations simulated for 100 seconds. It is simple to see how the density of satellites in the sky will drastically change in the near future if all constellations planned are deployed.
§.§ Radio frequencies
Satellite constellations transmit their downlink signals in frequencies allocated to the Fixed Satellite Service (FSS). Table <ref> contains some of the currently in-use and planned FSS bands and it is important to note the proximity to some ITU protected RAS bands immediately adjacent
or in very close proximity.
The close vicinity of the satellite's downlinks to radio astronomy bands is a matter of concern for radio astronomers and spectrum managers. As an example, the protection of the 10.6-10.7 GHz Radio Astronomy Service (RAS) band, which includes a passive band in 10.68-10.7 GHz protected by the footnote RR No. 5.340 in the ITU-R Radio Regulations (RR), was studied for the Starlink Ph1 and OneWeb ph1 constellations in <cit.>, with the conclusion that both systems should not use the first 250 MHz channel to protect the RAS band. These signals can not only impact sensitive observations in the RAS protected bands, but can also affect wideband receivers which include the frequency range of user downlinks. Such wideband receivers (from 1 to 14 GHz in the case of VGOS) are necessary to conduct cutting edge science or Geodesy <cit.>.
This paper focuses on the downlink frequency range 10.7 to 12.75 GHz where both OneWeb and Starlink have divided the band in 8 channels of 250 MHz each. The study can be replicated for higher frequency bands with the appropriate modification of satellite and telescope characteristics.
§ POTENTIAL IMPACT ON VGOS
By using large reflector antennas pointed towards the sky and wideband receivers covering the frequency range 1 to 14 GHz <cit.>,
VGOS telescopes can be impacted by downlinks of the large satellite constellations in different ways. In fact the VGOS bandwidth is wide while the protected Radio astronomy band is very narrow in and Starlink and OneWeb frequencies use a considerable portion of spectrum. The severity of this
impact depends on the interaction between the radio telescope beam and the satellite downlink beams. One of the most important aspects is how much
a correlated baseline can be affected, as the primary product of a VGOS observation. Nevertheless, the multi-dimensionality of this problem requires an analysis of the
complete signal reception mechanisms and how each part of the signal chain may be impacted.
In a typical VGOS schedule, targets are observed with durations in the order of seconds to tens of seconds, the position of the target in the local sky and the density of satellites deployed will define how much interference will be seen by the telescope. The instantaneous received power from all satellites above the horizon
may saturate the analog signal chain (low noise amplifiers, mixers, etc), causing non-linearities that would render the complete receiver band unusable, even if the
digitizer band is tuned to a completely different frequency than the satellite downlinks channels. If the RFI power is not as strong and the analog signal chain remains linear, then there
can be two possible scenarios:
* First scenario: when the observed band is outside of the satellite downlink frequency range, in which case out of band emissions from the satellites could be
a problem depending on their level. This work is not focusing on this, but <cit.> has studied that case.
* Second scenario: if the observing band falls within one satellite downlink band (250 MHz channels) or vice versa, strong RFI will be received by the VGOS antenna. This RFI can potentially be mitigated by correlation as long as the number of bits in the digitizer are enough to correctly digitize the signal. Since a VGOS digitizer has only two bits, the total integrated
RFI needs to be lower (practically at least 10 dB lower or 1/10) than the integrated noise power of the receiver <cit.>.
Non-linearities and lack of headroom for RFI are transient phenomena and can be considered in terms of a data-loss associated with the moments where one satellite is going through the main beam of the radio telescope. The issue of out of band emission is related to long integrations
and needs a comparison between the level of integrated RFI vs the integrated level of the astronomical source under observation. The following section describes a simulation method and presents a particular case for the Starlink phase 1, OneWeb phase 1 and Starlink phase 2 constellations to estimate data loss due to strong received power and the total aggregated RFI, the effects of the correlation is not included in this work as is currently under study by the authors.
§ SIMULATION METHODOLOGY
The simulation is based on the Equivalent Power Flux Density (epfd) concept (see <cit.>), where the satellite constellation is propagated for a defined time duration, obtaining the coordinates and attitude of every satellite for each time step. Then, the telescope antenna is pointed towards a defined sky-cell in azimuth and elevation and for each of the simulated time steps, the received power from all satellites above the horizon is calculated with the formula:
P_rx_(t,p)=∑_i=0^N_sat(PFD_sat_(i,t) * A_eff_RAS_(i,t,p))
where:
t = time step
p = pointing direction
i = satellite index
PFD_sat = Satellite power flux density in W/m^2 towards the telescope location
A_eff_RAS = Effective area of the telescope antenna in m^2 towards the satellite position
This calculation is iterated for a number of trials (typically hundreds to thousands), where each try has a random start time of the constellation and therefore contributes to a statistically representative result.
In situations where multiple frequencies are
calculated, like for example the case of OneWeb with its 16 fixed-beams antenna (see Figure <ref>), the number of channels is added to the result.
Therefore the final calculation results in a data cube with four dimensions, namely number of iterations,
number of pointing directions, number of time steps, and number of channels:
N_iters, N_pointing, N_time and N_channel.
Although the original epfd calculation as defined by the ITU uses telescope pointings in local coordinates (Alt,Az), this work considers pointings in celestial coordinates (Ra,Dec) as this allows to understand how celestial positions in different declinations can be impacted by satellite constellations transmissions.
§.§ Satellite position propagation
Using the Python package Cysgp4 <cit.> and the Astropy Coordinates package <cit.>, the position of the satellites in horizontal coordinates (Alt,Az) and Sky coordinates (Ra,Dec) are calculated for each timestep and each iteration (see Figure <ref>).
§.§ Satellite power flux density (PFD)
The PFD from each satellite in a constellation is modelled based on publicly available information (ITU documents and FCC filings). To calculate the power flux density towards the telescope site, the coordinates of the telescope in the satellite reference
frame are also calculated using the Python package cysgp4 <cit.>.
OneWeb satellites are modelled based on the information available in the ECC report 271 <cit.>,
with 8 channels in the 10.7-12.75 GHz. A fixed beam antenna pattern, like the OneWeb system, makes it simpler to
calculate the received power in a deterministic way.
The PFD from Starlink satellites is more complex to model since they
have an antenna array that can produce, and electronically steer, several beams in one or multiple frequency channels. The mean PFD from a Starlink satellite is modelled as a function of the elevation of the satellite, obtained from a
Monte Carlo simulation in where the steering angle, the number of beams and the position of satellite and observer was varied a large number of times. Starlink satellites are modeled as one
frequency channel at a time.
§.§ Radio Telescope antenna
The radio telescope antenna is modelled based on <cit.>. While this model is not a real measurement of the antenna pattern of a radio telescope, it is based on real measurements and is considered
as a worst case for compatibility studies. To obtain the gain towards
the satellite, the angle between the pointing direction and the position of the satellite is calculated.
The Effective Area of the antenna is calculated with the following equation:
A_eff = G_RAS*(λ^2/(4*π))
§.§ Correlation
Interferometry can greatly mitigate the effects of RFI, especially when the baselines are large like in the case of VLBI <cit.>. Although Thompson and others have studied the effect that long baselines have over single RFI transmitters (and stationary), the situation is not the same when potentially hundreds of transmitters using the same frequency and bandwidth are received simultaneously as can happen now.
For example in <cit.>, Petrachenko identifies the 10.7-12.75 GHz range as a usable frequency range as only Geostationary satellites were using that frequency at that time.
Now the received RFI signal at one antenna will be the sum of the signals from all satellites above the horizon (of course with different levels of attenuation). This analysis is deferred to a further update of this work.
§.§ Saturation Limit threshold
Digital processing operations in a radio telescope can be applied as long as the analog and digital signal chains behave in a linear manner;
strong enough signals will generate non-linearities corrupting the complete receiver band for the duration of the interference. Defining the level where
a receiver goes non-linear is not a simple task and will depend on each particular receiver. In the case of the VGOS receivers a conservative
value for total power of -50 dBm is considered to keep the analog signal chain within the linear regime.
If the received power is below this linearity threshold, the analog signal can then be correctly digitized with a bandwidth of 1 GHz. Two scenarios can be identified:
* Digitizing a frequency range outside of the 10.7-12.75 GHz, which should not have any complications since the signal chain behaves in a linear way and therefore this case will not be further studied;
* Digitizing in a frequency range within the 10.7-12.75 GHz. In this case is interesting to understand when the RFI produces a significant amount of power compared to the RMS noise of the receiver.
Given the distinct characteristic of VGOS systems using a 2 bit correlator, it is reasonable to consider that there is not much headroom in the digital
signal chain to accommodate for RFI, this work considers that any signal above or equal to the receiver's noise power will result in a data loss. This defines the second threshold as a spectral power flux density equal to the RMS noise of a 20 K receiver system (-215 dBW/Hz).
These two thresholds are used in the simulation; a first set of flags is produced when the total integrated power (considering the 8 channels of 250 MHz
for each constellation) is higher than -50 dBm (representing a total data loss) and the second one representing a data loss in the case of observing in the same frequency range as the satellite transmissions.
After these two flagging stages, low level RFI will still be present, it is of interest to understand how this will affect the correlation of the baseline. This will be further study in a future update to this work and compared to the thresholds defined in RA.769 <cit.>.
§.§ Metrics
Based on the threshold limits defined in the previous section, the following metrics are used:
* Full Band Data Loss (FBDL): percentage of time that the complete band is lost due to very strong RFI, where the total received power is >-50 dBm;
* Digitizer Data Loss (DDL): Percentage of the total observation time (single run multiplied by the number of iterations) that the instantaneous power
spectral density is above 10% of the integrated noise power of the receiver. This can be calculated as a function of the declination of the source;
* Average Equivalent Spectral Power Flux Density (aESPFD): average value of the equivalent Spectral Power Flux Density during the observation
time in each antenna. The eSPFD is calculated as the received spectral power flux density [W/m^2/Hz] divided by the maximum effective antenna area, and it is useful to compare to the SPFD (in units of Jy) of a celestial source
in the main beam of the antenna;
§ CASE STUDY SIMULATION
A specific study case was selected to understand the impact from several satellite constellations on two telescopes normally involved in VGOS observations, it is the intent to further expand this work into how correlation over the long baseline mitigates the RFI.
The VGOS stations in Sweden (Onsala Observatory) and Germany (Wettzell Observatory) were selected as the test stations, using the parameters in Table <ref>, and Starlink phase 1, OneWeb phase 1 and Starlink phase 2 as constellations see Table <ref>. The simulated observations were runned for 100 seconds in 1 second timesteps with 100 iterations.
Originally it was intended to use a real VGOS schedule, using real Ra, Dec of sources observed, but to get a more representative results of the impact as a
function of source declination the number of sources was increased artificially to 277 in a random fashion, see Figure <ref> for a plot of the sources distribution. Figure <ref> shows the view of the local sky in (Alt,Az) and how the celestial sources and the satellite constellation (in this case Starlink Phase1) move across the sky in that timeframe.
§ RESULTS
The results for each one of the selected metrics is summarized here for each constellation simulated.
§.§ Full Band Data Loss (FBDL)
Notably, the analog saturation threshold was not reached due to the combination of maximum PFD from the satellites (-98 dBW/m^2 in 250 MHz) and
maximum effective area of the VGOS antennas (106 m^2 or 20.3 dBm^2), as can be seen in Figure <ref>.
This shows that even with large constellations such as Starlink phase 2 the analog receivers would still behave in a linear fashion.
§.§ Digital Data Loss (DDL)
When considering an observation coinciding in frequency with the downlinks of satellites (i.e. in within the 10.7-12.75 GHz) the DDL varies as a function of declination of the observed source and observatory latitude. This effect is attributable to the different structures of each constellation's density of satellites around the Earth and the latitude of the observer. This shows that impact to VGOS stations (and radio telescopes in general) will strongly depend on the observatory latitude. See Figure <ref>.
§.§ Average Equivalent Spectral Power Flux Density (aESPFD)
After a certain percentage of the observed data was lost as DDL (see section <ref>, the aESPFD is calculated for each constellation as a function of declination. In this case the flagged percentage is calculated as the product of the flags from the previous section for each antenna.
Considering that the ITU-R RA.769 thresholds for harmful interference for VLBI are defined as -193 dBW/m^2/Hz, representing an ESPDF of 250 Jy in an antenna of 13 m diameter, the results show that VGOS observations could in principle be conducted inside the satellite downlink bands (considering the percentage of data lost). See Figure <ref>.
§ CONCLUSIONS
This paper proposed metrics to evaluate the impact of large satellite constellations on VGOS operations by a simil-epfd simulation for Starlink ph1 and ph2, and OneWeb ph1, and two European stations as receivers.
Through calculations and simulations it was proved that the maximum received power even in beam-to-beam coupling condition with satellites will not be enough to saturate the analog chain of a VGOS receiver.
As for the digitized part, the simulations show that observations in the same band as the downlinks from satellites can have a significant percentage of data loss due to strong signals compared to the thermal noise of the receiver. Nevertheless the results shows that the ESPFD for both antennas and all constellations is lower than the thresholds defined by ITU-R for VLBI. Observations outside of the satellite downlink bands should not be impacted by satellite downliks in this frequency range.
As further work the authors will continue investigating how correlation can help mitigate this signals from satellite constellations and how the aggregation of all constellations scales the impact.
§ ACKNOWLEDGEMENTS
The authors would like to thank the IVS Coordinating Center at NASA Goddard Space Flight Center (GSFC) for taking the archive of IVS sessions. The schedule used in this work is available at the https://ivscc.gsfc.nasa.gov/sessions/2022/vo2027 web page. We are grateful to Salvo Buttaccio, for the assistance with the VGOS schedule, to Dr. Benjamin Winkel for assistance with the use of the Cysgp4 Python package, and to Dr. Jose Antonio Lopez-Perez and Dr. Hayo Hase for useful discussions about VGOS receivers and operations.
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B. Petrachenko, "The Impact of Radio Frequency Interference (RFI) on VLBI2010"
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RECOMMENDATION ITU-R RA.769 "Protection criteria used for radio astronomical measurements"
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|
http://arxiv.org/abs/2307.05635v1 | 20230711083050 | Fundamental limits of overparametrized shallow neural networks for supervised learning | [
"Francesco Camilli",
"Daria Tieplova",
"Jean Barbier"
] | cs.LG | [
"cs.LG",
"cond-mat.dis-nn",
"cond-mat.stat-mech",
"cs.IT",
"math.IT",
"math.ST",
"stat.TH",
"68Txx, 68T07"
] |
Tree-Based Scenario Classification: A Formal Framework for Coverage Analysis on Test Drives of Autonomous Vehicles
Till Schallau1 0000-0002-1769-3486, Stefan Naujokat1 0000-0002-6265-6641, Fiona Kullmann1 0000-0001-5858-0659, and Falk Howar12 0000-0002-9524-4459
1TU Dortmund University, Dortmund, Germany
2Fraunhofer ISST, Dortmund, Germany
{till.schallau, stefan.naujokat, fiona.kullmann, falk.howar}@tu-dortmund.de
August 12, 2023
============================================================================================================================================================================================================================================================================================================================
< fcamilli,dtieplov,[email protected]>
We carry out an information-theoretical analysis of a two-layer neural network trained from input-output pairs generated by a teacher network with matching architecture, in overparametrized regimes. Our results come in the form of bounds relating i) the mutual information between training data and network weights, or ii) the Bayes-optimal generalization error, to the same quantities but for a simpler (generalized) linear model for which explicit expressions are rigorously known. Our bounds, which are expressed in terms of the number of training samples, input dimension and number of hidden units, thus yield fundamental performance limits for any neural network (and actually any learning procedure) trained from limited data generated according to our two-layer teacher neural network model. The proof relies on rigorous tools from spin glasses and is guided by “Gaussian equivalence principles” lying at the core of numerous recent analyses of neural networks. With respect to the existing literature, which is either non-rigorous or restricted to the case of the learning of the readout weights only, our results are information-theoretic (i.e. are not specific to any learning algorithm) and, importantly, cover a setting where all the network parameters are trained.
§ INTRODUCTION
Artificial neural networks (NNs) are universal approximators <cit.> with remarkable abilities for supervised learning tasks such as regression or classification. In particular, modern deep neural networks, originally inspired by multilayer perceptrons <cit.>, achieve exceptional performance in image classification or speech recognition <cit.> just to name a few examples. However, despite the important activity revolving around them, their theoretical understanding remains rather poor.
One reason for the lack of strong theoretical guarantees for realistic NN models is related to the complex interplay between at least three aspects, whose individual effects are hard to single out: their architecture, the structure inherent to the data sets on which they are trained, as well as the algorithms and optimization procedures used to do so. It is therefore of crucial interest to tackle well defined theoretical models which are rich enough to capture some of the features of real NNs while remaining theoretically tractable. In this work, we propose to analyse a teacher-student set-up from a Bayesian-optimal perspective, with random input data and dependent responses generated according to a rule based on a teacher NN. This setting has the advantage to disentangle the aforementioned three components of NN learning by allowing us to mostly focus on how the architecture of the NN used for learning (and data generation), and how the amount of accessible data, influence the prediction performance. More precisely, we are going to show that when learning a complex rule linking unstructured inputs to responses in the information-theoretic optimal way, and this in an overparametrized regime, then an explicit characterization of the prediction capabilities of the NN is possible. Our results being of an information-theoretic nature, they will not depend on a specific learning procedure. Moreover because the inputs will be structure-less, the conclusions drawn will essentially capture architecture-dependent features of the learning; in the present case, the effect of overparametrization.
A key challenge one has to face when analysing NNs is the presence of non-linear activation functions, whose role is essential for the network expressivity. Models with linear activations cannot capture non-linearities, but they serve as a starting point for deeper understanding <cit.>. The case of a narrow hidden layer was already studied more than thirty years ago <cit.> and more recently in <cit.>. However, in the more challenging regimes where all layers are large and of comparable sizes it was observed (for instance in <cit.>) that certain NNs models behave like finely tuned linear models regardless of the activation type, given sufficient regularity. From this observations, a whole set of Gaussian equivalence principles (GEPs) have emerged as valuable tools for handling non-linear activations in both rigorous and more heuristic approaches. GEPs leverage a well-known fact in high-dimensional probability: suitably rescaled low-dimensional projections of high-dimensional vectors with weakly correlated components exhibit Gaussian behavior. Classical results <cit.> and recent developments <cit.> support the validity of GEPs in various high-dimensional inference contexts, such as in the description of certain observables for shallow neural networks <cit.>.
However, the extent to which GEPs apply to the information-theoretic study of NNs where all weights are learned remains uncertain. Certain scaling regimes relating the number of data samples and network weights must cause GEPs to break down, as
NNs do not always behave like linear models <cit.>. This paper aims to bridge this gap by means of rigorous mathematical physics techniques developed in the study of spin glasses. We demonstrate the existence of a scaling regime for two-layer networks where GEPs are rigorously applicable, with the number of data playing a central role. As a result, we establish the information-theoretical equivalence between a two-layer NN and a generalized linear model, that hence share the same optimal generalization error.
Notations Bold notations are reserved for vectors and matrices. By default a vector is a column vector, and its transpose ^⊺ is therefore a row vector. Thus the usual L_2 norm ^2=^⊺ and ^⊺ is a rank-one projector. _A is an expectation with respect to the random variable A; is an expectation with respect to all random variables entering the ensuing expression. For a function F of one argument we denote F' its derivative. Notations like i≤ N always implicitly assume that the index i starts at 1. We often compactly write (⋯)^2=[(⋯)^2]≥ ((⋯))^2=^2(⋯) and similarly for other functions, we denote equivalently [f(⋯)] and f(⋯). For any functions f and g, f=O(g) means that there exists a constant K such that |f|≤ K|g|; in other words we simply denote O(g):=O(|g|) where in the r.h.s. we use the standard big O notation. Hence, taken for instance a Gaussian r.v. Z∼𝒩(0,1), f=O(Z) does not imply f =0, but rather | f|≤|f|≤ K|Z|.
Acknowledgements The authors were funded by the European Union (ERC, CHORAL, project number 101039794). Views and
opinions expressed are however those of the author only and do not necessarily reflect those of the European Union or the European
Research Council. Neither the European Union nor the granting authority can be held responsible for them. We wish to thank Marco Mondelli for suggesting meaningful references, and Rosalba Pacelli and Pietro Rotondo for fruitful clarifications on <cit.>.
Artificial Neural Networks (NNs) proved to be very efficient tools in classifying data, and, more in general, since they are universal approximators <cit.>, to approximate any non linear function provided they have enough free parameters to be tuned. A NN is a set of neurons, that can be thought as some real degrees of freedom, and of weighted connections, typically referred to as weights. Each neuron is equipped with a non-linear function, called activation, that is applied to the sum of the other neurons' values , weighted by the network weights. The way neurons and weights are organized defines the architecture of the network. A very popular one, that achieves state-of-the-art performances in several tasks (image classification, speech recognition and many more <cit.>), is the deep architecture, where neurons are arranged in consecutive layers, and the weights are non-zero only between neurons of two different adjacent layers <cit.>. Deep networks are greatly inspired by what is believed to be their building block: the perceptron <cit.>, or, more in general, generalized linear models (GLMs) <cit.>. Indeed, GLMs can be thought as a one layer NN, where all the neurons in the first layer are connected to a single (or few) output unit(s).
Be it deep or not, a NN needs to be trained, i.e. to adjust its weights, so to fit the target function. This is typically done through Empirical Risk Minimization (ERM), where the weights are selected so that they minimize a risk, a measure of the deviation of the NN model from the target function. Notice that this is in general a very high dimensional problem, since one favours large NNs, with lots of neurons and connections, in order to grant a certain expressivity. The empirical risk, and hence the final weights of the NN, is a function of the so-called training set, a set of data made of examples of inputs and correct outputs that the NN is supposed to reproduce. For our purposes, the input is a high-dimensional signal, such as the pixel values of an image, fed into the first layer of a deep NN. The output instead, located in the last layer/unit in a deep architecture, is low dimensional.
Given their vast range of applications, it is important to establish what are the fundamental limits of deep NNs. Their performance and reliability is generally measured via the so-called generalization error, that is the expected error the NN would make in classifying/fitting a new input. Recently, there has been a remarkable effort in studying this quantity when the NN is trained with ERM <cit.>. However, as intuition suggest, the learning or training procedure can deeply impact on the performance. It was indeed proved that, at least for GLMs, a naive ERM training does not attain the generalization error that would be instead accessible according to information theory. To be precise, one can design an empirical risk whose minimum is attained at weights that yield information theoretically optimal performance. However, the related minimization problem is non-convex <cit.>. See later for a discussion on related works for more details.
One of the main obstacles when facing the theoretical study of a deep NN, is the presence of the non-linear activation function, that is at the same time responsible for the expressivity of the network. Indeed, if the activations are all linear, the NN is simply a model of a linear function, which cannot fit non linearities, though being a starting point for deeper understanding <cit.>. Ref. <cit.> also studies the case with one hidden, narrow layer, whose non linearity can be re-absorbed with a suitable redefinition of activations. This is possible thanks to the fact that the narrow layer remains of finite size. In certain regimes however, with all the layers growing in size, experiments show that NNs behave as suitably tuned linear models, regardless of the type of the activation, provided it is regular enough. In particular, a key tool used to take care of non-linear activations, even rigorously in specific settings, are the Gaussian Equivalence Principles (GEPs). GEPs are inspired by a well known fact in high dimensional probability: low dimensional projections of high dimensional vectors with weakly correlated components, when properly rescaled, behave as Gaussians. There are a lot of beautiful classical results in this direction <cit.>, not to mention more recent developments <cit.>. GEPs appeared in recent literature applied to describe the prediction mean square errors in shallow neural networks <cit.>. As it shall be clearer in the following, GEPs allow to control the activations, evaluated on random variables, with their first moments, reducing notably the complexity of the analytical study.
However, to the best of our knowledge, on the rigorous side it is not clear yet to what extent GEPs are applicable to the information theoretical study of NNs, where all the weights have to be learnt. There must be some scaling regimes, in the number of data and network weights, in which GEPs break down, since NNs do not always behave as linear models. The goal of this paper is precisely to fill this gap. For a two-layer network, using rigorous Mathematical Physics techniques developed in the study of spin-glases, we show that there exists a scaling regime in which GEPs are rigorously applicable, and the number of data at disposal is shown to play a central role.
As a consequence, we prove the reduction of a two-layer NN to a GLM, or more precisely, their information theoretical equivalence. The fundamental limits of the GLM, including its information theoretically optimal generalization error, are thus shared by the two-layer NN.
§ SETTING AND MAIN RESULTS
§.§ Bayesian-optimal learning of a shallow network in the teacher-student set-up
We consider supervised learning with a Bayesian two-layer neural network in a teacher-student setup, with matched architecture for teacher (i.e., data-generating) and student (i.e., trained) models. To be precise, let the dataset be 𝒟_n= {(_μ,Y_μ)}_μ=1^n, with inputs _μ∈ℝ^d and responses Y_μ∈ℝ. The Y_μ could also be taken from ℝ^K with K independent of d,p,n without much changes. The teacher network generates the responses according to the following rule:
Y_μ=f(^*⊺/√(p)φ(^*_μ/√(d)); _μ)+√(Δ)Z_μ .
Here, ^*∈^p and ^*∈^p× d. The readout function f can include stochasticity, modeled through its second argument _μ which are independent and identically distributed (i.i.d.) random variables in ^k with k fixed, whose common distribution is denoted by P_A. Whenever it is not specified, real functions are applied component-wise to vectors, such as the non linearities f,φ. We assume the following regularity hypotheses:
A1) The activation function φ:↦, φ∈ C^3 is c-Lipschitz for some absolute constant c, is an odd function, and has bounded second and third derivatives.
A2) The readout function f as well as its first and second derivatives are P_A-almost surely bounded.
We draw the independent inputs _μ𝒩(0,I_d) from the standard Gaussian measure. Furthermore, Gaussian label noise Z_μ𝒩(0,1) is added in (<ref>), whose variance is tuned by Δ>0. Introducing the output kernel
P_ out(y| x):=∫ P_A(d)1/√(2πΔ)exp(
-1/2Δ(y-f(x;))^2
)
,
once can see that the random outputs are generated independently, conditionally on the teacher parameters ^*=(^*,^*) and the inputs, as
Y_μ∼ P_ out(·|^*⊺/√(p)φ(^*_μ/√(d)))
.
The probability distribution for the weights of the teacher network are drawn from a centered Gaussian factorized prior distribution. Taking them with different variances is possible but it does not add much to the result, so we consider them all equal to one for our purposes: a_i^*, W_ij^*𝒩(0,1). The same Gaussian law is used as prior distribution for the Bayesian student neural network model. In empirical risk minimization, a Gaussian prior would induce a L^2 norm regularization for the weights.
In this paper we will instead deal with Bayesian learning in the so-called Bayes-optimal setting, which is the proper framework to quantify the fundamental performance limits of neural networks. Concretely, the Bayes-optimal scenario corresponds to a realizable matched setting where the student network has exactly the same architecture as the teacher network used to generate the responses in the data 𝒟_n.
The analysis in this setting therefore yields the best information-theoretically achievable student's generalization performance whenever optimally trained, namely, when using Bayesian learning based on the posterior distribution of the student's parameters. As a consequence,
our results lay down fundamental upper bounds for the performance of any neural networks, Bayesian or not, trained in any possible manner (including empirical risk minimization rather than Bayesian learning) or, more generally, for any learning procedure for the supervised task at hand. Moreover, both the readout and internal hidden layer are trained, with a number of hidden units that can grow proportionally to the input dimension.
To the best of our knowledge, this is the first rigorous result of this kind in this challenging scaling regime.
More formally, a student network is said to be Bayes-optimal if it employs the same output kernel P_ out as used by the teacher network, or equivalently same number of layers and layers widths, readout f and activation φ, label noise variance Δ, as well as same prior law for its weights. Bayes-optimal learning is then based on the Bayes posterior of the network parameters =(,) which reads
dP(|𝒟_n)= 1/(𝒟_n)∏_μ=1^nP_ out(
Y_μ|^⊺/√(p)φ(_μ/√(d))
)D
where for brevity
D:=∏_i=1^pda_i/√(2π)e^-a_i^2/2∏_i=1^p∏_j=1^ddW_ij/√(2π)e^-W_ij^2/2=:D D .
We will often use the notation D for densities of objects whose entries are i.i.d. standard Gaussian variables. The normalization of the posterior distribution will be referred to as partition function:
(𝒟_n):=∫ Dexp(∑_μ=1^nu_Y_μ(s_μ))
where we have introduced the two further definitions u_y(x)=log P_ out(y| x) and
s_μ:=^⊺/√(p)φ(_μ/√(d)) , S_μ:=^*⊺/√(p)φ(^* _μ/√(d)) .
Note that the partition function is random through the randomness of the dataset 𝒟_n. Then, (optimal) Bayesian learning means that the predictor Ŷ_ Bayes(_ new) of the response associated with a fresh input test sample corresponds to the mean of the Bayes posterior distribution of the response given the training data:
Ŷ_ Bayes(_ new):=[Y_ new|𝒟_n,_ new]=∫ dY Y P_ out(Y|^⊺/√(p)φ(_ new/√(d))) dP(|𝒟_n) .
We will sometimes employ the language of statistical mechanics. In particular we interpret the posterior distribution as a Boltzmann-Gibbs measure over degrees of freedom which are the network weights. We shall denote the expectations w.r.t. the posterior with the so-called Gibbs brackets ⟨·⟩. For future convenience we introduce also its replicated version: for a function g dependent of k copies (_b)_b≤ k of the parameters,
⟨ g⟩^⊗ k:=1/(𝒟_n)^k∫∏_b=1^k D_b ∏_μ=1^nP_ out(
Y_μ|^⊺_b/√(p)φ(_b _μ/√(d))
) g((_b)_b≤ k) ,
that, with a slight abuse of notation, will still be denoted by ⟨·⟩. From the above definition we see that the replicated Boltzmann-Gibbs measure is factorized for a given realization of the dataset, interpreted as quenched randomness in the analogy with spin glasses <cit.>. Hence, replicas, namely i.i.d. samples from the posterior measure, are independent conditionally on 𝒟_n. However, when computing so-called quenched averages ⟨·⟩ a further expectation is taken w.r.t. the quenched data which couples the replicas.
One of the main object of interest is the free entropy (i.e., log-partition function) per sample, which is nothing else than minus the Shannon entropy H(𝒟_n) of the data distribution per sample:
f̅_n:=1/nlog(𝒟_n)=-1/nH(𝒟_n) ,
where the expectation is w.r.t. to the training data 𝒟_n={(_μ,Y_μ)}_μ=1^n. The normalization by n is natural given that the number of terms in the “energy” defined by the exponent in (<ref>) is precisely n. The data has a joint law that can be written in terms of the output kernel
dP(𝒟_n) =∏_μ=1^n(∏_j=1^ddX_μ j/√(2π)e^-X_μ j^2/2)dY_μ _^*,^*∏_μ=1^nP_ out(Y_μ| S_μ)
=: ∏_μ=1^n D _μ dY_μ _^*,^*exp(∑_μ=1^nu_Y_μ(S_μ)) .
Two observations are in order. First, the samples, indexed by μ, are not independent because the responses were all drawn from the teacher, even though the _μ's are independently generated. Second, except for the presence of differentials on the quenched variables this expression is very similar to the partition function (<ref>). This is due to the Bayes-optimality of the student network and has some pragmatic consequences, such as the Nishimori identities which will be key for the proof, see Appendix <ref>.
Finally, note that for the sake of simplicity we did not include trainable biases in the definition of our NN model. However, we believe that adding them would not change much to our analysis as long as, like the other trainable parameters, they are Gaussian distributed and the student network is again Bayes-optimal and uses same architecture and number of parameters as the teacher model.
§.§ An information-theoretically equivalent generalized linear model
We now introduce an another model known as generalized linear model (GLM) <cit.>, which can be represented as a one layer neural network and which is thus a generalization of the classical perceptron <cit.>. One particular instance of the GLM turns out to be deeply connected to the setting with shallow networks introduced in the previous section. In this model the responses are generated independently conditionally on the inputs as
Y_μ^∘=f(ρ^*⊺_μ/√(d)+√(ϵ)ξ_μ^* ; _μ)+√(Δ)Z_μ , or Y_μ^∘∼ P_ out(·|ρ^*⊺_μ/√(d)+√(ϵ)ξ_μ^*)
where ^*=(v_j^*)_j≤ d∈^d, v_j^*𝒩(0,1), ξ_μ^*𝒩(0,1) all independently and the rest is as before. With respect to the two-layer neural network, the non-linearity brought by the middle layer has been replaced by a linear model with an additional effective Gaussian noise ξ^*_μ. ρ and ϵ≥0 are two real parameters that will be specified later.
This connection between the Bayes-optimal learning of neural networks and this GLM was recently conjectured in <cit.> based on the replica method and the application of Gaussian equivalences. Our results vindicate their conjecture but for different scaling regimes relating the diverging parameters d,p,n. We are going to further comment on this point later on.
All the above construction can be repeated for the generalized linear model. From now on, quantities characterized by a ^∘ superscript will refer to the GLM. For starters, we denote the dataset generated through (<ref>) by 𝒟_n^∘:={(_μ,Y_μ^∘)}_μ=1^n.
Let
s_μ^∘=ρ^⊺_μ/√(d)+√(ϵ)ξ_μ , S_μ^∘=ρ^*⊺_μ/√(d)+√(ϵ)ξ_μ^*
and D=∏_μ Dξ_μ. The expectation under the GLM posterior of any bounded test function g of k “replicas” (i.e., conditionally on the data i.i.d. copies) (_b,_b)_b≤ k reads
⟨ g⟩^∘ ⊗ k:=1/^∘(𝒟_n^∘)^k∫∏_b=1^k D_b D_b ∏_μ=1^nP_ out(
Y_μ^∘|ρ^⊺_b_μ/√(d)+√(ϵ)ξ_bμ) g((_b,_b)_b≤ k) ,
with ^∘(𝒟_n^∘) the GLM posterior normalization. As before, the free entropy reads
f̅_n^∘:=1/nlog𝒵^∘(𝒟_n^∘)=1/nlog∫ D Dexp(∑_μ=1^n u_Y_μ^∘(s_μ^∘)) .
Finally, we write the distribution of the dataset, that is used for the quenched average in the above formula:
dP(𝒟_n^∘) =∏_μ=1^n(∏_j=1^ddX_μ j/√(2π)e^-X_μ j^2/2)dY_μ^∘ _^*,^*∏_μ=1^nP_ out(Y_μ^∘| S^∘_μ)
=: ∏_μ=1^n D _μ dY_μ _^*,^*exp(∑_μ=1^nu_Y^∘_μ(S^∘_μ))
and the optimal Bayesian predictor is
Ŷ^∘_ Bayes(_ new):=[Y^∘_ new|𝒟^∘_n,_ new]=∫ dY Y P_ out(Y|ρ^⊺_ new/√(d)+√(ϵ)ξ_μ) dP(,|𝒟^∘_n) .
§.§ Results
Our first theorem concerns the equivalence between the Bayes-optimal learning of the neural network and GLM models at the level of free entropy. Letting Z∼𝒩(0,1) we denote _𝒩(0,1)g:= g(Z).
Let
ρ:=_𝒩(0,1)φ' and ϵ^2:=_𝒩(0,1)φ^2-ρ^2 ,
and suppose Assum:phi and Assum:f hold. Then
|f̅_n-f̅^∘_n|=O(√((1+n/d)(n/p+n/d^3/2+1/√(d)))) .
From the previous statement we can identify the scaling regime for which the equivalence holds, namely, when the right hand side of (<ref>) goes to 0. From now on we shall denote by
lim g_d,p,n:=lim_i→∞ g_d_i,p_i,n_i
where (d_i,p_i,n_i)_i is any sequence of triplets of integers such that
lim_i→∞(1+n_i/d_i)(n_i/p_i+n_i/d_i^3/2+1/√(d)_i)=0 .
As a corollary we have the matching of the mutual informations between the dataset 𝒟_n and the network weights in the same limit. For the two-layer neural network the mutual information is related to the free entropy through the following expression (H is the Shannon entropy)
1/nI_n(^*;𝒟_n)=1/nH(𝒟_n)-1/nH(𝒟_n|^*)=-f̅_n+log P_ out(Y_1|^*⊺/√(p)φ(^*_1/√(d))) ,
whereas for the equivalent GLM we have
1/nI_n^∘(^∘*;𝒟_n^∘) =-f̅^∘_n+log P_ out(Y_1|ρ^*⊺_1/√(d)+√(ϵ)ξ_1^*) .
Considering that the teacher weights and inputs are Gaussian we have in law
^*⊺/√(p)φ(^*_1/√(d)) Z√(1/p‖φ(^*_1/√(d))‖^2)
with Z∼𝒩(0,1) and · is the standard L^2 norm for vectors. Therefore, it is clear that the randomness in ^* and _1 will disappear in the limit. A similar equality holds for the GLM:
ρ^*⊺_1/√(d)+√(ϵ)ξ_1^* Z√(ρ^2‖_1‖^2/d+ϵ) .
Our goal is now to show that the arguments under square root both tend to ρ^2+ϵ=_𝒩(0,1)φ^2 in the limit, and that we can plug this result inside the last terms in (<ref>) and (<ref>), with a control on the error we make. To this end, define
S_d(t)=√(tρ^2(‖_1‖^2/d-1)+ϵ+ρ^2) , or S_d(t)=√(t(1/p‖φ(^*_1/√(d))‖^2-_𝒩(0,1)φ^2)+_𝒩(0,1)φ^2)
and
Ψ(t):=∫ dY P_ out(Y| Z S_d(t))log P_ out(Y| Z S_d(t)) .
Using the properties later described in Lemma <ref> and the definition of P_ out in (<ref>), under assumptions (Assum:phi-(Assum:f one can readily verify that
|Ψ̇(t)|≤ C(f) |Z| |Ṡ_d(t)| ,
where C(f) is a constant depending on f. From this bound, by the fundamental theorem of integral calculus, we have
|Ψ(1)-Ψ(0)|≤ C(f)∫_0^1 dt |Ṡ_d(t)|≤ C(f)|‖φ(^*_1/√(d))‖^2/p-_𝒩(0,1)φ^2|/√(_𝒩(0,1)φ^2) for the 2-layer NN
ρ^2|‖_1‖^2/d-1|/√(_𝒩(0,1)φ^2) for the GLM .
The remainder for the two-layer NN is O(p^-1/2+d^-1/4) whereas for the GLM we have O(d^-1/2). Finally, thanks to the previous argument
1/nI_n(^*;𝒟_n) =-f̅_n+Ψ(_𝒩(0,1)φ^2)+O(p^-1/2+d^-1/4) ,
1/nI_n^∘(^∘*;𝒟_n^∘) =-f̅_n^∘+Ψ(_𝒩(0,1)φ^2)+O(d^-1/2) ,
where
Ψ(_𝒩(0,1)φ^2):=Ψ(0)=∫ dY P_ out(Y| Z√(_𝒩(0,1)φ^2))log P_ out(Y| Z√(_𝒩(0,1)φ^2)) .
Hence, we have just proved the information theoretical equivalence:
Under the same hypothesis of Theorem <ref> the following holds:
|1/nI_n(^*;𝒟_n)-1/nI_n^∘(^∘*;𝒟_n^∘)|=O(√((1+n/d)(n/p+n/d^3/2+1/√(d)))) .
The performance of the neural network is quantified using the generalization error on test data using the square loss. The Bayes-optimal generalization error thus reads
ℰ_n:=(
Y_ new-[Y_ new|𝒟_n,_ new]
)^2 .
The outer expectation is intended w.r.t. the training data set and the test sample which is generated independently from the training data according to the same teacher model. The GLM Bayes-optimal generalization error ℰ^∘_n is defined similarly but considering the GLM teacher-student setting described in the previous section.
As a consequence of the proof technique of Theorem <ref> it is possible to show the following equivalence at the level of the generalization error, proven in Section <ref>.
Under the same hypothesis of Theorem <ref> the following holds:
lim |ℰ_n-ℰ^∘_n|=0 ,
i.e., the shallow neural network and noisy GLM settings lead to the same Bayes-optimal generalization error in any high-dimensional limit such that (<ref>) holds.
Our results combined with those of <cit.> for the GLM provide explicit rigorous formulas for the mutual information and Bayes-optimal generalization error for the Bayesian neural network studied in this paper.
A remark is in order. The previous theorem states that the Bayes-optimal generalization error of a two-layer NN, which is trained on the dataset 𝒟_n generated by a teacher two-layer NN with same architecture, equals that of a noisy GLM trained on the dataset 𝒟_n^∘ generated by a matched teacher GLM. However, we cannot deduce from this that the two-layer NN trained using the GLM teacher dataset 𝒟_n^∘, or viceversa, achieves the Bayes-optimal generalization error. It would be interesting to investigate this aspect in the future.
§.§ Related works
There exist by now a whole zoology of theoretical models for NNs studied in the literature and it becomes increasingly challenging to cover them whole. We provide here a partial classification of the main models divided according to how scale their internal widths compared to the inputs dimension, and whether the internal weights are trainable or not, see Figure <ref>. For each class we provide a selection of relevant references without trying to be exhaustive.
Perceptrons and committee machines The perceptron (and its generalization, i.e., the GLM) are linear classifiers with a non-linear readout. Committee machines can be viewed as two-layer neural networks with a narrow hidden layer and a single neuron output layer. These models have been studied in teacher-student set-ups and with online learning since the nineties <cit.>. Despite their rich phenomenology with a so-called specialization phase transition where the model realizes the in-put-output rule is non-linear, these machines cannot capture the features of realistic data: they project high-dimensional data in a comparatively too low-dimensional space. The relevant regime for these models is n,d→∞ with n/d→α∈(0,∞) and p=O(1). Despite the fact that all the weights among layers have to be learnt, w.r.t. our setting, the middle layer remains finite while the input dimension and number of data points diverge together. Note that, at first sight, it might be surprising that our result does not imply equivalence when p is finite. However, for any p>1 such equivalences are not expected because a committee machine with two or more hidden units is not, in general, a linear classifier and can represent more complex non-linear relations. On the contrary, GEP-type of results are expected to provide reductions towards (generalized) linear models.
Mean-field regime In the series of works <cit.> the authors study the stochastic gradient descent (SGD) dynamics of multilayer neural networks. In contrast with the committee machine, here also the hidden layer can diverge in size (see <cit.>). This projection of a relatively low-dimensional signal into a very high-dimensional space has a regularizing property on the risk landscape. In particular, it causes the merging of possible multiple minima of the finite p risk. SGD is then able to reach a near optimum with controllable guarantees.
However, mean-field analyses of SGD dynamics differ from the information-theoretical one. Indeed, SGD produces a “one-shot estimator”, which is in general outperformed by the Bayes-optimal one. Also, online learning is generally considered while our analysis consider (optimal) learning from a large fixed data set. Furthermore, for the information-theoretical equivalence in Theorem (<ref>) to be valid, we need to control the size of the training set w.r.t. the network size, and in principle we can send both d,p→∞ with d/p finite, as long as the training set is not too big (n≪ p).
Frozen hidden weights: neural tangent kernel, random features and lazy training Neural tangent kernel (NTK) <cit.> is a linearization of, say, a two-layer neural network, which reduces its training to a linear regression on the readout weights. As specified in <cit.>, NTK describes well the neural network performance at the initial stage of learning using SGD when the network weights are virtually frozen around their initialization. In a similar fashion, in random feature models and lazy training regimes <cit.> the internal weights of the network are quenched, i.e., fixed once and for all. Other results <cit.> show that large-width NNs with purely random weights behave as gaussian processes. Finally, recent works based on random matrix theory consider linear-width NNs but again under the assumption that only the last layer is learned, while internal ones are random
<cit.>.
Even though some of the above results, and more recently <cit.>, extend to extensive width input and hidden layers, as well as extensive number of data, i.e. d,p,n→∞ with d,p,n all proportional, they hold in a setting that is fundamentally different from ours. In fact, we address the learning of all the parameters in the network, considering all of them as annealed variables, from a statistical mechanics perspective. Moreover, it is worth stressing again that we study the Bayes-optimal generalization error, and not the one coming from ERM. In this regard, it was shown in <cit.> that ERM with hinge or logistic losses can reach generalization errors that are close to Bayes-optimal in GLMs. In addition, as long as a suitable (though not convex) loss is taken into account, ERM can yield Bayes-optimal performances. However, this holds for GLMs, which have no hidden layer and thus only p parameters to be learnt.
Linear-width regimes A line of recent works <cit.> deals with the full training of the network as we do here. <cit.> in particular carries out a thorough study for linear neural networks. In <cit.>, instead, the authors conjecture the Bayes-optimal limits in the extensive width and data regime d,p,n→∞ all proportionally. Their computations are based on a combination of the heuristic replica method from spin glasses and a Gaussian equivalence principle, that allows to treat the non-linear activations in an efficient way. Despite GEPs have been proved rigorously in other contexts (for instance <cit.>), it is not obvious that they are directly applicable to the extensive width and data regime when the full training of the network is carried out. Indeed, it this not clear to us whether our proof can be extended to the whole regime considered in <cit.>; in particular, we cannot assess whether the equivalence results provided in the next section do hold in the fully proportional regime where d=Θ(n) and large (this is allowed by our bounds) and p=Θ(n) (which is instead prevented by our bounds) as considered <cit.>, in spite that we cannot prove it at the moment.
Gaussian equivalence principles are also present in random matrix theory literature <cit.> and find applications in the study of random features models.
Estimation in multi-layer generalized linear models Finally, we emphasize the difference between the learning problem considered in our work and the inference problem discussed in <cit.>, later extended in <cit.>, where the authors consider the task of reconstructing a vector from observations obtained from a multi-layer GLM with fixed, known, weight matrices. We believe that the proof of the concentration of the free entropy in <cit.>, which was a bottleneck for GLM extensions to the multi-layer setting initiated in <cit.>, can be adapted to our learning problem, yielding the Bayes-optimal generalization error GLM reduction for a deep network.
§ PROOF OF THEOREM <REF>
§.§ The interpolating model
Our proof is based on the interpolation method, introduced in the seminal papers <cit.>. This method is a very effective tool whenever a comparison between two high dimensional models is needed. The idea is that of introducing an interpolating model, for any t∈[0,1], that at its ends t=0 and t=1 matches the two models to be compared. In analogy with <cit.>, we shall interpolate at the level of the variables s_μ, S_μ:
S_tμ :=√(1-t)𝐚^*⊺/√(p)φ(𝐖^*𝐗_μ/√(d))+√(t)ρ𝐯^*⊺𝐗_μ/√(d)+√(tϵ)ξ_μ^* ,
s_tμ :=√(1-t)𝐚^⊺/√(p)φ(𝐖𝐗_μ/√(d))+√(t)ρ𝐯^⊺𝐗_μ/√(d)+√(tϵ)ξ_μ .
We thus introduce an interpolating teacher and interpolating student, such that the second is Bayes-optimal for any t∈[0,1]. This allows us to use the so-called Nishimori identities of Appendix <ref> uniformly in t.
The interpolating teacher network shall produce the μ=1,…,n conditionally indpendent responses
Y_tμ∼ P_ out(·|
S_tμ) ,
where the output kernel is unchanged since it depends only on f. Posterior means now read
⟨ g ⟩_t:=1_t∫ D𝐚D𝐖D𝐯Dexp[∑_μ=1^nu_Y_tμ(s_tμ)]g
for any observable g depending on ,,,, where
_t=∫ D𝐚D𝐖D𝐯Dexp[∑_μ=1^nu_Y_μ(s_tμ)] .
In the following we drop the subscript t to keep the notation light and simply use ⟨·⟩. We also introduce the compact notation
_(t)(·):=_^*_∖^*(·)=_^*𝔼_𝐖^*,𝐯^*,^*,{𝐗_μ}∫∏_μ=1^ndY_tμ
e^u_Y_tμ(S_tμ)(·) .
The free entropy of this interpolating model is thus
f̅_n(t):=1/n𝔼_(t)log_t ,
whence it can be verified that
f̅_n(0)=f̅_n , f̅_n(1)=f̅_n^∘ .
Due to the identity f̅_n(1)-f̅_n(0)=∫_0^1 d/dtf̅_n(t) dt, a sufficient condition to prove our Theorem <ref> is to show that d/dtf̅_n(t) is uniformly bounded by the same order as in the statement. A direct computation shows
d/dtf̅_n(t)=-A_1+A_2+A_3+B
where
A_1:=1/2n_(t)log_t
∑_μ=1^nu^'_Y_tμ(S_tμ)𝐚^*⊺/√((1-t)p)φ(𝐖^*𝐗_μ/√(d)) ,
A_2:=1/2n_(t)log_t
∑_μ=1^nu^'_Y_tμ(S_tμ)ρ𝐯^*⊺𝐗_μ/√(td) ,
A_3:=1/2n_(t)log_t
∑_μ=1^nu^'_Y_tμ(S_tμ)√(ϵ/t)ξ_μ^* ,
B:=1/n_(t)⟨∑_μ=1^nu^'_Y_tμ(s_tμ)ds_tμ/dt⟩ .
We will control each term individually. For that we will need a number of Lemmas wich we provide now.
§.§ Lemmata
We collect here some Lemmas that shall be used intensively in the following. For the convenience we postpone proofs to the Appendix.
Recall the definition u_y(x):=log P_ out(y| x). We denote u'_y(x):=∂_x u_y(x). Furthermore, let
U_μν:=δ_μν u”_Y_tμ(S_tμ)+u'_Y_tμ(S_tμ)u'_Y_tν(S_tν) .
Under Assumptions (Assum:phi and (Assum:f the following statements hold:
[u'_Y_tμ(S_tμ)| S_tμ]=[U_μν| S_tμ,S_tν]=0 ,
[(u'_Y_tμ(S_tμ))^2| S_tμ] , [U_μν^2| S_tμ,S_tν]≤ C(f) ,
for a positive constant C(f) depending solely on the readout function.
It is worth to point out a simple observation that for μ=ν we have U_μμ=P^''_ out(Y_tμ| S_tμ)/P_ out(Y_tμ), where
P^'_ out(y| x):= ∂_x P_ out(y| x) , P^''_ out(y| x):= ∂_x∂_x P_ out(y| x) ,
from what immediately follows
[(P^''_ out(Y_tμ| S_tμ)/P_ out(Y_tμ| S_tμ))^2| S_tμ]≤ C(f) .
The following lemma will play a crucial role, and it contains all the approximations due to the law of large numbers. We introduce here a convenient notation for the pre-activations:
α_μ:=^*_μ/√(d) .
Hence, conditionally on the inputs (_μ)_μ≤ n, the α's have covariance
1/p[α_μ^⊺α_ν|_μ,_ν]:= 1/p_^*(^*_μ)^⊺/√(d)^*_ν/√(d)=_μ^⊺_ν/d .
Let φ̃ be either φ or the identity function. Under assumptions (Assum:phi and (Assum:f the following estimates hold for any choice of μ,ν≤ n:
_^*φ'(α_μ i)= ρ+O(‖_μ‖^2/d-1) ,
_^*φ^2(α_μ i)=_𝒩(0,1)φ^2+O(‖_μ‖^2/d-1) ,
_^*φ(α_μ i) φ̃(α_ν i)=ρ_𝒩(0,1)φ̃'_μ^⊺_ν/d+O(_μ^⊺_ν/d(‖_μ‖^2/d-1))+O((_μ^⊺_ν/‖_ν‖^2)^2)+O((_μ^⊺_ν)^2/‖_ν‖^2d) ,
_^*φ'(α_μ i)φ'(α_ν i)= ρ^2+O(‖_μ‖^2/d-1)+O(_μ^⊺_ν/‖_ν‖^2) ,
_^*φ^2(α_μ i)φ̃^2(α_ν i)=_𝒩(0,1)φ^2_𝒩(0,1)φ̃^2
+O(‖_μ‖^2/d-1)+O(_μ^⊺_ν/‖_ν‖^2) .
The final key ingredient is the concentration of the free entropy, that we prove in Section <ref>.
Under assumptions (Assum:phi and (Assum:f there exists a non-negative constant C(f,φ) such that
𝔼_𝐚^*𝕍_∖^*(1nlog_t)
=𝔼(1nlog_t-𝔼_∖^*1nlog_t)^2
≤ C(f,φ)(1d+1n) .
§.§ Proof of Theorem <ref>
We split the proof of Theorem <ref> into different Lemmas for the sake of readability. If not differently specified, all the following Lemmas hold under the same hypotheses of Theorem <ref>. The first one concerns the B contribution to the derivative of the free entropy (<ref>).
B=0.
The random variable inside the brackets in (<ref>) is a function of the data Y_tμ and of a sample from the posterior through s_tμ. Hence we can use the Nishimori identities to get rid of the brackets, replacing s_tμ with the ground truth version S_tμ (from now on we denote with an upper dot Ṡ:=dS/dt the t-derivative):
B=1/n_(t)∑_μ=1^nu'_Y_tμ(S_tμ)Ṡ_tμ=1/n∑_μ=1^n_(t)[_(t)[ u'_Y_tμ(S_tμ)| S_tμ]Ṡ_tμ]
where we used the tower rule for expectations. The latter is identically zero thanks to Lemma <ref>.
We split A_1 into two other contributions A_1=A_11+A_12 where
A_11:=1/2n√(1-t)_(t)log_t
∑_μ=1^nu^'_Y_tμ(S_tμ)(𝐚^*⊺/√(p)φ(𝐖^*𝐗_μ/√(d))-ρ𝐚^*⊺𝐖^* 𝐗_μ/√(pd)) ,
A_12:=1/2n√(1-t)_(t)log_t
∑_μ=1^nu^'_Y_tμ(S_tμ)ρ𝐚^*⊺𝐖^* 𝐗_μ/√(pd) .
Let us simplify these terms by Gaussian integration by parts. In A_12, integrating by parts w.r.t. 𝐖^* yields
A_12 =ρ/2n_(t)log_t∑_μ,ν=1^nU_μν𝐚^*⊺(𝐚^*∘φ'(𝐖^*𝐗_ν/√(d)))/p𝐗_μ^⊺𝐗_ν/d
with U_μν defined in Lemma <ref> and ∘ denotes the entry-wise (Hadamard) product. Concerning A_11, because of the non-linearity, we can only integrate by parts w.r.t. 𝐚^* and obtain
A_11 =1/2n_(t)log_t∑_μ,ν=1^nU_μν[ φ(α_μ)^⊺φ(α_ν)-ρα_μ^⊺φ(α_ν)/p] .
The off-diagonal μ≠ν and diagonal terms in the previous equations play two very different roles, and they shall be treated separately in the following.
The following holds:
A_11^ off:=1/n_(t)log_t∑_μ≠νU_μν[φ(α_μ)^⊺φ(α_ν)-ρα_μ^⊺φ(α_ν)/p]=O(√((1+n/d)(n/p+n/d^3/2))) .
Let us start noticing that, for any smooth function F(α_μ,α_ν) we have
_∖^*U_μνF(α_μ,α_ν)=
_∖^*[_∖^*[U_μν|𝐖^*,^*,ξ^*,𝐗]F(α_μ,α_ν)]=0
thanks to Lemma <ref>. As a consequence, with ^* fixed, we can modify A_11^ off as follows without changing its value:
A_11^ off=_∖^*(f_n-_∖^*f_n)∑_μ≠νU_μν[φ(α_μ)^⊺φ(α_ν)-ρα_μ^⊺φ(α_ν)/p]
with f_n:=log_t/n. In the following we shall simply write u'_μ in place of u'_Y_tμ(S_tμ), and φ_μ instead of φ(α_μ) for brevity. We are now in position to use Cauchy-Schwartz's identity:
(A_11^ off)^2≤𝕍_∖^*[f_n] ∑_μ≠ν∑_λ≠η_∖^* U_μν U_λη[φ_μ^⊺φ_ν-ρα_μ^⊺φ_ν/p][φ_λ^⊺φ_η-ρα_λ^⊺φ_η/p] .
Note that when all the four Greek indices are different from one another we get the highest combinatorial factor of O(n^4). However, using the conditional independence of the responses and Lemma <ref>, the expectation sets them to 0. Hence, the only contributions from the double sums come from μ=λ and ν=η, or μ=η and ν=λ, which gives twice the same quantity. Thus
(A_11^ off)^2≤𝕍_∖^*[f_n] 2/p^2∑_μ≠ν_∖^* (u'_μ u'_ν)^2 ∑_i,j=1^p[φ_μ iφ_ν iφ_μ jφ_ν j
-2ρα_μ iφ_ν iφ_μ jφ_ν j+ρ^2α_μ iφ_ν iα_μ jφ_ν j] .
The double sum on i,j comes from the square of a scalar product. Lemma <ref> then allows to bound [U_μν^2| S_tμ,S_tν] by a constant.
Let us treat the off-diagonal terms (i≠ j) first. We call Lemma <ref>, in particular (<ref>), to simplify the first term in (<ref>):
_∖^*∑_i≠ j,1^pφ(α_μ i)φ(α_ν i)φ(α_μ j)φ(α_ν j)= p(p-1)__μ,_ν(_^*[φ(α_μ 1)φ(α_ν 1)])^2
= p(p-1) [ρ^4 (_μ^⊺_ν/d)^2+O(_μ^⊺_ν/d(_μ^⊺_ν/‖_ν‖^2)^2)+O((_μ^⊺_ν/d)^2_μ^⊺_ν/‖_ν‖^2)+O((_μ^⊺_ν/d)^2(‖_ν‖^2/d-1))] .
The first term in the square brackets corresponds to the square of the leading term in (<ref>) with φ̃=φ. The other two terms are obtained as cross products between the leading term in (<ref>) and the remainders.
Exploiting the fact that the norm of a Gaussian vector concentrates with exponential speed, i.e.,
ℙ(|‖_μ‖^2/d-1|≥ h)≤exp(-dL h^2/2) , L>0 ,∀ h>0 ,
one can conclude that
_∖^*∑_i≠ j,1^pφ(α_μ i) φ(α_ν i)φ(α_μ j) φ(α_ν j)=p(p-1)[ρ^4/d+O(1/d^3/2)] .
We now turn to the second term of (<ref>): using again a Gaussian integration by part for the second equality below followed by Lemma <ref> we get
ρ__μ,_ν_^* ∑_i≠ j^pα_μ iφ_ν iφ_μ jφ_ν j=ρp(p-1)__μ,_ν_^*[α_μ 1φ_ν 1]_^*[φ_μ 1φ_ν 1]
= ρp(p-1)__μ,_ν_μ^⊺_ν/d_^*[φ'_ν 1]_^*[φ_μ 1φ_ν 1]
= p(p-1)__μ,_ν[ρ^2_μ^⊺_ν/d+O(_μ^⊺_ν/d(‖_μ‖^2/d-1))]
×[ρ^2_μ^⊺_ν/d+O((_μ^⊺_ν/‖_ν‖^2)^2)+O((_μ^⊺_ν)^2/‖_ν‖^2d)+O(_μ^⊺_ν/d(‖_μ‖^2/d-1))]
which shows that
ρ__μ,_ν_^* ∑_i≠ j^pα_μ iφ_ν iφ_μ jφ_ν j=p(p-1)
[ρ^4 /d+O(1/d^3/2)] .
Finally, for what concerns the off-diagonal terms i≠ j, we deal with the last term of (<ref>):
ρ^2∑_i≠ j^p__μ,_να_μ iφ_ν iα_μ jφ_ν j =ρ^2p(p-1)__μ,_ν_^*[α_μ 1φ_ν 1]_^*[α_μ 1φ_ν 1]
=p(p-1)ρ^2 __μ,_ν(_μ^⊺_ν/d)^2_^*φ'_μ 1_^*φ'_ν 1
=p(p-1)[ρ^4/d+O(1/d^3/2)]
where we used integration by parts and the approximation Lemma <ref>. From this computation we see that, remarkably, the leading orders of the off-diagonal terms i≠ j in (<ref>) cancel each other, leaving the more convenient rate O(1/d^3/2). More precisely, there exists an absolute constant K such that
(A_11^ off)^2≤𝕍_∖^*[f_n] 2K/p^2∑_μ≠ν_∖^*{∑_i=1^p[φ^2_μ iφ^2_ν i
-2ρα_μ iφ^2_ν iφ_μ i+ρ^2α_μ i^2φ_ν i^2]+O(p^2/d^3/2)} .
From the previous bound we see that we cannot hope that the same cancellation occurs in the diagonal terms i=j. Using again the results from Lemma <ref> one can show that
(A_11^ off)^2≤𝕍_∖^*[f_n] 2K/p^2∑_μ≠ν[O(p)+O(p^2/d^3/2)] .
The statement is thus proved after we take care of the remaining expectation over ^* using Theorem <ref>:
|_^*A_11^ off|≤_^*√((A_11^ off)^2)≤√(_^*𝕍_∖^*[f_n]O(
n^2/p+n^2/d^3/2))
=O(√(n/p+n/d^3/2+n^2/dp+n^2/d^5/2)) .
The previous result is telling us that the number of data points n can grow as fast as o(d^3/2) with the size of the input layer, but has to be much smaller than p, the size of the hidden layer. Furthermore, treating the difference φ(α_μ)^⊺φ(α_ν)- ρα_μ^⊺φ(α_ν) altogether is fundamental to obtain the scaling n^2d^-3/2 instead of n^2d^-1. We also stress that the pre-activations α_μ in the hidden layer have correlations among them that scale as d^-1/2. If d is not big enough they cannot be considered weakly correlated.
From Lemma <ref> we thus infer that
A_11 =1/2n_(t)log_t∑_μ=1^nP_ out”(Y_tμ| S_tμ)/P_ out(Y_tμ| S_tμ)[ ‖φ(α_μ)‖^2-ρα_μ^⊺φ(α_μ)/p]+O(√((1+n/d)(n/p+n/d^3/2))) .
For the term A_3 we use integration by parts with respect to the variables Gaussian ξ_μ^*:
A_3=ϵ/2n_(t)log_t
∑_μ=1^n((u^'_Y_tμ(S_tμ))^2+u^''_Y_tμ(S_tμ))
=ϵ/2d_(t)log_t∑_μ=1^nP^''_ out(Y_tμ| S_tμ)P_ out(Y_tμ| S_tμ) .
Hence
A_3-A_11=1/2n_(t)log_t∑_μ=1^nP^''_ out(Y_tμ| S_tμ)P_ out(Y_tμ| S_tμ)[ϵ-‖φ(α_μ)‖^2-ρα_μ^⊺φ(α_μ)/p]
+O(√((1+n/d)(n/p+n/d^3/2))) .
The following asymptotics holds:
A_3-A_11^ diag:=1/2n_(t)log_t∑_μ=1^nP^''_ out(Y_tμ| S_tμ)P_ out(Y_tμ| S_tμ)[ϵ-‖φ(α_μ)‖^2-ρα_μ^⊺φ(α_μ)/p]
=O(√((n/d + 1)(1/p + 1/√(d)))) .
Define
C:=1/n_∖^*log_t∑_μ=1^nP^''_ out(Y_tμ| S_tμ)/P_ out(Y_tμ| S_tμ)[ϵ-‖φ(α_μ)‖^2-ρα_μ^⊺φ(α_μ)/p] .
First, thanks to Lemma <ref>
_∖^*[P^''_ out(Y_tμ| S_tμ)/P_ out(Y_tμ| S_tμ)|^*,^*,_μ,ξ^*_μ]=0 ,
and this allows us to center the f_n=log_t/n with its mean without changing the value of C. After using Cauchy-Schwartz we have
C^2 ≤𝕍_∖^*[f_n]∑_μ,ν=1^n_∖^*[_∖^*[
P^''_ out(Y_tμ| S_tμ)/P_ out(Y_tμ| S_tμ)P^''_ out(Y_tν| S_tν)/P_ out(Y_tν| S_tν)|^*,^*,_μ,_ν,ξ^*_μ,ξ^*_ν]
×(ϵ-‖φ(α_μ)‖^2-ρα_μ^⊺φ(α_μ)/p)(ϵ-‖φ(α_ν)‖^2-ρα_ν^⊺φ(α_ν)/p)
] .
Thanks to the observation in (<ref>), only the diagonal terms μ=ν will survive in the double sum on the r.h.s. of the previous inequality. Furthermore recall (<ref>). Hence the bound on C^2 becomes
C^2 ≤𝕍_∖^*[f_n]C(f)n__1,^*(ϵ-‖φ(α_1)‖^2-ρα_1^⊺φ(α_1)/p)^2 .
Following an integration by part and Lemma <ref> we have
_^*α_1 iφ(α_1 i) = _^*φ'(α_1 i)‖_1‖^2/d= ‖_1‖^2/d(_𝒩(0,1)φ'+O(‖_1‖^2/d-1))
=_𝒩(0,1)φ'+O( ‖_1‖^2/d-1) .
From this and the approximation Lemma it follows that (letting ^2(⋯)=((⋯))^2)
__1_*[‖φ(α_1)‖^2-ρα_1^⊺φ(α_1)/p] =_𝒩(0,1)φ^2-^2_𝒩(0,1)φ'+O(__1|‖_1‖^2/d-1|)
=ϵ+O(d^-1/2) ,
__1_*[‖φ(α_1)‖^2-ρα_1^⊺φ(α_1)/p]^2 =1/p__1_^*(
φ^4(α_11)-2ρφ^3(α_1 1)α_1 1+ρ^2α_1 1^2φ^2(α_11)
)
+p-1/p__1(_^*^2φ^2(α_1 1)-2ρ_^*φ^2(α_1 1)
_^*φ(α_11)α_11+ρ^2_^*^2φ(α_11)α_11)
=ϵ^2+O(p^-1)+O(d^-1/2) .
Hence, we finally have
__μ,^*(ϵ-‖φ(α_μ)‖^2-ρα_μ^⊺φ(α_μ)/p)^2=O(p^-1)+O(d^-1/2) .
Plugging this and the bound in Theorem <ref> into the inequality for C^2 we readily get the statement.
Now the remaining goal is to prove that A_2-A_12→0. Using integration by parts in A_2 w.r.t. ^* we obtain
A_2=ρ^2/2n_(t)log_t∑_μ,ν=1^nU_μν_μ^⊺_ν/d .
Recall also formula (<ref>) for A_12.
The following asymptotics holds:
A_12-A_2=ρ/2n_(t)log_t∑_μ,ν=1^nU_μν_μ^⊺_ν/d[𝐚^*⊺(𝐚^*∘φ'(α_ν))/p-ρ]=O(√(( 1+n/d)(n/dp+n/d^3/2))) .
Conditional on ^* define
C:=1/n_∖^*log_t∑_μ,ν=1^nU_μν_μ^⊺_ν/d[𝐚^*⊺(𝐚^*∘φ'(α_ν))/p-ρ] .
As before, thanks to Lemma <ref>, we can center the random variable log_t with its expectation _∖^*log_t, without affecting C. We can thus use Cauchy-Schwartz's inequality, obtaining
C^2≤𝕍_∖^*[f_n]_∖^*∑_μ,ν=1^n∑_λ,η=1^nU_μνU_λη_μ^⊺_ν/d[𝐚^*⊺(𝐚^*∘φ'(α_ν))/p-ρ]_λ^⊺_η/d[𝐚^*⊺(𝐚^*∘φ'(α_η))/p-ρ]
for a given ^*. Thanks again to Lemma <ref> the only terms that survive in the above quadruple sum are those with μ=ν=λ=η, and μ≠ν, λ≠η but with μ=λ, ν=η or vice versa. Up to constants everything can be summed up as follows:
C^2≤ K 𝕍_∖^*[f_n]_∖^*∑_μ,ν=1^n(_μ^⊺_ν/d)^2[𝐚^*⊺(𝐚^*∘φ'(α_ν))/p-ρ]^2
where we used again Lemma <ref>. Now, expanding the square and computing the ^* average of φ' via Lemma (<ref>) we readily get
C^2≤ K' 𝕍_∖^*[f_n]∑_μ,ν=1^n__μ,_ν(_μ^⊺_ν/d)^2[(‖^*‖^2/p -1)^2+O(‖_ν‖^2/d-1) (‖^*‖^4/p^2+‖^*‖^2/p)]
where K' is a suitable positive constant.
Denoting the double sum by D we have
|A_2-A_12|≤ K”_^*√(𝕍_∖^*[f_n] )√(D)≤ K”√(_^*𝕍_∖^*[f_n] _^*D)=O(√((1/n+1/d)(n^2/dp+n^2/d^3/2))) .
Putting the results of all the Lemmas in this section together we get that the time derivative of the interpolating free entropy is bounded by
d/dtf̅_n(t) =O(√((1+n/d)(n/p+n/d^3/2)))_A_11^ off+ O(√((1+n/d)(1/p + 1/√(d))))_A_3-A_11^ diag+O(√((1 +n/d)(n/dp+n/d^3/2)))_ A_12-A_2
=O(√((1+n/d)(n/p+n/d^3/2+1/√(d)))) .
All the bounds in this section are uniform in t∈[0,1]. This finishes the proof of Theorem <ref>.
The last convergence speed contains subleading terms w.r.t. to the other rates, or terms that are already taken into account, in fact:
O(√((1/n+1/d)(n^2/dp+n^2/d^3/2))=O(√((1+n/d)(n/dp+n/d^3/2))) .
The (1+n/d)n/d^3/2 is already contained in other rates, whereas the term coming from n/(pd) is sub-leading w.r.t. n/p, also present in other rates. Concerning instead the middle contribution in (<ref>), the part coming from
n/d(1/p+1/√(d))=n/pd+n/d^3/2
§ CONCENTRATION OF THE FREE ENTROPY
Here we prove that the free entropy of the interpolating model
concentrates, i.e., Theorem <ref>. To simplify the notations we use C(f,φ) for a generic non-negative
constant depending only on f and φ. We recall that the partition function is defined as
_t=∫ D𝐚 D𝐯D𝐖∏_μ=1^nDξ_μexp[∑_μ=1^nlog P_ out(Y_tμ|s_tμ)] ,
where
Y_tμ =f(S_tμ;_μ)+√(Δ)Z_μ ,
S_tμ =√(1-t)𝐚^*⊺/√(p)φ(𝐖^*𝐗_μ/√(d))+√(tρ)𝐯^*⊺𝐗_μ/√(d)+√(tϵ)ξ_μ^* ,
s_tμ =√(1-t)𝐚^⊺/√(p)φ(𝐖𝐗_μ/√(d))+√(tρ)𝐯^⊺𝐗_μ/√(d)+√(tϵ)ξ_μ .
We prove Theorem <ref> in several steps, first we show concentration with respect only to {Z_μ}_μ, {ξ^*_μ}_μ and {𝐗_μ}_μ using classical Poincare-Nash inequality, then with respect to {_μ}_μ using the corollary of Efron-Stein inequality, and then finally with respect to 𝐖^* with ^*. For this we rewrite
𝔼(1/nlog_t-𝔼_^*,𝐖^*,𝐗,ξ^*,,1/nlog_t)^2=
= 𝔼(1/nlog_t-𝔼_𝐗,ξ^*,1/nlog_t)^2
+𝔼(1/n_𝐗,ξ^*,log_t-_𝔼_𝐗,ξ^*,1/nlog_t)^2+
+𝔼(𝔼_ψ1/nlog_t-𝔼_𝐯^*,𝐖^*𝔼_ψ1/nlog_t)^2
where by 𝔼_ψ we denoted the joint expectation with respect to , ξ^*, 𝐗, and . Also for brevity, in what follows, by writing , ξ^*, 𝐗, and we mean the sets {Z_μ}_μ, etc.
We recall two classical concentration results, whose proofs can be found in <cit.>, Chapter 3.
Let ξ=[ξ_1,…,ξ_K]^⊺ be a real Gaussian standard random vector. If g:^K→ is a continuously differentiable function, then
g(ξ)≤∇ g(ξ)^2 .
Let ξ=[ξ_1,…,ξ_K]^⊺ be a random vector with i.i.d. elements taking values in some space 𝒜. If function g:𝒜^K→ satisfies
sup_1≤ i≤ Ksup_x_1,…,x_K, x'_i∈𝒜|g(x_1,…,x_i,…, x_K)-g(x_1,…,x_i^',…, x_K)|
≤ C
for some C>0, then
{g(ξ)}≤14KC^2.
In what follows we will denote P^y(y|x):=∂ P_ out(y|x)/∂ y and P^x(y|x):=∂ P_ out(y|x)/∂ x.
There exists a non-negative constant C(f,φ) such that
𝔼(1/nlog_t-𝔼_𝐗,ξ^*,1/nlog_t)^2≤C(f,φ)n .
Since ξ_μ^*, Z_μ and all elements of vectors 𝐗_μ are jointly independent for all μ, we have thanks to Proposition <ref>
_𝐗,ξ^*,(1/nlog_t)≤1/n^2∇log_t^2
=1/n^2∑_μ=1^n(∂log_t/∂ξ^*_μ)^2
+1/n^2∑_μ=1^n∑_i=1^d(∂log_t/∂ X^i_μ)^2
+1/n^2∑_μ=1^n(∂log𝒵_t/∂ Z_μ)^2=:I_1+I_2+I_3 .
For the sake of brevity we drop index out and write P_μ=P_ out(Y_t,μ|s_t,μ) in what follows. Gibbs brackets ⟨·⟩ are defined as in (<ref>).
After taking derivative in the first term we obtain
|∂log_t/∂ξ^*_μ|=|⟨P_μ^y/P_μ⟩ f^'(S_t,μ;_μ)√(tϵ)|
≤ c√(tϵ)C(f)(|Z_μ|^2+1) .
Last inequality is due to boundedness of f^' and Lemma <ref>. One can see that the only randomness left is in Z_μ. Since it is Gaussian, the average of polynomial is bounded by some uniform constant c. We obtained that each term in I_1 is bounded by constant, the number of terms is n, from this it follows immediately that I_1≤ C(f)/n.
The second type of partial derivative will give us
∂log_t/∂ X^i_μ=⟨P_μ^y/P_μ⟩ f^'(S_t,μ;_μ)(√(1-t)(𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*)_i/√(pd)+√(tρ)v^*_i/√(d))
+⟨P_μ^x/P_μ(𝐚∘φ^'(𝐖𝐗_μ/√(d))𝐖)_i/√(pd)⟩√(1-t)
+⟨P_μ^x/P_μv_i/√(d)⟩√(tρ) .
Plugging this into I_2 and using the simple inequality (a+b)^2≤2( a^2+ b^2) in order to square each term of the r.h.s. separately, we notice that it appeared the terms
⟨P_μ^x/P_μ(𝐚∘φ^'(𝐖𝐗_μ/√(d))𝐖)_i/√(pd)⟩^2 , ⟨P_μ^x/P_μv_i/√(d)⟩^2 ,
which depend only on Y_tμ, S_tμ and s_tμ. This allow us to use Nishimori identity by removing the brackets and adding * to , and . Then evaluating each ratio with P_ out using Lemma <ref> we get
I_2≤C(f) (1-t)/n^2∑_μ=1^n((|Z_μ|^2+1)^2𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*^2/pd)
+C(f)tρn((|Z_μ|^2+1)^2^*^2d) .
after what we notice that factors (|Z_μ|^2+1)^2 are independent of others and its expectation can be bounded with positive constant. In the end we obtain
I_2≤C( f)/n^2∑_μ=1^n𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*^2/pd
+C( f)n .
To finish the proof we notice that when we expand the norm appearing above all non zero terms will contain only squares, e.g., (a^*_iφ^'_iW^*_ij)^2, and so be positive. This gives as opportunity to bound φ^' with C(φ) in each term and calculate its expectation which is simply C(φ). The number of such terms is exactly pd, this gives us simple bound
I_2≤ C(f,φ)/n.
Finally, the derivatives with respect to Z_μ are of the form
∂log_t/∂ Z_μ=√(Δ)⟨P^y_μ/P_μ⟩ .
Similarly to what done above we bound I_3 with the help of Lemma <ref>
I_3≤C(f)/n .
Next step would be to prove the concentration of function _𝐗,,ξ^*log_t/n with respect to using Proposition <ref> while keeping 𝐖^* and 𝐯^* fixed.
There exists a constant C(f,φ)>0 such that
(1/n_𝐗,,ξ^*log_t-_1/n_𝐗,,ξ^*log_t)^2≤C(f,φ)/n .
We consider h()=_𝐗,,ξ^*log_t/n a function of all the elements _μ,i of _μ for 1≤ i≤ k and 1≤μ≤ n. We denote by ^' a vector such that ^'_μ,i=_μ,i for μ≠ν, i≠ j and ^'_ν,j is a random variable with distribution P_A, independent of all others.
According to Proposition <ref> it is sufficient to prove that
|h(^')-h()|<C(f,φ)/n .
If we denote by H (and H^') the Hamiltonian corresponding to _t (and _t with ^') one can see that
h(^')-h()
=1/n_𝐗,,ξ^*log⟨ e^H-H^'⟩_H≥1/n_𝐗,,ξ^*⟨ H-H^'⟩_H ,
the last step being true due to Jensen inequality. On the other hand h(^')-h()
≤_𝐗,,ξ^*⟨ H-H^'⟩_H^'/n.
We recall the definition (<ref>) of P_ out(Y_tμ, s_tμ) and similarly we obtain
H-H^'≥1/2Δ⟨ (f(S_tν;_ν^')-f(s_tν;)+√(Δ)Z_ν)^2-(f(S_tν;_ν)-f(s_tν;)+√(Δ)Z_ν)^2⟩_G^'
and
H-H^'≤1/2Δ⟨ (f(S_tν;_ν^')-f(s_tν;)+√(Δ)Z_ν)^2-(f(S_tν;_ν)-f(s_tν;)+√(Δ)Z_ν)^2⟩_G ,
where ⟨·⟩_G (or with G^') defined as
⟨·⟩_G=∫ P_A(d)e^-1/2Δ(Y_tν-f(s_tν;))^2(·)∫ P_A(d)e^-1/2Δ(Y_tν-f(s_tν;))^2
or with Y^'_tν where _ν is changed to ^'_ν. Since f is bounded we immediately obtain |H-H^'|≤ C(f)(|Z_μ|^2+1) and (<ref>). Now, due to the Proposition <ref>, the statement of the Lemma is proved.
The last part is to prove the concentration
of function g=_ψlog_t/n with respect to 𝐖^*,^*.
There exists a constant C(f,φ)>0 such that
(g-_𝐖^*,^*g)^2≤C(f,φ)/d .
Due to Poincare-Nash inequality we have
(g-_𝐖^*,^*g)^2≤∑_i,j^p,d(∂ g/∂ W^*_ij)^2
+∑_i^d(∂ g/∂ v^*_i)^2 .
Let us first deal with the partial derivatives with respect to W^*_ij
∂ g/∂ W^*_ij
=1/n∑_μ^n_ψ(⟨P_μ^y/P_μ⟩ f^'(S_tμ,_μ)√(1-t)a_i^*φ^'_iX_μ^j/√(pd)) ,
where φ^'_i=φ^'(𝐖^*_i𝐗_μ/√(d)).
Before integrating by parts with respect to X_μ^j let us notice that in the sum over μ all terms are the same since we are taking the expectation over all i.i.d. vectors 𝐗_μ, _μ, Z_μ and ξ_μ, it means that we can disregard the sum and just multiply by n directly.
Then we have
∂ g/∂ W^*_ij
=_ψ[∂ S_tμ/∂ X^j_μ√(1-t)a_i^*φ^'_i/√(pd)(-⟨P_μ^y/P_μ⟩^2f^'(S_tμ;_μ)^2+⟨P_μ^yy/P_μ⟩ f^'(S_tμ;_μ)^2+⟨P_μ^y/P_μ⟩ f^''(S_tμ;_μ))]
+_ψ[√(1-t)a_i^*φ^'_i/√(pd)(-⟨P_μ^y/P_μ⟩⟨P_μ^x/P_μ∂ s_tμ/∂ X^j_μ⟩ +⟨P_μ^yx/P_μ∂ s_tμ/∂ X^j_μ⟩)f^'(S_tμ;_μ)]
+_ψ[√(1-t)a_i^*φ^''_iW^*_ij/√(p)df^'(S_tμ;_μ)] .
Due to the Lemma <ref> absolute values of ratios of P's are bounded with C(f)(|Z_μ|^2+1). In the first term of expression above one can easily get rid of Z_μ since _ψ⟨(1+|Z_μ|^2)⟩<C. On the other hand derivatives of f and φ, in view of (Assum:f, remain bounded with non-negative constant C(f,φ), so combining all above and plugging into latter expression along with derivatives of S_tμ and s_tμ we obtain
| ∂ g/∂ W^*_ij|
≤C(f,φ) /√(d)_ψ|(𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*)_j/√(pd)a_i^*/√(p)|
+C(f,φ) /√(d)_ψ|v^*_j/√(d)a_i^*/√(p)|
+C(f,φ) /√(d)_ψ|(|Z_μ|^2+1)a_i^*/√(p)⟨(𝐚∘φ^'(𝐖𝐗_μ/√(d))𝐖)_j/√(pd)⟩|
+C(f,φ) /√(d)_ψ|(|Z_μ|^2+1)a_i^*/√(p)⟨v_j/√(d)⟩|
+C(f,φ)/√(d)_ψ|a_i^*W^*_ij/√(pd)| .
After using repeatedly (a+b)^2≤ 2 a^2+2 b^2 along with Jensen inequality one can show
∑_i,j^p,d| ∂ g/∂ W^*_ij|^2
≤C(f,φ) /d([𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*^2/pd^*^2/p]
+[(|Z_μ|^2+1)^2^*^2/p⟨𝐚∘φ^'(𝐖𝐗_μ/√(d))𝐖^2/pd⟩]
+[(|Z_μ|^2+1)^2^*^2/p⟨^2/d⟩]
+2) .
Two terms of the form [b⟨ c⟩] we bound by using Cauchy-Schwartz inequality, Jensen's inequality and Nishimori identity consecutively:
[b⟨ c⟩]≤^1/2[b^2]^1/2[⟨ c⟩^2]
≤^1/2[b^2]^1/2[⟨ c^2⟩]
≤^1/2[b^2]^1/2[ c^2] .
This allows us to rewrite (<ref>) as
∑_i,j^p,d| ∂ g/∂ W^*_ij|^2
≤C(f,φ) /d(^1/2[𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*^4/p^2d^2]
+C) .
What is left is to notice that in 𝐚^*∘φ^'(𝐖^*𝐗_μ/ √(d))𝐖^*^4/(p^2d^2) all non zero terms will have only even powers so we can bound φ^' with a constant in all of them, which gives immediately
∑_i,j^p,d| ∂ g/∂ W^*_ij|^2
≤C(f,φ) /d .
Now we consider the partial derivative with respect to v_i^*
∂ g/∂ v^*_i=1/n∑_μ^n_ψ[⟨P^y_μ/P_μ⟩ f^'(S_t,μ;_μ)√(tρ) X_μ^i/√(d)] .
As in the case with W_ij^* it is necessary to integrate by parts also with respect to X_μ^i since blind bounds will not give us the desired order. The result will be very similar to the previous calculation just this time we don't have factors φ^'_i and a^*_i. After similar simplification (bounds on ratios of P's, etc.) we obtain
| ∂ g/∂ v^*_i|
≤C(f,φ) /√(d)_ψ|(𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*)_i/√(pd)|
+C(f,φ) /√(d)_ψ|v^*_i/√(d)|
+C(f,φ) /√(d)_ψ|(|Z_μ|^2+1)⟨(𝐚∘φ^'(𝐖𝐗_μ/√(d))𝐖)_i/√(pd)⟩|
+C(f,φ) /√(d)_ψ|(|Z_μ|^2+1)⟨v_i/√(d)⟩| .
Similarly to the previous case it is easy to see that
∑_i^d| ∂ g/∂ v^*_i|^2
≤C(f,φ) /d([𝐚^*∘φ^'(𝐖^*𝐗_μ/√(d))𝐖^*^2/pd]
+[^*^2/d]
+[(|Z_μ|^2+1)^2⟨𝐚∘φ^'(𝐖𝐗_μ/√(d))𝐖^2/pd⟩]
+[(|Z_μ|^2+1)^2⟨^2/d⟩]) .
In order to be able to apply Nishimori identity to the last two terms we have first to use Cauchy-Shchwarz inequality to separate factors with noise Z_μ from the Gibbs bracket. Due to the fact that φ^' is bounded we obtain
∑_i^d| ∂ g/∂ v^*_i|^2
≤C(f,φ) /d .
This combined with (<ref>) finishes the proof.
§ PROOF OF PROPOSITION <REF>
The proof is implied by that of Theorem <ref>. We introduce a further set of data 𝒟̃_n:={(_ν,Ỹ_ν)}_n+1≤ν≤ n(1+ε) with responses
Ỹ_ν=√(λ)Y'_ν+Z̃'_ν , Y'_ν∼ P_ out(·| S_ν)
where λ≥ 0 and n+1≤ν≤ n(1+ε) for some ε≥ 0, Z̃_ν' are i.i.d. Gaussian variables independent of everything else, and where S_ν is defined as in (<ref>) but for the new inputs, with same teacher as used to generate 𝒟_n. We now define a proxy for the Bayes-optimal generalization error given the original and new data:
ℰ_n(λ,ε):=1/nε∑_ν=n+1^n(1+ε)[( Y'_ν-[ Y'_ν|𝒟_n∪𝒟̃_n])^2] .
This would recover the true definition of generalization if we set λ=0 in it. The quantity ℰ_n(λ,ε) can be obtained through the I-MMSE relation <cit.>
1/n∂/∂λI_n(';√(λ)'+'|,(_μ)_μ≤ n(1+ε))=ε/2ℰ_n(λ,ε) .
Following the general arguments in <cit.>, the mutual information on the l.h.s. is concave in λ. Moreover the proof of Theorem <ref> can be readily extended to take into account the additional side information (<ref>): indeed, the proof is exactly the same as before except that the “channel” (<ref>) generating the nε responses is slightly different than the original one P_ out. The new channel is equivalent to the original one once we rescale the readout f by √(λ) f and the noise variance Δ as λΔ+1 in (<ref>). This channel still verifies the same hypotheses. One then just need to keep track of the indices with responses generated according to the basic model (<ref>) and those from the rescaled channel (<ref>). In particular, it is possible to show the asymptotic equivalence of the quantities I_n(';√(λ)'+'|,(_μ)_μ≤ n(1+ε))/n and I^∘_n(';√(λ)'+'|,(_μ)_μ≤ n(1+ε))/n, where the second one refers to the equivalent GLM, with S_ν^∘=ρ^*⊺_ν/√(d)+√(ϵ)ξ_ν^*. Then by concavity one can use Griffiths' Lemma (see for instance <cit.>) to exchange the derivative with the limit n→∞ almost everywhere:
lim_n→∞1/n∂/∂λI_n(';√(λ)'+'|,(_μ)_μ≤ n(1+ε))=∂/∂λlim_n→∞1/nI_n^∘(';√(λ)'+'|,(_μ)_μ≤ n(1+ε)) .
The exchange fails when the model on the r.h.s., i.e. the GLM, presents a phase transition. Then one has to send both λ,ε→0. The authors of <cit.> showed that these limits commute with the n→∞ limit for the GLM. So the r.h.s. yields the optimal generalization error for the GLM proved in <cit.> that is therefore shared by the Bayesian two-layer neural network under study.
§ NISHIMORI IDENTITY
The Nishimori identities are a very general set of symmetries arising in inference in the Bayes-optimal setting as a consequence of Bayes rule. They were initially discovered in the context of the gauge theory of spin glasses <cit.>, which possess a sub-region of their phase space, called Nishimori line, where the most relevant thermodynamic quantities can be exactly computed, and we can generally count on replica symmetry, namely the concentration of order parameters <cit.>.
To introduce them, consider a generic inference problem where a Bayes-optimal statistician observes that is a random function of some ground truth signal ^*: ∼ P_Y|X(^*). Then the following holds:
For any bounded function f of the signal ^*, the data and of conditionally i.i.d. samples from the posterior ^j∼ P_X| Y( ·|), j=1,2,…,n, we have that
⟨ f(,^*,^2,…,^n)⟩=⟨ f(,^1,^2,…,^n)⟩
where the bracket notation ⟨ · ⟩ is used for the joint expectation over the posterior samples (^j)_j≤ n, is over the signal ^* and data .
The proof follows directly from Bayes' rule. An elementary proof can be found in <cit.>.
§ PROOF OF LEMMA <REF>
Let us start with proving the auxiliary Lemma where we combine all necessary bounds concerning derivatives of P_ out(y|x). In what below C(f) is a general constant that depends on f and also might depend on Δ. Below, upper indices represent partial derivatices, e.g., P_ out^x(y|x)=∂_x P_ out(y|x) and P_ out^xx(y|x)=∂_x∂_x P_ out(y|x).
Let y=Y_tμ=f(S_tμ;_μ)+√(Δ)Z_μ. Under assumption (Assum:f there exists constant C(f) such that
max{|P_ out^y(y|x)P_ out(y|x)|,
|P_ out^x(y|x)P_ out(y|x)|,
|P_ out^yy(y|x)P_ out(y|x)|,
|P_ out^yx(y|x)P_ out(y|x)|,
|P_ out^xx(y|x)P_ out(y|x)|}<C(f)(|Z_μ|^2+1) .
For convenience we recall here the definition of P_ out(y|x)
P_ out(y| x)=∫ dP_A()1/√(2πΔ)exp(
-1/2Δ(y-f(x;))^2
) .
It is easy to see that the ratio of any of these derivatives of P_ out with P_ out can be rewritten using an average
⟨·⟩_:=∫ dP_A()(·) e^
-1/2Δ(y-f(x;))^2/∫ dP_A()e^
-1/2Δ(y-f(x;))^2 .
After some algebra we get
P_ out^y(y|x)/P_ out(y|x) =⟨ -1/Δ(y-f(x;))⟩_ ,
P_ out^x(y|x)/P_ out(y|x) =⟨1/Δ(y-f(x;))f^'(x;)⟩_ ,
P_ out^yy(y|x)/P_ out(y|x) =⟨1/Δ^2(y-f(x;))^2⟩_-1/Δ ,
P_ out^yx(y|x)/P_ out(y|x) =⟨ -1/Δ^2(y-f(x;))^2f^'(x;)+1/Δf^'(x;)⟩_ ,
P_ out^xx(y|x)/P_ out(y|x) =⟨(1/Δ^2(y-f(x;))^2-1/Δ)f^'(x;)^2+1/Δ(y-f(x;))f^''(x;)⟩_ .
Since all expressions have a similar form we will treat only the last one, all others can be bounded in the same way. We have
|P_ out^xx(y|x)/P_ out(y|x)|≤⟨(1/Δ^2(y-f(x;))^2+1/Δ)f^'(x;)^2⟩_+⟨1/Δ|y-f(x;)||f^''(x;)|⟩_ .
When y=f(S_tμ;_μ)+√(Δ)Z_μ, since f is bounded along with its first two derivatives (see Assum:f we obtain immediately
|P_ out^xx(y|x)/P_ out(y|x)|≤
C(f)(|Z_μ|^2+1) .
With such bound one can see that after averaging such ratios (with or without ⟨·⟩ as in (<ref>)) with respect to Z_μ, we simply obtain a uniform bound C(f).
Now let us return to the proof of Lemma <ref>.
By definition one has [u'_Y_tμ(S_tμ)| S_tμ]=∫ dy P^x_ out(y| S_tμ)=0. For U_μν instead, we first need to realize that U_μμ= P^xx_ out(Y_tμ| S_tμ)/ P_ out(Y_tμ| S_tμ) which implies [U_μμ| S_tμ]=∫ dy P^xx_ out(y| S_tμ)=0. Concerning the off-diagonal instead, conditionally on S_tμ, S_tν the remaining disorder in the Y's is independent, so for μ≠ν we have [U_μν| S_tμ,S_tν]=∫ dy P^x_ out(y| S_tμ)∫ dy' P^x_ out(y'| S_tν)=0. For the boundedness of the derivatives it is sufficient to notice that
(u'_Y_tμ(S_tμ))^2=(P^y_ out(Y_tμ|S_tμ)/P_ out(Y_tμ|S_tμ))^2≤ C(f)(|Z_μ|^4+1) ,
the last inequality being true due to Lemma <ref> and the fact that Y_tμ=f(S_tμ;_μ)+√(Δ)Z_μ. Now it is immediate that after taking the expectation conditioned on S_tμ we obtain a bound C(f). In order to deal with the last quantity U^2_μν we rewrite
U^2_μν=(δ_μν(P^xx_ out(Y_tμ|S_tμ)/P_ out(Y_tμ|S_tμ)-(P^x_ out(Y_tμ|S_tμ)/P_ out(Y_tμ|S_tμ))^2)+P^x_ out(Y_tμ|S_tμ)/P_ out(Y_tμ|S_tμ)P^x_ out(Y_tν|S_tν)/P_ out(Y_tν|S_tν))^2 .
With the help of Lemma <ref> one can see immediately that U^2_μν≤ C(f)P(Z_tμ,Z_tν), where P is some polynomial with even degrees. Once again, after taking expectation we get a bound by a positive constant C(f).
§ PROOF OF LEMMA <REF>
For the reader convenience we repeat the statement of the Lemma below.
Recall ρ:=_𝒩(0,1)φ' and ϵ^2:=_𝒩(0,1)φ^2-ρ^2. Let φ̃ be either φ or the identity function. Under assumptions (Assum:phi and (Assum:f the following estimates hold:
_^*φ'(α_μ i)= ρ+O(‖_μ‖^2/d-1) ,
_^*φ^2(α_μ i)=_𝒩(0,1)φ^2+O(‖_μ‖^2/d-1) ,
_^*φ(α_μ i) φ̃(α_ν i)=ρ_𝒩(0,1)φ̃'_μ^⊺_ν/d+O((_μ^⊺_ν/‖_ν‖^2)^2)+O((_μ^⊺_ν)^2/‖_ν‖^2d)+O(_μ^⊺_ν/d(‖_μ‖^2/d-1)) ,
_^*φ'(α_μ i)φ'(α_ν i)= ρ^2+O(‖_μ‖^2/d-1)+O(_μ^⊺_ν/‖_ν‖^2) ,
_^*φ^2(α_μ i)φ̃^2(α_ν i)=_𝒩(0,1)φ^2_𝒩(0,1)φ̃^2
+O(‖_μ‖^2/d-1)+O(_μ^⊺_ν/‖_ν‖^2) .
Starting from (<ref>), using the fundamental theorem of integral calculus we get
|_^*φ'(α_μ i)-ρ|=|φ'(z√(‖_μ‖^2/d))-ρ| ≤∫_0^1 ds |z|/2|φ”(z√(s‖_μ‖^2/d+1-s))||‖_μ‖^2/d-1|/√(s‖_μ‖^2/d+1-s)
where the average is only over z∼𝒩(0,1). If we use the bound on the second derivative φ”≤K̅ we decouple completely s from z and we can compute both the average (|z|≤√( z^2)=1) and the integral, obtaining
|_^*φ'(α_μ i)-ρ|≤K̅|‖_μ‖^2/d-1|/‖_μ‖/√(d)+1≤K̅|‖_μ‖^2/d-1| .
Let us now focus on (<ref>). In the same spirit as the previous point:
|_^*φ̃^2(α_μ i)-_𝒩(0,1)φ̃^2| =|φ̃^2(z√(‖_μ‖^2/d))-_𝒩(0,1)φ̃^2| ≤K̅∫_0^1 ds |zφ̃(⋯)||‖_μ‖^2/d-1|/√(s‖_μ‖^2/d+1-s)
with φ̃'≤K̅ and the argument of φ̃ is z(s(‖_μ‖^2/d-1)+1)^1/2.
Here, before being able to integrate over s we need to bound the expectation |zφ̃(⋯)|. Recall that φ is Lipschitz, thus a simple bound is given by |φ̃(⋯)|≤K̅|(⋯)| using φ̃'≤K̅ and that φ̃(0)=0 as it is odd.
This yields immediately:
|_^*φ̃^2(α_μ i)-_𝒩(0,1)φ̃^2| ≤K̅^2|‖_μ‖^2/d-1| .
Consider now (<ref>). In what follows we drop the i-index for brevity. Let
α_μ⊥ν:=α_μ-α_να_μα_ν/^2 α_ν=α_μ-α_ν_μ^⊺_ν/‖_ν‖^2 ,
that is independent of α_ν. Now we expand φ around α_μ⊥ν:
_^*φ(α_μ)φ̃(α_ν)
=_^*φ'(α_μ⊥ν)_^*φ̃'(α_ν)_μ^⊺_ν/d+1/2_^*φ”(p)φ̃(α_ν)α_ν^2(_μ^⊺_ν/‖_ν‖^2)^2 .
The zero-th order is zero because φ is odd, and we have performed Gaussian integration by parts in the first order. p is a point in between α_μ⊥ν and α_ν. Now we expand again φ'(α_μ⊥ν) around the initial point α_μ:
_^*φ'(α_μ⊥ν)=_^*φ'(α_μ)-_^*φ”(p)α_ν_μ^⊺_ν/‖_ν‖^2=_^*φ'(α_μ)+O(_μ^⊺_ν/‖_ν‖^2)
where p has the same meaning as before, and we used φ”≤K̅. At this point it suffices to use (<ref>) on _^*φ'(α_μ) and _^*φ̃'(α_ν) and the estimate is proved.
Let us move to (<ref>). As for the previous point, we follow the orthogonalization procedure:
_^*φ'(α_μ)φ'(α_ν) =_^*φ'(α_μ⊥ν)_^*φ'(α_ν)+
_^*φ”(p)φ'(α_ν)α_ν_μ^⊺_ν/‖_ν‖^2
=_^*φ'(α_μ⊥ν)_^*φ'(α_ν)+O(_μ^⊺_ν/‖_ν‖^2) .
Now we use (<ref>) and (<ref>) to conclude:
_^*φ'(α_μ)φ'(α_ν)=ρ^2+O(‖_μ‖^2/d-1)+O(_μ^⊺_ν/‖_ν‖^2)
as in the statement.
Now we move to (<ref>). As before, we expand φ^2(α_μ) around α_μ⊥ν:
E_^*φ^2(α_μ)φ̃^2(α_ν)=_^*φ^2(α_μ⊥ν)φ̃^2(α_ν)+2_^*∫_0^1 dsφ(α_μ,ν(s))φ'(α_μ,ν(s)) φ̃^2(α_ν)α_ν_μ^⊺_ν/‖_ν‖^2
where α_μ,ν(s)=α_μ⊥ν+sα_ν_μ^⊺_ν/‖_ν‖^2. The integral on the r.h.s. can be bounded in different ways. For instance, one can first integrate the α_ν by part, recalling that α_μ⊥ν is independent of it, and then exploit the fact that both φ and φ̃ are Lipschitz. This yields the O(_μ^⊺_ν /‖_ν‖^2) in the statement.
The leading term _^*φ^2(α_μ⊥ν)φ̃^2(α_ν) can be split into _^*φ^2(α_μ⊥ν)_^*φ̃^2(α_ν) thanks to the orthogonalization. Expanding φ^2(α_μ⊥ν) around α_μ we get
_^*φ^2(α_μ⊥ν)=_^*φ^2(α_μ)-2∫_0^1ds _^*φ(α_μ,ν(s))φ'(α_μ,ν(s))α_ν_μ^⊺_ν/‖_ν‖^2
with α_μ,ν(s) as above. The integral contributes again with the same order as the one above, therefore
_^*φ^2(α_μ)φ̃^2(α_ν)=_^*φ^2(α_μ)_^*φ̃^2(α_ν)+O(_μ^⊺_ν/‖_ν‖^2) .
Finally, it only remains to apply (<ref>) to both the factors in the leading contribution on the r.h.s., which yields the missing remainder O(‖_μ‖^2/d-1) in the statement.
|
http://arxiv.org/abs/2307.03951v1 | 20230708105908 | High precision tests of QCD without scale or scheme ambiguities | [
"Leonardo Di Giustino",
"Stanley J. Brodsky",
"Philip G. Ratcliffe",
"Xing-Gang Wu",
"Sheng-Quan Wang"
] | hep-ph | [
"hep-ph",
"hep-th"
] |
sort compress
|
http://arxiv.org/abs/2307.07599v1 | 20230714194528 | JWST/CEERS Sheds Light on Dusty Star-Forming Galaxies: Forming Bulges, Lopsidedness and Outside-In Quenching at Cosmic Noon | [
"Aurélien Le Bail",
"Emanuele Daddi",
"David Elbaz",
"Mark Dickinson",
"Mauro Giavalisco",
"Benjamin Magnelli",
"Carlos Gómez-Guijarro",
"Boris S. Kalita",
"Anton M. Koekemoer",
"Benne W. Holwerda",
"Frédéric Bournaud",
"Alexander de la Vega",
"Antonello Calabrò",
"Avishai Dekel",
"Yingjie Cheng",
"Laura Bisigello",
"Maximilien Franco",
"Luca Costantin",
"Ray A. Lucas",
"Pablo G. Pérez-González",
"Shiying Lu",
"Stephen M. Wilkins",
"Pablo Arrabal Haro",
"Micaela B. Bagley",
"Steven L. Finkelstein",
"Jeyhan S. Kartaltepe",
"Casey Papovich",
"Nor Pirzkal",
"L. Y. Aaron Yung"
] | astro-ph.GA | [
"astro-ph.GA"
] |
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa, 277-8583, Japan Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA Physics & Astronomy Department, University of Louisville, 40292 KY, Louisville, USA Department of Physics and Astronomy, University of California, 900 University Ave, Riverside, CA 92521, USA INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA Dipartimento di Fisica e Astronomia "G.Galilei", Universitá di Padova, Via Marzolo 8, I-35131 Padova, Italy INAF–Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122, Padova, Italy Department of Astronomy, The University of Texas at Austin, Austin, TX, USA Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA ESA/AURA Space Telescope Science Institute Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA
We investigate the morphology and physical properties of a sample of 22 IR-selected dusty star-forming galaxies at Cosmic Noon (z ∼ 2), using James Webb Space Telescope Near Infra-Red Camera images obtained in the Extended Groth Strip field for the Cosmic Evolution Early Release Science survey.
The exceptional resolution of the NIRCam images allows us to spatially resolve these galaxies up to 4.4μm and identify their bulge/core even when very extinguished by dust.
Based on red-green-blue images using the F115W, F200W and F444W filters, we divide each galaxy in several uniformly colored regions, fit their respective Spectral Energy Distribution and measure dust attenuations, stellar masses, star formation rates and ages. After classifying each region as star-forming or quiescent, we assign galaxies to three classes, depending on whether active star-formation is located in the core, in the disk or in both.
(i) ∼ 70% of our DSFGs have a compact highly dust attenuated star-forming core that can contain up to 80% of the star-formation of the galaxy but only 20-30% of its stellar mass, and is always surrounded by a larger, less attenuated massive disk (no blue nuggets);
(ii) 64% (27%) of disks are significantly (strongly) lopsided, likely due to asymmetric cold gas accretion, major mergers and/or large scale instabilities;
(iii) 23% of galaxies have a star-forming core embedded in a quiescent disk, they are undergoing outside-in quenching, often facilitated by their strong lopsidedness inducing small and large scale instabilities;
(iv) some galaxies host highly heterogeneous disks in term of RGB colors: these are driven by in-homogeneous dust attenuation;
and (v) we find surprising evidence for clump-like substructures being quiescent and/or residing in quiescent regions.
This work demonstrates the major impact JWST/NIRCam has on understanding the complexity of the evolution of distant massive galaxies.
JWST View of DSFGs at Cosmic Noon
Le Bail et al.
JWST/CEERS Sheds Light on Dusty Star-Forming Galaxies: Forming Bulges, Lopsidedness and Outside-In Quenching at Cosmic Noon
Aurélien Le Bail1 Emanuele Daddi1 David Elbaz1 Mark Dickinson2 Mauro Giavalisco3 Benjamin Magnelli1 Carlos Gómez-Guijarro1 Boris S. Kalita4,5,6 Anton M. Koekemoer7 Benne W. Holwerda8 Frédéric Bournaud1 Alexander de la Vega9 Antonello Calabrò10 Avishai Dekel11 Yingjie Cheng12 Laura Bisigello13,14 Maximilien Franco15 Luca Costantin16 Ray A. Lucas7 Pablo G. Pérez-González16 Shiying Lu1 Stephen M. Wilkins17,18 Pablo Arrabal Haro2 Micaela B. Bagley15 Steven L. Finkelstein15 Jeyhan S. Kartaltepe19 Casey Papovich20,21 Nor Pirzkal22 L. Y. Aaron Yung23NASA Postdoctoral Fellow
August 12, 2023
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§ INTRODUCTION
Until recently, the existence of the so-called galaxy Main-Sequence, a correlation that the majority of star-forming galaxies observe in the stellar mass (M_*) versus star formation rate (SFR) plane up to redshift 3 (MS, e.g., ) and its tight scatter has been interpreted as evidence that star formation in most galaxies is a fairly ordered process ().
The ‘consensus’ is that galaxies on the MS are forming stars in a quasi steady state inside gas-rich stellar disks (e.g., ) whereas galaxies above the MS undergo a starburst, driven by stochastic processes such as major mergers, whose typical signature is compact star formation (e.g., ).
However, recent studies at z ∼ 1 - 3 have shown that some massive (M_*≥ 10^11 M_⊙) MS galaxies have a stellar distribution typical of late type galaxies but where the star formation only occurs in a compact nucleus (). The origin of these compact SF sub-mm galaxies (SMGs) observed with the Atacama Large Millimeter Array (ALMA) is yet to be fully understood. Three main scenarios to form the compact sub-mm nucleus are : (1) gas fueled to the core via violent disk instabilities (VDI) and clump migration, (2) a starburst induced by a major merger or (3) accretion and/or minor mergers (e.g. ). These compact SF nuclei could be an indication of an early quenching phase ().
Besides the compact nucleus, high-z SF galaxies are observed to have giant SF clumps (radius ∼ 1kpc). The origin of these clumps has been investigated by many studies (). <cit.> suggests that they can either be in-situ clumps, originating from VDI (e.g. ), in this case they are young and star-forming, or they can be ex-situ clumps, originating from minor mergers, in that case they will be older and with a low gas fraction and low specific star-formation rate (sSFR). A recent simulation showed that the formation of such long-lived giant clumps is only possible with a gas fraction of at least 50% (). This large gas fraction is necessary to induce VDI that will produce clumps that will migrate toward the center, creating strong gas nuclear inflow and triggering an evolution of the structure of the galaxy, leading to a morphological evolution (). This scenario is also favored by some observations (). More recently, <cit.> studied a local galaxy as proxy for high-z galaxies, confirming that the giant SF clumps mostly originate from a fragmentation of the disk, induced by VDI and not accretion or minor mergers. With its high spatial resolution, the James Webb Space Telescope's (JWST) near-IR Camera (NIRCam) is able to better resolve such giant SF clumps and could help constraining this scenario.
It is thus becoming clear that the galaxies within the MS scatter are not all largely unperturbed gas-rich disks. The compact SF cores, as well as the giant clumps, independently of their formation history, imply complex phenomenology at play, much different than local SF galaxies in the MS that are typically well behaved spirals.
Recently, emphasis has been brought onto other kind of asymmetries characterising high redshift SF galaxies.
<cit.> discovered strong lopsidedness affecting the three massive SF galaxies in a z=2.91 group core. They suggested a link between the lopsidedness of a galaxy in a dense environment to gas accretion and minor mergers. The lopsidedness would then be a marker of the point of impact of the accretion stream, following <cit.> who investigated the origins of lopsidedness in simulated galaxies. Their conclusion is that it is very unlikely that the lopsidedness is the result of internal mechanisms but is more likely to be linked to the assembly history and the environment of the galaxy, to asymmetric gas accretion and to minor merger and interactions with neighbouring galaxies. This is also the conclusion of studies on lopsidedness of galaxies in the local universe ().
<cit.> studied a galaxy in a dense environment with SF off-center substructures. They interpreted it as either forming spiral arms following a minor merger, an interaction with a neighbouring galaxy or a lopsided structure resulting from the point of impact of the cold gas accretion stream.
<cit.> reported JWST MIRI observations of GN20, an extremely luminous sub-mm galaxy residing in a z=4.05 protocluster (). They reveal a massive extended disk surrounding the sub-mm compact nucleus, displaying strong lopsidedness.
As of today, the lopsidedness has only been studied in dense environments and serendipitously. Observing lopsided disk in less crowded environment and inferring their prevalence in complete samples could shed further light on their presumed origin from interactions and accretion, and clarify whether a massive hosting dark matter halo is, or not, required.
By probing the rest-frame optical to near infrared (near-IR) at Cosmic Noon, JWST/NIRCam has a unique ability to fill the gap between the sub-mm compact nucleus observed with ALMA and the larger galactic disk observed in the optical and will help critically examining the competing scenarios. As an example, <cit.> recently studied substructures within a dusty star forming galaxy (DSFG) at z ∼ 3 imaged with both ALMA and JWST. From NIRCam images, they showed that the ALMA substructures are also visible at 4μm, demonstrating the direct link that one can draw between near-IR and sub-mm emissions. This suggests that the long wavelength channel of NIRCam might be a good tracer of compact obscured star formation in MS DSFGs.
The present study is part of the Cosmic Evolution and Epoch of Re-ionization Survey (CEERS[<https://ceers.github.io>]; ERS 1345, PI: S. Finkelstein) which is one of the Early Release Science (ERS) programs of the JWST () that observed a part of the Extended Groth Strip (EGS) Hubble Space Telescope (HST) field with NIRCam (). EGS is too far North to be observed with ALMA and there is no high resolution imaging with the Northern Extended Millimeter Array (NOEMA) yet. However, the high sensitivity and exquisite spatial resolution of NIRCam towards 5μm can be used as a surrogate to identify the most obscured and massive regions within galaxies, hence those most likely vigorously star-forming.
Understanding how DSFGs are formed and evolve is crucial to get the larger picture of galaxy formation and evolution, and it could be a key element to explain the quenching of galaxies at and after Cosmic Noon. To this aim, JWST/CEERS allows a major step forward. Indeed <cit.> already showed that JWST reveals the diversity of morphologies of galaxies at high redshift. JWST high spatial resolution and sensitivity is able to detect faint disks that were previously undetectable with HST. Moreover, a recent study by <cit.> uses JWST/NIRCam to probe the dust attenuation and sSFR of a lensed DSFG at z = 2.3. They demonstrate the power of JWST/NIRCam to precisely measure these properties at sub-galactic scales, allowing them to conclude that despite a more dust attenuated bulge, the color gradient of this galaxy is mainly driven by an early stage of inside-out quenching. This makes JWST/NIRCam the best instrument to investigate the morphological evolution of DSFGs around Cosmic Noon, in terms of compact star formation, giant clumps and galaxy structure.
The paper is organized as follows. In Sect. <ref> we present the data used in this study and the sample selection process. In Sect. <ref>, we detail the methods used to analyse each galaxy individually. In Sect. <ref>, we outline the main results of the analysis. Finally, in Sect. <ref>, we discuss the possible implications of the results in terms of formation and evolution of DSFGs at Cosmic Noon.
In this work, we adopt H_0 = 70km s^-1 Mpc^-1, Ω_M = 0.3, Λ_0 = 0.7, and a Chabrier IMF (). When necessary, we converted stellar masses and SFR from Salpeter IMF () to Chabrier IMF by subtracting 0.24dex.
§ DATA
§.§ CEERS Imaging
For the purpose of this study, we used the NIRCam imaging of CEERS, reduced using a customized pipeline by the CEERS collaboration (). It includes images in 7 filters: F115W, F150W, F200W, F277W, F356W, F410M and F444W for an average 5σ depth of 28.6 AB mag (See Table 3 of <cit.> for more details, each filter/pointing as a slightly different depth). The Point-Spread-Function (PSF) Full-Width at Half-Maximum (FWHM) of those filters range from 0.040” to 0.145” for F115W and F444W respectively[<https://jwst-docs.stsci.edu/jwst-near-infrared-camera/nircam-performance/nircam-point-spread-functions>]. For this study, we used the CEERS imaging from the June 2022 pointings, which represent 40% of the total area covered by NIRCam for CEERS between June and December 2022. We used the background subtracted images as we wanted to measure precise photometry.
As we needed to extract galaxy properties based on spectral energy distributions (SEDs), we decided to complement shorter wavelengths by taking advantage of the existing HST imaging in the field.
We used the publicly available HST data products version 1.9, available through CEERS. These mosaics were derived from HST archival data, but with improved calibration compared to the default pipeline products, and have astrometry tied to Gaia-EDR3 (). As described in the accompanying data release, the mosaics were created from the combination of HST programs 10134, 12063, 12099, 12167, 12177, 12547, 13063, and 13792, and the reduction and calibration followed a similar procedure to those described in <cit.>.
We used two filters; F606W and F814W with a PSF FWHM of 0.115” and 0.110” respectively (). We did not use the HST/WFC3 images as these bands are redundant for bright galaxies, as they are covered by JWST/NIRCam images which are deeper and with better spatial resolution.
§.§ The “Super-deblended" FIR catalog
The goal of this paper is to study the morphology and SF activity of DSFGs. We select galaxies based on their IR detection in the state-of-the-art super-deblended far-IR (FIR) catalog of the EGS field (Le Bail et al., in preparation).
FIR emission is a secure tracer of star formation (once the AGN components are removed), while optical/near-IR classification of SF galaxies is subject to larger uncertainties especially in the presence of dust. Hence, our FIR selection ensures the galaxies under scrutiny are truly highly SF.
The super-deblending is based on a well-established technique (). It is a multi-wavelength fitting technique meant to optimize the number of priors fitted at each band to extract the deepest reachable information. They used images from Spitzer (24μm (FIDEL, )), Herschel (100μm and 160μm (PEP, ), 250μm, 350μm, 500μm (HerMES, )), SCUBA2 (850μm (S2CLS, ), 450μm and 850μm from <cit.>) and AzTEC (1.1mm from <cit.>). The key was to obtain an adaptive balance as a function of wavelength between the density of priors fitted, the quality of the fit, and the achievable deblending given the PSF sizes. They started with the deepest images and fitted band after band toward shallower images. Extensive Monte-Carlo simulations ensured that the uncertainties associated to the flux measurements were “quasi-Gaussian” (see ; A. Le Bail et al. in preparation).
§.§ Sample definition
We selected all sources securely detected in the FIR catalog (see Sect. <ref>) that fell in the CEERS/NIRCam regions observed in June 2022. Since short wavelength channels have a slightly different field of view than long wavelength channels, we checked that the sources are observed in all of them and that they were not too close to the edge of the images so that there were not partially cut.
In detail, we require the galaxies to have SNR_FIR > 5, where SNR_FIR is the signal-to-noise ratios (SNR) added in quadrature from 100μm to 1.1mm (Le Bail et al. in preparation) and have at least one detection (SNR> 3) in a Herschel/SPIRE band after deblending (required to reliably measure SF components in case of AGNs).
The implication of the IR selection is that we don’t have a stellar mass complete sample of SF galaxies (e.g., complete above some mass threshold), and we have instead something closer to a (redshift-dependent) SFR limit. We are aware that we are missing SF galaxies below our IR detection threshold, as we wish to focus to highly (and securely) star-forming galaxies.
We also limited the sample to galaxies within 1.5<z<3.0, as we are willing to focus on galaxies at “Cosmic Noon”, as recalled in the Introduction.
To get accurate redshift estimates, we used the recent redshift compilation produced by <cit.>, which includes photometric redshifts based on CANDELS () as well as grism-based redshifts from 3D-HST () and spectroscopic redshifts from the MOSDEF survey ().
This sample comprised a total of 26 IR-detected sources. From these, 4 had to be rejected after a clean up. After close inspection, three galaxies were in a blended region and/or close to a much brighter IR source, making the Herschel measurements less reliable. The last rejected source hosted an AGN (clear radio excess, ∼ 10× brighter than what is expected for the radio continuum based on IR emissions, and X-ray detected: ID15327, RA = 215.82825, Dec = 52.80844, z_phot = 1.61, log_10(L_AGN/L_⊙) ≳ 11.3), hence the majority of its IR luminosity does not come from SF regions which are the main objects of this study.
This left us with a clean sample of 22 FIR-bright DSFGs around Cosmic Noon. We illustrate in Fig. <ref> the distribution of the sample in terms of stellar mass estimated in the pre-JWST era () and total IR luminosity (Le Bail et al. in preparation, calculated based on the equations in <cit.>) versus redshift (). We also show the distance from the MS () with a 0.6 dex total scatter () defined as Δ _MS = SFR_IR/SFR_MS. <cit.> uses a Salpeter IMF (), we converted stellar masses and SFRs from Salpeter IMF to Chabrier IMF by subtracting 0.24 dex. The red shaded region corresponds to the pure starburst region as defined in <cit.> (log_10(SFR_IR/SFR_MS) > 0.6 dex), we have two galaxies in our sample classified as pure starburst. The rest is mostly either within the scatter of the MS, but above its average trend, i.e. above the MS but below the starburst regime.
In Figs. <ref>, <ref> and <ref>, we show RGB cutouts of our sample of galaxies using the F115W, F200W and F444W filters of NIRCam. The galaxies are separated in three classes, as discussed in detail in the next Section.
figuresection
figuresection
§ METHODS
In this Section, we detail the methods used to analyze each galaxy, taking one of the objects (ID15371) as an example, to better clarify the procedure that we applied to all galaxies.
For each galaxy, we started by creating cutouts in each band (HST/ACS F606W, F814W and JWST/NIRCam F115W, F150W, F200W, F277W, F356W, F410M, F444W).
We show the cutouts of a DSFG in Fig. <ref> where one can already see by eye a difference between the disk visible in all bands and the center of the galaxy invisible in the HST images but getting brighter at longer wavelengths, justifying the need to study each component individually rather than the galaxy as a whole.
One of the first steps was to see if we could identify a bulge and a disk in each galaxies just like for ID15371, as discussed below.
JWST/NIRCam images have a spatial resolution ranging from 0.040” at 1.15μm up to 0.145” at 4.4μm. The larger 4.4μm PSF allows a resolution in physical size down to 1.23 (1.12) kpc for a galaxy at redshift 1.5 (3). This means that we were able to spatially resolve galaxy substructures down to a radius ∼ 0.6kpc. This made the resolution of F444W perfect for this study as we know the sizes of compact SF regions and giant clumps to be ∼ 1kpc ().
§.§ Measuring galaxy sizes
Several studies have shown that the regions of star-formation, either traced by the dust emission at 1.1mm observed with ALMA or by the radio continuum emission detected by the Very Large Array (VLA), are more compact than the optical size of the galaxy ().
JWST, with its sensitivity of the near and mid-IR, can detect both the obscured star-forming central part of each galaxy invisible with HST and the less obscured larger system, invisible with ALMA or VLA and bridge the gap.
To investigate this, we measured the total near-IR half-light radius (R_e,NIR) of each galaxy in the closest band to 1.6μm rest-frame (F410M or F444W filter depending on the redshift).
This rest-frame wavelength was chosen as it is a known tracer of the stellar mass of galaxies and is not affected by dust attenuation (). Moreover a recent study using NIRCam/CEERS data showed the excellent agreement between the near-IR size and the stellar mass size of galaxies around Cosmic Noon ().
We measured R_e,NIR from a curve of growth method, given that in all cases the PSF has a negligible effect (much smaller than any R_e,NIR).
The R_e,NIR was defined as the radius of a circular aperture, centered at the center of mass (barycenter) of the galaxy, which encompassed half of the total flux density of the galaxy at the considered wavelength.
To estimate the uncertainty, we used the fact that we typically have a 5% uncertainty on the measurement of the total flux of the galaxy (see Sect. <ref> for more details on the photometry measurements). We also measured the bias introduced when using a circular aperture for edge-on galaxies (like ID23510 in Fig. <ref>) by comparing the fluxes encompassed in an elliptical aperture and a circular aperture. The difference is about 5%. Hence, by changing the total flux of the galaxy within 10% we can estimate the uncertainty on R_e,NIR for which 50% of the total flux is encompassed.
We also measured the total optical half-light radius (R_e,O) of each galaxy in the closest band to 550nm rest-frame following the same procedure to compare it with R_e,NIR.
§.§ Identification of cores/bulges
Depending on the redshift, the F444W filter of NIRCam probes the rest frame near-IR between 1.1μm and 1.8μm which is a good tracer of stellar mass (). Hence, inspection of galaxy morphologies in this filter allowed us to search for the center of mass of each galaxy in our sample, or lack there-of, as a well defined peak in the F444W images. We were able to clearly identify a peak in the flux distribution of this filter for every galaxy. Depending on the galaxy, the peak was more or less pronounced, but always confidently there. We then defined a region in each galaxy encompassing the peak, as the core or the bulge of the galaxy. The regions are defined by eye as the peak is easily identifiable in every galaxies, the limit of the core is where the flux coming from the red F444W filter doesn't dominate anymore the RGB (F115W, F200W, F444W) color. Generally, a bulge is often defined in the literature as a quiescent central component with a high Sersic index (e.g., n∼ 4), and is a common component in local massive galaxies. In our study we did not attempt obtaining Sersic fits of separate components, and, more importantly, we anticipated that in many cases the central concentrations would not be quiescent, actually, most of them were highly SF and attenuated. We decided thus to call the central concentrations as cores when they were SF and bulges when they were quiescent.
They are represented by the regions delimited by the red dotted lines in all galaxies in Fig. <ref>, <ref> and <ref>.
We emphasize that for most of our sample, it would not have been possible to identify the center of mass only based on HST images (see e.g. ID15371 in Fig. <ref> as an obvious example).
This demonstrates once again the power of JWST when it comes to studying high-z DSFGs.
§.§ Lopsidedness
Having defined the core/bulge of each galaxy, we considered the rest to be the disk. Hence, we could obtain an evaluation of the lopsidedness for each galaxy. We considered it to be an important property to investigate because a lot of galaxies in our sample are obviously highly lopsided already by visual inspection (see for example ID11887, ID13776, ID18278, ID18694 in Fig. <ref> and <ref>). To quantitatively study this phenomenon, we defined two parameters: the eccentricity, defined as:
E = √((X_core - X_disk)^2 + (Y_core - Y_disk)^2/R_disk^2),
where (X_core,Y_core) and (X_disk,Y_disk) are the coordinates of the central core of the galaxy and of its disk respectively, while R_disk is the radius of the disk. The center of the core was simply defined as the pixel with the maximum flux density in the F444W filter. The center of the disk was defined as the barycenter of the disk measured in the rest-frame optical band (F150W or F200W depending on redshift). We measure it in the optical and not in the near-IR because the disk is less attenuated than the core, hence brighter than the core at these wavelengths. To not be biased by the core, we applied a circular mask centered on (X_core,Y_core) with a radius defined by the closest pixel to the center that has a F444W flux density less than half the core center flux density. Finally, R_disk was calculated using a circular aperture centered on (X_disk,Y_disk) encompassing half of the disk flux density.
This quantifies the eccentricity of the disk with respect to the core/bulge compared to its size and is a-dimensional.
The other quantity that we defined to probe the lopsidedness of the galaxies is the asymmetry. The asymmetry was calculated for the F444W NIRCam filter as we are trying to probe the mass distribution asymmetries and, as previously mentioned, F444W is the best tracer of the stellar mass distribution. We calculated the asymmetry by rotating each image by 180^∘ and subtracting it from the original image, the center of rotation was (X_core,Y_core) from Eq. <ref>. The asymmetry is defined as:
A = ∑_i=0^N| F_i - F_i^180^∘|/F_tot,
where F_i and F_i^180^∘ are the flux of the i-th pixel and its 180^∘ symmetric counterpart with respect to the center of the central core/bulge as defined in Equation <ref>. F_tot is the total flux of the galaxy. Since we worked on background subtracted images, we considered the background asymmetry to be negligible. This quantity describes how smoothly and how symmetrically the stellar mass is distributed around the central core/bulge of the galaxy and is also a-dimensional. Usually, the lopsidedness is probed using a Fourier decomposition (e.g. ). We decided to use a different, simpler method; the asymmetry, that has already been used in gas velocity space and was found to correlate well with the Fourier analysis of stars ().
§.§ Clumpiness
After identifying the core or bulge of each galaxy, we investigated the surrounding disk-like structures. Some of the galaxies have a smooth disk, others have a much more perturbed/complex disk morphology showing a large number of clumps (see Fig. <ref>, <ref> and <ref>).
We did not embark in a physical study of the clumps in this work. Our goal for this paper is to assess the presence or not of clumps in the disks and have an idea of how fragmented the disks are. Hence, we did not try to derive any physical properties of the individual clumps.
We decided to measure a clumpiness index, defined as the number of clumps in the disk of each galaxy. We counted the number of clumps visually identifiable in the RGB (F115W, F200W, F444W) image, making sure that the bulge/central concentration was not counted as a clump. This number varies from 0 up to 7 for the clumpiest galaxy. To be counted as a clump, the feature had to be compact compared to the galaxy size, and either had to have a different RGB color from the surroundings and/or appear as a local brighter spot. The clumps appear most clearly at the shortest wavelength (F115W or F200W filters), as expected (). For ID15371, we identify 4 clumps, there are shown by the white ellipses in the left panel of Fig. <ref>.
§.§ Spatially resolved photometry
To quantitatively study our galaxies, we needed photometry measurements. We decided to divide our galaxies in several components. For the simplest cases we only had the core/bulge and the disk, and when the disk had several clump/patches with different colors in the RGB image, we broke it down to several circular or elliptical regions. Each region was designed so that it had, qualitatively, a homogeneous (F115W, F200W, F444W) color. The division of the disk is once again done by visual inspection. We emphasize that we seek to study each region that has a different color, hence, if several clumps are close and with a similar RGB color, we consider them to be part of the same disk component.
Moreover, due to the spatial resolution of the PSF-matched images, we did not want to design too small regions that could lead to biased flux measurements. We tried to respect a balance between the size of the component we defined (not too close to the PSF size) and the homogeneity of the RGB color inside it.
We emphasize that the components are not necessarily concentric as most of the galaxies are not radially symmetric and are not limited in number. If we observed, for example, two blue disconnected patches in a galaxy, we defined them as two different components and fitted them individually. In the case of ID15371, we divided the galaxy in three regions, the red central core/bulge, the bluer disk and an intermediate region, that is still part of the disk but close to the red core and with intermediate colors (see Fig. <ref>).
In terms of rest-frame colors, since our sample of galaxies is distributed across z ∼ 1.5 to z ∼ 3, F115W probes the rest-frame near-UV/blue (300-460nm), F200W probes the rest-frame green/red (500-800nm) and F444W probes the rest-frame near-IR (1110-1780nm). The scatter in rest-frame wavelength is less or equal to the band-width of each filter. This means that we globally probed consistent colors between galaxies.
By dividing each galaxy in sub-galactic regions, there was a risk that small regions get close to the PSF FWHM of some filters. Hence, leading to an underestimation of the flux at the longest wavelengths, and an artificial deformation of the SED. To avoid this, we decided to work on PSF-matched images using the broader PSF of the F444W filter.
In Fig. <ref>, we show RGB images of the DSFG ID15371 using (F115W, F200W, F444W) before and after PSF-matching. To make sure that we didn't underestimate stellar masses and SFR when running the SED fitting, we chose regions larger than the PSF FWHM (0.145”).
In Fig. <ref>, <ref> and <ref>, for each galaxy we overlay the delimitation of the different components we decided to study separately based on their color (those RGB images are showed before PSF-matching).
After having defined the regions to study, we measured the flux in each band for each region. To do so, we summed the value of each pixel in each region of the science image. The pixels were counted only once, meaning that the flux in the smaller regions (like the red ellipse for ID15371) was not included when calculating the flux of larger regions (like the green ellipse for ID15371, see Fig. <ref>).
Our goal was to fit the SED of the different components of each galaxy. For the properties that we later extracted from these SEDs to be reliable, it was crucial that we had reliable uncertainties on the fluxes.
To estimate the flux uncertainties, we re-normalized the errors propagated via the Root Mean Square (RMS) images. The uncertainty was defined as:
df = f_λ,N×√(∑_i=0^Nσ_i^2),
where the sum was made on all pixels in the region, σ_i is the RMS of the pixel i, and N the total number of pixel in the region. We decided to define f_λ,N, a normalisation factor that takes into account extra noise, e.g. from the correlated signal between pixels that is particularly important for the long wavelength filters that were drizzled from a pixel size of 63mas to 30mas. To calculate this factor, we measured the flux dispersion in ∼ 20 empty regions of the science image for several apertures in each band. We then compared this value to the RMS calculated from the RMS image in apertures of the same size and the normalisation factor is defined as their ratio. To be conservative, we never applied a factor leading to lower uncertainties. These factors are generally small, (f_λ,N∼ 1.5 at most).
§.§ SED Fitting
To characterize our sample of galaxies, we needed to have access to their resolved M_* and SFR. To this aim, we fitted each galaxy component SED using the Code Investigating GALaxy Emission (CIGALE, ).
We used a single declining exponential model also known as “τ model" to model the star formation history of each galaxy.
We adopted the <cit.> model for computing the spectral evolution of single stellar populations with a fixed solar metallicity of Z = 0.02 which is reasonable for M_*∼ 10^10-12M_⊙ DSFGs following the Mass-Metallicity relation ().
After testing with and without including nebular emissions, we decided not to include them as, for our sample, they lead to higher χ ^2 with no noticeable effect on the extracted properties (A_V, SFR, M_* and redshift).
Some galaxies showed possible signature of strong emission lines, visible as green patches/clumps in Figs. <ref>, <ref> and <ref>. However, including them had a negligible effect on the estimation of the SFR since it usually had a 50% uncertainties. We discuss this in more detail in Sect. <ref>.
We used a modified <cit.> dust attenuation law and the <cit.> dust emission models update from 2014 to predict FIR flux densities. The idea behind the modification of <cit.> model is that young stars embedded in their birth cloud suffer from additional attenuation compared to stars that have broken out and escaped into the ISM, and that the attenuation curves associated to the birth cloud and the ISM must be different. In practice, this is modelled
by assuming two different power-law attenuation curves of the form A(λ) ∝λ ^δ: one for the birth cloud with a slope of δ _BC = -1.3, and one for the ISM with a slope of δ _ISM = -0.7. Because radiation from young stars has to travel through both the birth cloud and the ISM to escape the galaxy, the spectrum of stars younger than 10Myr are attenuated by both the birth cloud and ISM curves. Stars older than 10Myr are only attenuated by the ISM curve ().
For the redshift, we used the <cit.> catalog, as well as the latest redshift catalog published by <cit.>. We encountered three different cases:
* If we had a high-quality spectroscopic redshift, then we used it and fixed it. We have 5 galaxies with a spectroscopic redshift.
* If we had a grism-based redshift from 3D-HST, we downloaded the spectrum and examined its quality, actual features detected, the redshift probability distribution and defined the redshift and its uncertainty accordingly. We have 10 galaxies for which we find a high-quality grism-based redshift.
* If we only had photometric data, we allowed (1+z) to vary within ± 10%. We have 7 galaxies with a photometric redshift.
In Fig. <ref>, we show the best SED models corresponding to each region of our example galaxy defined in Fig. <ref>.
To be able to extract reliable information from the SED fit, it was crucial to check the fit quality. To be conservative and have reasonable χ ^2, we decided to limit the photometric accuracy of each band to S/N = 20.
However, if the CIGALE fit returns high χ ^2 values, there is a possibility that the input flux uncertainties are still underestimated. In that case, we increased the uncertainties by adding up 10% of the flux to the error in each band.
To consider the fit acceptable, we want the reduced χ ^2 such as χ _red^2≤ 1.67, which is the reduced critical value corresponding to a significance level of 10% in the χ ^2 test for 8 degrees of freedom.
To estimate the robustness of the best model, we studied the χ ^2 distributions associated to the 3 main free input parameters: the dust attenuation, the age of the stellar population and the e-folding time. In Fig. <ref>, the upper-left panel shows the χ ^2 distribution associated with the different values of the dust V-band attenuation A_V of the stellar continuum used to fit the SED of the red core of the DSFG ID15371. The upper-right panel shows the same information for t/τ with t and τ being the age of the oldest stars and e-folding time of the stellar population used to define the star formation history of the galaxy. Taking the width of these distribution at χ ^2_min + 1 and χ ^2_min + 2.7 give us the 68% and 95% confidence interval respectively (), illustrated by the horizontal thick and thin dashed lines in Fig. <ref>. The fact that we see only a portion of the distribution for t/τ comes from the fact that the age is getting close to the age of the Universe, allowing larger t would not make physical sense.
We can use the same reasoning for the properties extracted from the SED like the M_* or the SFR averaged over the last 10Myrs. We show an example in the lower panels of Fig. <ref>.
Just by looking at Fig. <ref>, we can already conclude that the red core of the DSFG ID15371 is dusty (A_V ∼ 2.73) and weakly star-forming (SFR ∼ 18 - 40 M_⊙/yr and t/τ ≫ 1).
As a sanity check, we estimated the SED of the whole galaxy by summing up the SEDs of all the components. We then compared this SED with the near-IR and FIR flux densities measured in the super-deblended catalog (, Le Bail et al. in preparation) to make sure that they were consistent. If the FIR flux densities are brighter than predicted by the SED fitting, it can be a hint that this galaxy is in a starburst episode and/or that there is a deeply attenuated component that is not visible even at 4.44μm. It can also be due to the presence of an AGN that boosts the FIR flux, this can be confirmed by a radio excess or an X-ray detection (Le Bail et al. in preparation, ). We recall that we removed from the sample only one galaxy where we knew that the FIR luminosity was dominated by the AGN luminosity (see Sect. <ref>) but kept those where the AGN luminosity didn't dominate the FIR luminosity.
On the contrary, if the SED predicts a FIR flux density brighter than the one measured, it means that there is a problem in the fitting possibly linked to the grid of the input parameters. In Fig. <ref>, we show the comparison between the total SED of the galaxy ID15371 and the FIR flux densities. For this galaxy, the flux densities are consistent with the predicted FIR SED meaning that there is no hidden component. This is the case for all the galaxies in our sample except one (ID13107 for which we have a FIR detection brighter than the SED model, pointing toward either a deeply attenuated component or an AGN even though there is no AGN signature in X-ray or radio). However for 3 galaxies (ID13098, ID13776 and ID31281), the measured 100μm flux is boosted compared to the SED predicted flux, possibly a signature of a hot AGN, 2 of them have an X-ray detected AGN (). By observing Fig. <ref>, one can notice that the predicted IRAC fluxes are fainter than the actual measurement. This observation is not true for every galaxy, we measure the flux in the NIRCam F356W and F444W which probe the same wavelength as IRAC channel 1 and 2 to be fainter for this galaxy. This is mostly a sign of blending in the earlier IRAC imaging.
A caveat of this SED fitting method is that we used the same SFH and parameters for all regions, some with very different properties. We chose to use the simple tau model because of the meaning of t/τ regarding the star-forming activity of the galaxy. We decided to make a two-pass SED fitting, in the first pass, the goal was to separate the star-forming from quiescent regions. In the second pass, we fitted the star-forming regions with a nearly constant SFR (by imposing τ≫ t).
This allowed to have a good estimate of the recent SFR. Moreover, by comparing it to the far-IR SFR from the super-deblended catalog (Le Bail et. al, in preparation.) and to the relative position of each component with respect to the Main Sequence or the quiescent quadrant of the UVJ color-color diagram (see Sect. <ref> for more detail on these last two points), we had a confirmation of the star-forming activity of each galaxy component. For the quiescent regions, there can be a degeneracy between the age and the dust attenuation, to tackle this, we imposed t ≫τ.
We estimated that the good quality of the photometry in the rest-frame near-IR and the two-pass SED fitting procedure allowed us to get robust estimates of both the stellar mass and the SFR of each component.
To tackle the great diversity of galaxies, we decided to divide them into several classes as defined in the next Section.
§.§ Classification
From the CIGALE SED fitting, we derived an estimation of the M_*, the SFR and the dust attenuation (A_V) of each component of the galaxies. For the galaxy ID15371, in the upper panel of Fig. <ref>, one can see the three components respective M_* and SFR plotted on the MS (). All the components of the DSFG ID15371 have some ongoing star-formation, with the red core being on its way to quenching but still slowly star-forming.
Using the best SED models provided by CIGALE, we also estimated the rest-frame U, V and J AB magnitudes. We used <cit.> U and V filters, and for the J band, we used the 2MASS J relative spectral response curve. In Fig. <ref>, we display all the regions of our galaxies on the (V-J, U-V) plane. We recover the sSFR effect: when moving from the lower right corner to the upper left corner, the sSFR decreases (). This makes the UVJ color-color diagram ideal to separate SF galaxies from quiescent galaxies. We note that the galaxies with sSFR ≲ 0.1Gyr^-1 are all in the quiescent region defined by <cit.> and delimited by the black dashed line in the Figure. The colored dotted lines delimit the regions defined by <cit.>.
For the DSFG ID15371, we have confirmation in the UVJ diagram that all the components are star-forming (lower panel of Fig. <ref>). Moreover, the three components are aligned on the diagonal of the diagram which is the signature of a gradient of dust attenuation from the center towards the outer parts (). Indeed, from the SED fitting, we had A_V,red = 2.70 ± 0.11 > A_V,green = 2.09 ± 0.23 > A_V,blue = 0.75 ± 0.11.
Generally, to estimate if a region was SF or quiescent, we used the UVJ color-color diagram (is the component in the quiescent quadrant or not ?), the position relative to the MS (is the component on/above the MS or well below it ?, what is its position compared to the other regions of the same galaxy ?) and, as we used a simple exponential model for the star-formation history, the value of t/τ is also an indicator of the star-formation activity. If t/τ≫ 1, then the peak of star-formation is firmly in the past, and the component is on its way to quenching. On the contrary, if t/τ≲ 1, the galaxy is still actively star-forming. Based on these three pieces of information, we were able to discriminate between SF and quiescent regions. Here, we defined a region as quiescent if below the Main Sequence and the other galaxy components (by ∼ 0.6dex). Hence, some regions that we classified as quiescent are not completely passively evolving and could still be slowly star forming. Most of the time, the three indicators are in agreement, however, in some cases the results were ambiguous: the regions where all three indicators where not in agreement represent less than 5% of all the studied regions. In that case, we first looked at their position relative to the MS to see if it was consistent with the t/τ values from the best models and it always was. The inconsistency of the UVJ color-color diagram can be explained in several ways: the UVJ diagram uses only a part of the information (3 rest-frame bands) contrary to the other probes that uses the full SED. More importantly, real situations exist where the UVJ diagram characterizes correctly the presence of star formation but this star formation is suppressed as exemplified by the sub-MS location (suppressed with respect to the ensemble average given the mass) and by the t/τ (suppressed with respect to the past star formation history of this galaxy). That is the case for the central region of ID15371 (see Figs. <ref> and <ref>). In the rare cases where the t/τ value didn't allow any conclusion (t/τ∼ 1), we decided based on the MS and the UVJ color-color diagram that were consistently pointing either toward star-formation or quiescence. In all the cases, we were able to classify the regions as quiescent or star-forming.
As a result of this analysis, we had 22 vastly different galaxies with various morphologies, colors (see Figs. <ref>, <ref> and <ref>) and star-formation activity. We found that the variety of features could be meaningfully re-conducted to three galaxy groups:
* Type I: SF disks with a red SF core, characterized by the fact that all their regions are SF. Some have a complex multi-color clumpy disk morphology in the RGB (F115W, F200W, F444W) image. They all have a dust attenuated red SF core.
* Type II: Quenched disks with a SF core, characterized by a dust attenuated red SF core and a quenched disk (in one case, partially quenched).
* Type III: SF disks with a quenched bulge, characterized by a quenched central bulge while the disk is still star-forming. These are similar to local spirals.
For the disks with several components, they usually were all either SF of quiescent. There was only one galaxy (ID18278) where only a fraction of the disk was quiescent (green region in Fig. <ref>), we decided to include it to the Type II as the quiescent part encompasses 16% of the disk stellar mass and could be considered as an early stage of quenching. 4 galaxies hosts X-ray AGNs that do not dominate the FIR emissions; 1 is a Type I galaxy (ID30186), 2 are Type II galaxies (ID13098 and ID13776) and the last one is a Type III galaxy (ID23205) ().
After having classified our sample of 22 galaxies, we had 10 Type I galaxies, 5 Type II and 7 Type III. The RGB cutouts of our sample are separated following the three Types, with Figs. <ref>, <ref> and <ref> showing the Type I, II and III galaxies respectively. This is summarized in the top panel Fig. <ref> where each wedge size is proportional to the number of galaxies of the considered Type. We illustrate each Type with a pictogram, the color red representing quiescent regions and the color blue representing star-forming regions. The color of each wedge is linked to the Type, in all Figures in the rest of this paper, the red markers will represent Type I galaxies, green markers Type II and blue markers Type III.
The lower panels of Fig. <ref> summarize the properties of each Types by looking at the connection between sSFR and A_V and color (in mag AB) gradients. The first observation is that cores/bulges are systematically redder than disks and there is a strong correlation between A_V gradient and color gradient (Pearson coefficient = 0.83, p-value = 2e-6) while there is no correlation between sSFR gradient and color gradient (Pearson coefficient = 0.27, p-value = 0.23). This means that the color differences that we observe in Figs. <ref>, <ref> and <ref> trace dust density in-homogeneities and not older/younger stellar population.
The Type I galaxies (in red) do not have a noticeable sSFR gradient (sSFR_core∼ 1.2× sSFR_disk), but have a strong A_V gradient, hence, the fact that the cores of Type I galaxies appear much redder than the disks in Fig. <ref> is due to their high dust density; the blue regions are low A_V regions.
For the Type II galaxies, we observe the sSFR gradient we expected, the core is star-forming while the disk is quenched (sSFR_core∼ 6.5× sSFR_disk), they have the strongest dust gradient because of their highly dust attenuated core and their quenched disk that has low level of dust attenuation. We note that the sSFR gradient should make the core appear bluer than the disk (because of the younger stellar population in the core), however we observe the exact opposite. The color gradients we observe in Fig. <ref> are dominated by the dust attenuation gradient.
Eventually, Type III galaxies have low attenuation both in their quenched bulge and star-forming disk, hence have a weak A_V gradient. Their sSFR gradient is however strong, as expected of the opposite sign compared to Type II (quenched bulge and star-forming disk, sSFR_core∼ 0.2× sSFR_disk). In Fig. <ref> the color gradients mostly trace the age difference between the stellar populations of the (redder) bulge and the (bluer) disk.
We note that, the strong gradients we observe, both in sSFR and A_V justify the need to divide our galaxies in three Types to illustrate the three possible sSFR gradient and to divide them in several sub-galactic regions because of the huge dust gradient. Moreover, as expected by the selection criteria detailed in Sect. <ref>, we did not have any fully quiescent galaxy in our sample.
§ RESULTS
In this section, we present the results of the analysis of the 22 galaxies in our sample, distinguishing among the three classes we just defined in the previous Section.
We first looked at the properties of the whole galaxies in Sect. <ref> and then at the resolved properties at a sub-galactic level in Sect. <ref>.
In Table <ref>, we give the main properties of our sample of 22 galaxies.
In the following, we compared the behaviour of the different Types of galaxies. To assess the significance of the trends, we compared the difference between the mean of a property for each Type with the error on the mean. We emphasis that we also checked the median value and that it doesn't affect the observed trends. In the Figures, each star-shaped marker is the mean and the error bar is the error of the mean (defined as err_mean = rms/√(N) with rms the root mean square of the distribution and N the number of galaxy in each Type).
§.§ General properties
§.§.§ Main Sequence galaxies
To characterize the different Types of galaxies, we first looked at their typical redshift, M_* and sSFR_IR. The redshifts and M_* were extracted from the SED fitting procedure described in Sect. <ref> while the sSFR_IR was computed by dividing the SFR_IR of each galaxy by the sum of the M_* of each component with the SFR_IR taken from the super-deblended catalog (Le Bail et al., in preparation).
In Fig. <ref>, a redshift trend is appearing: the Type I galaxies with their SF core and SF disk are on average at higher redshift (z= 2.32 ± 0.15) than the Type II galaxies with their SF core and quiescent disk (z = 1.94 ± 0.11), that are themselves at a slightly higher redshift than the Type III galaxies (z = 1.80 ± 0.09), analogs to the spiral galaxies we observe in the local universe with a quiescent bulge within a SF disk. The difference in redshift is 2σ between the Types I and II and 3σ between Type I and III. This suggests that this redshift trend is real and opens the possibility of an evolutionary link between class I and II/III.
All of our galaxies have a M_* > 10^10M_⊙ with an average of M_* = 8.2^+2.2_-1.7× 10^10M_⊙ (left panel of Fig. <ref>). There is no correlation between the Types and the M_*, all Types have a similar average M_*.
By comparing the sSFR_IR of our galaxies with the MS of <cit.> (right panel of Fig. <ref>), we confirmed that typically these galaxies are MS galaxies, consistently with Fig. <ref> and Sect. <ref>. The MS sSFR at a fixed redshift was calculated by taking the mean M_* of our sample which is <M_*> = 10^10.92M_⊙. Moreover, the typical sSFR_IR is observed to decrease at lower redshift, as expected from the cosmic trend.
The Type III galaxies, which have a quenched bulge, have the weakest sSFR_IR on average (sSFR_IR = 0.75^+0.18_-0.14Gyr^-1 for quenched bulges versus sSFR_IR = 2.01^+0.81_-0.56Gyr^-1 for SF cores). They also are at lower redshift than the others. This suggests that they are more evolved than other classes.
§.§.§ Galaxy near-IR sizes
The presence of highly obscured cores at the center of galaxies, like for ID15371 (see Fig. <ref>), can let us believe that we are studying the counterparts of the ALMA compact SF SMGs. Indeed, SMGs are known to be compact, dust obscured and with a high star formation efficiency.
The galaxies hosting a SF region at their center (Type I and II) tend to be slightly more compact in the near-IR, with R_e,NIR = 2.34 ± 0.37kpc, than the galaxies with a quenched bulge (Type III), with R_e,NIR = 2.93 ± 0.42kpc (this is tentative as there only is a 1σ difference, see Fig. <ref>).
The Type II galaxies and their quiescent disk are on average the most compact galaxies in the near-IR with a typical size of 2.19 ± 0.30kpc.
In Fig. <ref>, we compare the R_e,NIR to the M_*-R_e relation from <cit.> based on rest-frame optical measurements. Most of our 22 galaxies are more compact in the near-IR than in the optical, with ∼ 40% being below the M_*-R_e relation scatter. We also checked that the optical sizes of our galaxies are compatible with the M_*-R_e relation.
This demonstrates than in our galaxies, the dust, traced by the near-IR emissions, is more concentrated than the stellar light, traced by the optical emissions.
This is a confirmation of an already well established fact ().
However, we note that the Type I galaxies have very comparable optical and near-IR sizes (∼ 15% difference in size on average), their star-forming core is not as concentrated as for the other galaxies of the sample.
We discuss in Sect. <ref> how the Type I and II galaxies might relate to the ALMA SMGs.
However, studying the half-light radius is not enough as a large fraction of the galaxies in our sample are not symmetric (see Figs. <ref>, <ref> and <ref>).
§.§.§ Widespread Lopsidedness
As one can see in Figs. <ref>, <ref> and <ref>, some galaxies are strongly lopsided (marked with a `L'). They are asymmetric and/or their red central region is off-centered with respect to the disk. This lopsidedness appears to be quite common among Type I and II galaxies. In Figs. <ref>, <ref> and <ref>, the marked galaxies are the 6 most lopsided galaxies, 3 are Type I (30% of the sample) and 3 are Type II (60% of the sample). The Type III galaxies look much more symmetric, these galaxies have a quenched bulge and are on average at lower redshift, they had presumably more time to evolve and stabilize their disk. To verify this, we investigate the lopsidedness of each galaxy.
As explained in Sect. <ref>, for each galaxy we calculated its asymmetry (A) and eccentricity (E). Type III galaxies appear to be much less lopsided, they have a low eccentricity (9.8± 2.5 %) and asymmetry (22.8± 3.0%) while Type I and II galaxies, which show comparable lopsidedness, tend to be much more asymmetric (33.0± 3.5%) and off-centered (30.3± 4.0%) (see upper panel of Fig. <ref>). The difference has a 4.3σ and 2.2σ significance for the eccentricity and asymmetry respectively. In the upper panel of Fig. <ref>, we show the eccentricity vs the asymmetry. We considered the Type III galaxies as not lopsided, and used their typical eccentricity and asymmetry as a proxy for measurement errors and systematic effects. The thin black dotted line shows the threshold to define a galaxy as weakly lopsided (A + E > 0.37, this value corresponds to the average [A+E]+1σ of Type III galaxies). We have 14 galaxies that are at least weakly lopsided, representing 64% of the sample. If the galaxies are above the thick black dashed line, meaning that A+E>0.70 (this value corresponds to the average 2× [A+E]+1σ of Type III galaxies), we consider them as strongly lopsided, we encircled them in Fig. <ref> and they are visible in Figs. <ref>,
<ref> and <ref> with a L. We have 6 strongly lopsided galaxies, representing 27% of the sample. Usually, a strong asymmetry is linked to a strong eccentricity, however we have galaxies with a low level of asymmetry but with a highly off-centered disk. All the strongly lopsided galaxies (circled in black) have high eccentricity.
In other words, we observe a lack of strong asymmetry with low eccentricity.
The position of the average lopsidedness of Type I and II galaxies in Fig. <ref>, indicates that being lopsided might be a typical property of these galaxies.
In the lower panels of Fig. <ref>, Type III galaxies, which are more evolved and have a quiescent bulge have low level of asymmetry. On the contrary, Type I and II galaxies have a higher level of asymmetry. We observe (1) a lack of galaxies with a compact disk and high asymmetry and vice-versa, (2) a lack of galaxies with a high core mass fraction and high asymmetry and vice-versa and (3) the galaxies with a quiescent bulge with high mass fraction have low asymmetry. This is consistent with the observation of galaxies in the local universe, indeed, present-day late-type galaxies with more extended disks and lower central stellar mass density are typically more lopsided than early-type galaxies with smaller disks and higher central stellar mass density (). It seems that as the core grows in mass from accretion, the disk gets smaller and loses its lopsidedness, leading to Type I spiral-like galaxies.
Thanks to the spatial resolution of JWST, we had access to sub-galactic scales, which is crucial to understand the morphology and evolution of DSFGs.
§.§ Resolved properties
For each galaxy, each component has been classified either as star-forming or quiescent (see Sect. <ref>). In Fig. <ref>, we show that the quiescent regions are massive (M_*≳ 10^10M_⊙) and have a relatively low dust attenuation with an average of A_V∼ 1.6 and maximum at A_V∼ 3 while SF regions have an average of A_V∼ 2.3 maximum at A_V∼ 5.4. The SF regions follow a correlation (with a Pearson coefficient of 0.62, p-value = 9e-8), the more massive components are more attenuated. This is consistent with the idea that the stellar mass is the main driver of dust attenuation in SF galaxies ().
In the following Sections, we present the results regarding the core/bulge and disk of our galaxies.
§.§.§ Cores and bulges properties
We first looked at the red central region of each galaxy, as defined in Figs. <ref>, <ref> and <ref>. In the left panel of Fig. <ref>, we show the dust attenuation versus the M_* of the red star-forming cores (in red and green) and quiescent bulges (in blue). As mentioned above, the dust attenuation of SF cores (Type I and II) correlates with its M_*: the more massive the core, the more dust attenuated (with a Pearson coefficient of 0.75, p-value = 0.001). Also, the bulges are less attenuated than SF cores, consistent with the fact that they are quiescent and host an evolved stellar population where the dust might have been consumed/destroyed.
Figure <ref> also shows a trend in redshift. On average, the bulges are slightly more massive (M_*^B) than the SF cores (M_*^C) but with only a 1.5σ significance.
The SF cores of Type II galaxies (M_*^II) and those of Type I galaxies (M_*^I) are consistent within errors.
M_*^B = 3.75_-0.81^+1.04× 10^10 M_⊙≳
M_*^C = 1.81_-0.65^+1.19× 10^10 M_⊙
M_*^II = 2.60_-1.19^+2.19× 10^10 M_⊙≈
M_*^I = 1.26_-0.53^+0.92× 10^10 M_⊙
The weak trend between the M_* of higher-z SF cores and lower-z bulges is consistent with the idea of a bulge that grows in mass with time, fed by accretion from the disk, clump migration or minor/major mergers.
We compared the M_* and SFR fraction of the red cores and bulges with respect to the host galaxy (right panel of Fig. <ref>). For Type I galaxies, the red core M_* represents only 21.6 ± 4.0% of the M_* of the galaxy. This fraction is smaller than for the other galaxies of the sample where the red core represent 34.4 ± 6.2% for Type II (∼ 2σ difference) and 35.9± 3.6% for Type III (∼ 3σ difference) of their total M_*. This can be linked to the redshift trend, the Type I galaxies being at higher redshift, their core could still be at an early stage of growth. It also explains their lowest R_e,IR/R_e,O = 0.89± 0.14, as their M_* is much less concentrated in the central region that the other two Types.
As expected from the definition of our Types of galaxies, the Type II galaxies have a red core with a SFR fraction (64 ± 18%) significantly greater than the M_* fraction (34.4 ± 6.2%) since the disk is mostly quenched, while the Type III galaxies have a red bulge M_* fraction (35.9 ± 3.6%) significantly more important than the SFR fraction (9.8 ± 3.4%) as the bulge is quenched.
Some of these cores/bulges appeared to be compact, we decided to investigate them further in the next Section.
§.§.§ Compact cores and bulges
All of our galaxies have a central core/bulge appearing in the near-IR (filter F410M or F444W). For some galaxies, the core/bulge has a clear clump-like morphology, is much brighter than the surroundings and is clearly delimited (e.g. ID13776 and ID23205 in Fig. <ref>). We identified compact cores in 17 galaxies out of 22: 6 Type I galaxies with a compact core (60% of our sample), 4 Type II (80% of our sample) and all 7 Type III galaxies of our sample have a compact bulge. We decided to investigate further these compact cores/bulges by dividing the in two categories, the SF cores (from Type I and II) and the quiescent bulges (from Type III).
To do so, we measured the half-light radius of the compact cores and bulges defined as the radius of a circular aperture encompassing half of the flux of the core, we applied a similar technique as described in Sect. <ref>. The SF cores tend to be slightly more compact than the quiescent bulges (0.76 ± 0.03kpc vs 0.84± 0.04kpc, with a ∼ 1.5σ significance, see Fig. <ref>). The markers with a black circle are the compact core with an X-ray detection, possibly tracing an AGN. 3 of them are found in SF cores, and 2 are in the most massive galaxies with the largest SF cores. Even if the definition of the compact core is somehow arbitrary, and that there could be some level of contamination from the disk, this goes in the same direction as <cit.> simulations. They found that without AGN feedback, the SF core would undergo a compaction event while the presence of AGN winds would prevent such compaction by evacuating the gas and precipitate the quenching of the core.
We also note that the quiescent bulge tends to be larger in more massive galaxies. <cit.> found that the most compact cores of SMGs are those where there is both star formation and an AGN. This is not what we observe for two of the SF cores hosting an X-ray AGN (showed with the encircled markers in Fig. <ref>), it is possible that in these galaxies the AGN has strong feedback and the system is quite evolved and ready to quench. The third SF core hosting an X-ray AGN is however compact and the presence of the AGN could facilitate this compaction.
The sizes of the SF compact cores are compatible with those measured in the sub-mm (See Sect. <ref> for more details).
After analyzing the cores of our galaxies, we decided to investigate their differences with respect to the disk, especially the reasons of the redness of the core compared to the surroundings.
§.§.§ NIRCam color variations within the disks
In Sect. <ref>, we showed that the main driver of the color gradient between the cores and disks is the dust attenuation.
When looking at Figs. <ref>, <ref> and <ref> we noticed that some disks are also highly in-homogeneous in terms of color. To investigate the physical processes responsible for the color variations we observed in the disks, we compared the color variations with the dust attenuation and sSFR variations in a similar way we did in Sect. <ref> when we investigated the gradients between the cores and the disks. When measuring the variations, we always measured the differences between a redder part of the disk and a bluer part (in other words Δ (F115W - F444W) > 0 in AB mag). We compared all the components of the disks, meaning that if a disk was divided in 3 patches, there are 3 markers in Fig. <ref> comparing the first and second, second and third and first and third component respectively. In the two upper panels of Fig. <ref>, we first clearly identify a correlation between the color variations and the dust extinction variations (Pearson coefficient = 0.78, p-value = 6e-11) consistent with the expectation that the redder regions are those with the greatest A_V ().
However, we do not identify any correlation between color variations and sSFR variations (Pearson coefficient = 0.16, p-value = 0.29), some color variations are even inconsistent as when sSFR_redder > sSFR_bluer, we are comparing a red patch hosting a younger stellar population (more star-forming) with a bluer patch hosting an older stellar population (less star-forming), hence the colors should be the other way around.
This two observations demonstrate that the color variations we observe within the disks in Figs. <ref>, <ref> and <ref> are driven by dust. NIRCam colors at z ∼ 2 trace dust, red spots are highly extinct while blue spots are weakly dust attenuated. This is consistent with previous studies based on NIRCam images (e.g. <cit.>) .
As the clumps could play an important role in the color variations, it is important to investigate their abundance.
§.§.§ Clumpy disks
As one can see in Figs. <ref>, <ref> and <ref>, some galaxies are very clumpy.
The clumpiness does not seem to be linked to a particular Type of galaxy. Most of the clumps are observed in the shortest wavelength, consistent with <cit.> who state that the number of clumps decrease when moving toward longer wavelength.
In Fig. <ref>, we investigate the the possible link between the clumpiness and the disk and the core of the galaxy. In the left panel,we show the distribution of the number of clumps observed in each disk versus the SFR of the disk (defined as the sum of the SFR of the regions delimited in Figs. <ref>, <ref> and <ref>) separating the SF disks from the quiescent disks. There is no apparent correlation between the star-forming activity of the disk and the number of clumps. The fact that we observe clumps in quiescent disks is quite surprising as they usually are supposed to be place of local starburst (). We discuss the implication of this result in Sect. <ref>.
In the left panel of Fig. <ref>, we study the impact of the fraction of stellar mass in the core (in blue) or bulge (in red) on the number of clumps. The galaxies with a quiescent bulge, that we know to be at lower redshifts (see Sect. <ref> and Fig. <ref>), have a higher fraction of their mass in their bulge (35.9%± 3.6%) than the galaxies with a star-forming core have in their core (25.8%± 3.7%) with a ∼ 2σ significance. They also tend to have a smaller number of clump: 1.7± 0.8 clumps on average for a galaxy with a bulge and 2.8± 0.6 clumps on average for a galaxy with a star-forming core (1.1σ significance). The plot also shows that by looking at galaxies with a star-forming core (in blue in Fig. <ref>), the ones with the smallest M_* fraction at their core are also the clumpiest. We see here both the effects of the redshift, lower redshift galaxies have less clumps and of the central M_* fraction, higher fraction leads to less clumps.
One could argue that the fact that galaxies with a star-forming core are at higher redshift than those with a quiescent bulge (Type III) means that we probe shorter rest-frame wavelength, hence, we have a higher probability of observing clumps in their disk (). However, the range of redshift that we are probing here is quite narrow, and the clumps that we count are the brightest and visible in several filters. These galaxies actually are clumpier.
§ DISCUSSION
In this Section, we first discuss the green patches/clump that are visible in the RGB cutouts in Figs. <ref>, <ref> and <ref> in Sect. <ref>. Then, we discuss the presence of blue clumps inside quiescent disks in Sect. <ref>. We investigate the possible link between the compact SMGs observed with ALMA and our DSFGs in Sect. <ref>. In Sect. <ref>, we discuss the origin and consequences of lopsidedness and its abundance. Eventually, in Sect. <ref>, we discuss two possible evolutionary paths that could lead to the formation of Type II galaxies.
§.§ Bright emission lines
When looking at Figs. <ref>, <ref> and <ref>, one can notice that some of the disks have different colors, with a blue and a green part.
The green clumps/patches are visible in all Types of galaxies. Considering their redshift, they probably are due to bright H_α or [O_III] emission lines which are known tracer of star-formation. The H_α line will fall in the green filter (F200W) for galaxies with a redshift between 1.67 and 2.39 and the [O_III] emission line will fall in the green filter for galaxies with a redshift between 2.52 and 3.47. On the 7 galaxies where we identify green patches, 2 are consistent with H_α emission from a star-forming region (ID15371 and ID29608) and 3 are consistent with [O_III] emission from a star-forming region (ID18694, ID23510 and ID23581). For the 2 remaining galaxies, it is more surprising, as the green patches/clump are observed in the quiescent disks of Type II galaxies.
For the ID13107 galaxy (z = 2.21 ± 0.02), the green patch is close to the center of the galaxy, it is then possible that the H_α line is produced by the accretion disk of an AGN sitting at the center of the galaxy that becomes bright in this region because of a much weaker dust attenuation than in the core. Even though we have no radio or X-ray signature of an AGN in this galaxy, as mentioned before, the predicted SFR from the SED fitting is not enough to explain the FIR flux density observed with Herschel for this galaxy. This convinced us that there could be an AGN at the core of this galaxy.
For the ID18278 galaxy (z = 1.805), the situation is different, the green patch is in the outer region and composed of clumps. These clumps could have actually been ionized by the hot evolved low-mass stars () with an enhanced H_α line due to shocks from the minor merger. Indeed, these clumps are old (age of oldest stars = 2.5 ± 0.5Gyr) and have a very low sSFR, consistent with the ex-situ clumps defined in <cit.>.
§.§ Origin of dusty patches within disks
In Sect. <ref>, we demonstrated that the color gradient is linked to the strong A_V gradient. The fact that the core is much more attenuated than the disk is expected because the SFR surface density is higher in the core than the disk, hence is the dust surface density and the dust column density.
However, the patchy distribution of dust within the disks is more surprising. From the lower panel of Fig. <ref>, we observe a correlation between dust density and sSFR for Type II and Type III galaxies (Pearson coefficient = 0.62 and 0.83 with p-value = 0.04 and 0.01 respectively). Meaning that for these galaxies, the patches could be linked to not yet quenched regions in the disks of Type II galaxies and partly quenching disk for Type III. The patches could then find their origin in internal instabilities, or interactions with the local environment. For Type I galaxies, we do not observe this correlation (Pearson coefficient = 0.35, p-value = 0.07). For these galaxies, the patches could be correlated either to metallicity, higher metallicity leading to higher dust column, or to geometry.
We investigated the origins of the patchy distribution by looking for correlations between the greatest difference in A_V in each disk and the redshift, the fraction of stellar mass in the core/bulge, the fraction of SFR in the core/bulge, the lopsidedness and the environment. We found no correlation (all p-value > 0.2). We then looked for a correlation between the number of patches/components of each disk (as defined in Figs. <ref>, <ref> and <ref>) and the same parameters. The only correlation we found, that is visible in Fig. <ref>, is with the mass fraction in the core (Pearson coefficient of -0.60, p-value = 0.003), the number of patches/components gets smaller when the mass is more concentrated in the core of the galaxy. This is especially true for the galaxies with a star-forming core (Type I and II, with a Pearson coefficient of -0.67 and a p-value of 0.006, while Type III have a Pearson coefficient of 0.14 with p-value of 0.76). This correlation is expected from <cit.>; when the central gravitational potential well is deep enough, it stabilizes and homogenizes the disk. This correlation is consistent with the one we observed for the clumps (See Sect. <ref> and Fig. <ref>). However, if this (anti-)correlation justify why we do not see patches in Type III galaxies, it doesn't clear up the mystery of their origin.
We would need spectroscopy to understand better what is happening in those disks, and even there the mystery would remain of why the disks are so in-homogeneous in dust attenuation, whether it is due to metallicity or geometry differences (and why these would persist over homogeneous patches within a disk, as opposed e.g. to simple radial gradients).
§.§ Clumps in DSFGs
In all the Types of galaxies, we identified the presence of clumps. We observed that galaxies at lower redshift tend to have less clumps, this suggests that the clumps either get destroyed within the disk and are not replaced by new clumps, or migrate toward the core and participate to its mass growth possibly triggering enhanced star formation. They might also be lower mass/less luminous, hence below our detection threshold.
We do not see any evidence of recent major mergers in our galaxies, suggesting that most of the clumps we observe are originating from a fragmentation of a gas rich unstable SF disk, consistently with <cit.> and <cit.> that showed that large scale instabilities in gas-rich galaxies can create such star-forming giant molecular clumps.
We also noted that the most clumpy high-redshift galaxies also have the least concentrated cores, with less than 20% of their stellar mass at the center of the galaxy (see Fig. <ref>) and, on the contrary, the least clumpy galaxies at lower redshift have nearly 40% of their stellar mass in the quiescent bulge. We also showed in Fig. <ref> that galaxies at later times have higher core mass fractions. This suggests that either, as the clumps migrate through the disk, they feed the central core, making it grow in mass or that, as the central gravitational potential well gets deeper, the disk is stabilized, the VDI are destroyed, and the galaxy can have a smoother spiral-like disk. Our observation are consistent with the simulations from <cit.> that showed a well defined dynamical center is necessary to stabilize the disk and put an end to bursty star-formation. Also, we are in agreement with the new JWST results from <cit.>, pointing to an increased galaxy fragmentation with decreasing bulge/core mass fraction.
When looking at Fig. <ref>, one can clearly identify clumps in the Type II galaxies. The blue clumps of these quiescent disks (ID13107, ID18278 and ID13776 in Fig. <ref>) are due to a low dust attenuation and not a high sSFR. Indeed, the disk has A_V = 1.0± 0.2 while the central SF core has A_V = 3.5± 0.2. We recall that the blue colors in NIRCam color cutouts for these redshifts are typically a signature of low dust attenuation. This could indicate that clumps are not only formed in highly star-forming regions.
§.§ Are we observing compact SMGs counterparts ?
In most of our IR-luminous galaxies, a central compact clump-like highly dust attenuated SF red core is present. While it is nearly invisible in the optical rest-frame, it becomes bright in the near-IR (see Figs. <ref> and <ref>). As we showed in Sect. <ref>, they are surrounded either by a SF (Type I) or a quiescent (Type II) disk with much lower dust attenuation. We identify 10 of those (see Table <ref> and Sect. <ref>) in our sample.
When we measured the size of these red compact SF cores, we found that the average R_e,NIR was about 0.76kpc (Fig. <ref>). This size is compatible with the sizes measured with ALMA for the compact SMGs : 0.6± 0.2kpc in <cit.>, ∼ 0.73kpc in <cit.> or 1-2kpc across in <cit.>. The NIRCam sizes tend to be slightly larger than the ALMA sizes, this is not due to a spatial resolution issue, but to the heavy dust obscuration of the core.
Moreover, compact SMGs at z ∼ 2-3 are characterized by a SFR ≥ 100M_⊙ yr^-1 (). 7 out of the 10 galaxies where we identified a compact star-forming core have a total SFR compatible with this criteria (see Table <ref>). 5 of them having SFR ≳ 100M_⊙ yr^-1 in the core alone, 1 has SFR ≥ 50M_⊙ yr^-1 in the core and the remaining galaxy has a lower SFR in their core.
To confirm the possibility of SMGs counterpart, we can use the FIR super-deblended catalog in the EGS (Le Bail et al., in preparation.). 6 out of the 10 galaxies are detected at 2σ in SCUBA2/850μ m among which 3 are 3σ detected. 3 out of the 4 undetected galaxies at 850μm are in the shallower part of the FIR catalog. Moreover, if we look at the predicted flux at 1.1mm for these galaxies, the mean predicted flux is 0.80mJy, and 4 of them are predicted to be brighter than 1mJy at 1.1mm. A total of 5 galaxies have either a 3σ detection in SCUBA2/850μm or a prediction > 1.1mJy at 1.1mm (ID13776 and ID21190 from the Type II class and ID16544, ID29608 and ID30186 from the Type I class). They correspond to the 5 galaxies measured with a SFR≳ 100M_⊙ yr^-1 in their core.
All these elements convinced us that we have at least 5 or 6 galaxies that are good candidates of compact SMGs counterparts, they are equally distributed between Type I and II.
Contrary to what is observed with ALMA, these compact cores are not isolated, they all are surrounded by a larger disk. The fact that their is a huge dust gradient between the core and the disk, as we showed in Sect. <ref> might explain why we do not this the latter in sub-mm surveys: the core is bright in the rest-frame near-IR while the disk is bright in the rest-frame optical. The presence of a disk confirms <cit.> and <cit.> who both stated that the compact SMGs are obscured part of a larger system.
The fact that some galaxies in our sample have highly extinct cores could link them the so-called HST-dark galaxies. We compared our sample with the HST-dark and HST-faint galaxies in the same field from <cit.>. Our galaxies are in general agreement with the SFG at z < 4 in <cit.>, especially with the fact that we observe highly dusty patches out to large radii. Four of the galaxies in our sample are classified as HST-faint (ID16544, ID18694, ID23581 and ID26188).
All are Type I galaxies, which seems logical because quiescent regions have lower A_V, hence are brighter in HST. One of them (ID23581) has A_V,min > 3, hence expected to be HST faint/dark, while the remaining three galaxies have A_V,min∼ 1.5, which is the average A_V,min of the sample. It is more surprising that those three galaxies are HST faint/dark.
However, these 4 galaxies are actually the galaxies at the highest redshift of the sample (2.7 < z < 2.9), with photometric redshift from our SED fitting procedure consistent with the ones from <cit.> and the ones from the super-deblending (Le Bail et. al, in preparation).
There is a chance that their HST faintness comes more from their higher redshift than their high level of dust (at least for 3 of them).
§.§ Relation to Blue Nuggets simulations
In the cosmological simulations from <cit.>, the typical high-redshift and low-mass galaxy is a gas-rich, star-forming, highly perturbed, and possibly rotating system, fed by intense streams from the cosmic web. When the stellar mass is in the ballpark of ∼ 10^10M_⊙, the galaxy undergoes a major, last, wet compaction into a `Blue Nugget', starting with a compact gaseous star-forming system that rapidly turns into a compact stellar system.
The galaxies that we observe are all above this ∼ 10^10M_⊙ threshold. However, non of them look like a blue nugget except possibly ID13098. We discuss the specific case of ID13098 in Sect. <ref>. The other ones that are in the range of mass where the wet compaction should happen do have a compact dusty star forming core, but they also have a much larger star-forming disk.
Moreover, the more massive galaxies could be undergoing a rejuvenation event after blue nugget phase as it is suggested by <cit.>.
However, when comparing the t_50 of the disk and core, we find no evidence that the star-forming disks are younger than the cores. The fact that we do not observe any blue nuggets (or a single one) might be due to their low-mass, or low SFR, or that the previous observations were not deep enough to detect the low-luminosity disks. It may be possible that the most massive galaxies undergo a different quenching mechanism that lower-mass galaxies.
§.§ Investigating the lopsidedness
Galaxy lopsidedness has not so far attracted much attention at high redshift, probably because of a lack of spatial resolution and/or incomplete data since the most obscured part of the galaxies are not visible with pre-JWST telescopes. However, the spatial resolution of NIRCam shows that it is a common features of DSFGs around the Cosmic Noon. Indeed, we showed in Sect. <ref> that being lopsided seem to be the typical morphology of Type I and II galaxies (see Figs. <ref>, <ref> and <ref>).
<cit.> investigated the origins of lopsidedness in field galaxies and concluded that it is very unlikely the result of internal mechanisms but rather linked to the history and environment of the galaxies. With the NIRCam images, we have access to the spatially resolved morphology of these galaxies, and can try to better understand the causes of the lopsidedness.
Among the lopsided galaxies showed in Figs. <ref> and <ref>, some have a clear compact central core and a rather homogeneously colored disks (e.g. ID11887, ID13776), others are mostly clumpy galaxies with a less compact core (e.g. ID18694, ID18278).
For the first category, even if we don't have the kinematics to confirm it, it seems that the galaxies have a stable disk, with no major merger features. This means that the lopsidedness of these galaxies, is probably due to accretion and minor mergers. This accretion would be happening via streams of cold gas that asymmetrically feed more generously one side of the galaxy making it grow larger than the opposite side. Moreover, the fact that these galaxies are clumpy (see Sect. <ref>) and that their disk is highly heterogeneous (see Sect. <ref>) favors the idea of accretion or minor mergers that could create clumps or patches in the disks with different SFRs or A_V. However, the fact that Type I galaxies have a star-forming disk and Type II a quiescent disk means that the properties of gas transport in Type I and Type II galaxies are different.
In Type I galaxies, the disk acquires its gas via accretion streams or minor mergers and forms stars, but the gas also goes to the core, which is SF as well. <cit.> showed via simulations that the strong lopsidedness could be the result of gas accretion if it as asymmetric enough and that the lopsidedness from accretion is relatively long-lived (∼ 3Gyr), hence easily observable. This has also been confirmed by a recent study based on the TNG50 simulation () where they conclude that the lopsidedness in local galaxies originates from accretion over several Gyr while symmetric galaxies formed earlier and within a shorter timescale.
In Type II galaxies, on the other hand, while the gas keeps going to the core and keeps it SF, the disk is quenched. This would seem to suggest that the gas does not stay in the disk, but goes straight to the center. A possible explanation would be that Type II galaxies have larger inflows or very powerful outflows that blow away and/or shock the gas in the disk (confirming this would require spectroscopy). It could also be that in Type II galaxies the accreted gas has a more radial accretion, with little angular momentum and goes straight into the central regions. Or, for some reason, the gas rapidly looses its angular momentum and abandon the disk and falls into the center. This would, depending on the direction of accretion, feed the lopsidedness. This effect has already been suggested by <cit.> where they were able to link the lopsidedness of 3 galaxies at z ∼ 3 in a dense environment to cold gas accretion using Lyman-α emissions. The strong lopsidedness of these galaxies, would then be a tracer of the point of impact of the accretion streams.
For the clumpier galaxies, the disk is star-forming and not homogeneous. <cit.> showed with simulations that gas-rich disks are able to survive major mergers and that the following enhanced star-formation is not entirely happening in the core of the galaxy, but a substantial fraction takes place in the disk too. This is compatible with our Type I galaxies, the fact that their SF disks are clumpy and heterogeneous in terms of dust and sSFR could be a signature of a recent major merger (). Moreover, <cit.> mention that the presence of a gas-rich disk contributes to reducing the efficiency of bulge formation, which is compatible with the non-compact core observed in some of these galaxies.
Usually major mergers features are short lived, but the clumps we observe could be preserved due to Toomre instabilities. Indeed <cit.> showed, via simulations, that a galaxy with a gas fraction greater than 50% will have strong disk instabilities leading to the formation of long-lived giant clumps and strong nuclear inflow affecting the structure of the galaxy and possibly introducing lopsidedness. It has already been observed in a local galaxy used as proxy for high redshift galaxies (). A major merger could then result in a clumpy galaxy with a perturbed structure, which is what we have in Fig. <ref> for some Type I galaxies. The color variations between clumps/regions in the galaxies could be tracers of the original galaxy they were a part of before the merging as they trace the dust attenuation. However, a major merger is not necessarily required, indeed, <cit.> studied a lopsided galaxy at z∼ 3 and concluded that its lopsidedness did not originate from interaction with the environment but from internal, large scale instabilities, that could, in the end, form bars or spiral arms.
The lopsidedness of these galaxies could also be the signature of the bulge angular momentum build-up. Indeed, either via accretion, minor mergers, major mergers, internal instabilities and tidal effects, the lopsidedness will break the disk balance, consequently creating a torque on the bulge of the galaxy resulting in an angular momentum loss.
The significance of the difference of lopsidedness between Type III galaxies and the rest of the sample means that, by some mechanism, the galaxies become much more symmetric after the Cosmic Noon. Indeed, we recall that our Type III galaxies have z = 1.80± 0.09 while Types I and II have z = 2.19 ± 0.14. This could be due to increasing virialization with passing of time, also due to the stabilising effect of the larger bulge mass fraction (see lower right panel of Fig. <ref>).
§.§ Where do Type II galaxies come from ?
The Type II galaxies (see Sect. <ref> and Fig. <ref>) have an unusual behavior. They have a compact star-forming core embedded in a quiescent disk, and represent ∼ 23% of the galaxies of our sample, so are relatively common. <cit.> studied such galaxies in a crowded environment at z ∼ 3 and linked the quiescence of the disk to its strong lopsidedness which rapidly fuels the gas to the core of the galaxy. In our sample of Type II galaxies, 3 have a strong lopsidedness, 1 is only weakly lopsided and has an off-center core while 1 is not lopsided at all. This means that even if lopsidedness can be a driver of outside-in quenching, it is not the only one.
Based on our observations, we have three possible scenarios that could explain the observed suppression of star-formation in the disk.
The first scenario is the one developed by <cit.> with the lopsidedness either coming from a major merger strong enough to result in this off-centered core or from asymmetric accretion of gas via streams and minor mergers, feeding the disk preferentially on one side. The strong lopsidedness resulting from this is enough to explain the quenching of the disk as it greatly facilitates the transportation of the gas toward the core ().
The second scenario is a wet compaction event leading to an apparent outside-in quenching. ID13098 is in the correct range of stellar mass and redshift to be in a `blue nugget' phase () where the galaxy undergo a wet compaction caused by gas-rich mergers or smoother gas streams, leading to an episode of high central star-formation and outside-in quenching. The presence of the low-luminosity quiescent disk might indicate that the compaction is not completely done yet. If it is a blue nugget, the outside-in quenching may not be final as when the gas has been consumed at the center and the bulge has grown, a star-forming ring can form in the disk via accretion of new gas-reach material from the inter-galactic medium leading to an inside-out quenching in the post-blue nugget phase.
The last scenario is an actual outside-in quenching linked to the strong lopsidedness but not resulting from a major merger. In Fig. <ref>, we show that the Type I galaxies are the most star-forming and at the higher redshift on average. They also have a stellar mass consistent with the Type II galaxies. This means that there could be an evolutionary path between Type I and Type II galaxies driven by VDI and lopsidedness. The idea is that the star-forming clumps of the Type I galaxies will migrate toward the center of mass of the galaxy (). By doing so, they will fuel strong gas nuclear inflow creating a compact SF core (). On their way to the center of the galaxy, the clumps will accrete the gas of the disk and could leave a completely gas deprived disk and a compact SF core. When growing, the SF core will prevent the formation of new clumps in the disk by stabilizing it () while the lopsidedness could be conserved due to the large scale instabilities. In this scenario, Type II galaxies are observed in a process of outside-in quenching.
demonstrated that the local ULIRG Arp220 is composed of a central starburst and a larger quiescent disk. The starburst has been triggered by a major merger. The galaxy is classified as shocked post-starburst galaxy which is a stage prior to post-starburst.
In that case, it appears that shocks induced by the merger forced the outer disk in this galaxy to turn quiescent. This is close to first scenario we described with the outside-in quenching originating from a major merger.
In our case, in the four remaining galaxies, two have clumpy heterogeneous disks (ID13107 and ID18278, see Fig. <ref>), the different properties of the patches, either linked to dust or sSFR (see Sect. <ref>) favors the idea of asymmetric accretion streams and minor mergers as the source of lopsidedness for these two galaxies. The ID13776 has a clumpy but more homogeneous disk, but highly off-centered. The eccentricity of this galaxy can both originate from asymmetric accretion making the disk grow on one side or from a major merger strong enough to shift the disk. In the same way, it is hard to conclude for the last galaxy (ID21190) which is not lopsided and seem to have a smooth homogeneous disk.
§.§ The role of environment
A way to discriminate between the scenarios of outside-in quenching and the origin of the lopsidedness of galaxies is to look at their local environment. To this aim, we use the environment density measurements from <cit.>. They measure the density contrast of galaxies with a magnitude brighter than 26 AB mag in the H-band. The density contrast is defined as the number density enhancement with respect to the average density in the vicinity of the galaxy (local density/background density). In Fig. <ref>, we compare the local density contrast of our sample with the general population of galaxies in the EGS field.
The star markers in Fig. <ref> are the Type II galaxies. They do not sit in any particular kind of environment, they are relatively close to the median of the general population showed by the blue dotted line. This suggest that outside-in quenching can happen both in dense environment via major mergers but also in lighter environment via internal effects. The galaxy at lower density is ID13098 that we discuss in Sect. <ref>. The fact that this galaxy is relatively isolated favors the scenario of wet compaction as the origin of its outside-in quenching. For the other galaxies, the local density is insufficient to discriminate between scenarios as they do not sit in strongly over/under crowded environments but show that they all are likely.
The color of the markers trace the lopsidedness of the galaxies. There is no obvious difference between the lopsided galaxies and the general population. We do not see any signature that could link the environment to the lopsidedness. The fact that we see lopsided galaxies not only in dense environment and that most of them have a regularly looking disk favors the idea that lopsidedness originates from accretion and/or VDI. However, this is only a tentative explanation, these measurements are not strong enough to say if environment could be a driver of lopsidedness. The circular marker showing a weakly lopsided galaxy in a high density environment is ID30186. This galaxy is the brightest galaxy of a group of ∼ 16 members at z_spec = 1.85, is undergoing a major merger and is surrounded by quiescent intra-halo light ().
Discriminating further between the different scenarios would require spatially resolved spectroscopy to study the kinematics of each of these galaxies, and especially of the disk of each of them, to see if their disk is rotating, which would favor accretion and minor mergers, or if they are dominated by dispersion velocity favoring the scenario of major mergers and VDI.
§ SUMMARY
In this paper, we used the new set of images in the near-IR from JWST/NIRCam in the EGS field from the CEERS collaboration to investigate the formation and evolution of DSFGs at Cosmic Noon.
To start with, we selected a sample of DSFGs based on their FIR emissions and around Cosmic Noon (1.5 < z < 3.0). We ended up with 22 galaxies in the CEERS field.
We studied each galaxy on a sub-galactic scale by dividing them in different regions based on their NIRCam (F115W, F200W, F444W) colors, taking advantage of the spatial resolution. Using the available photometry from HST and JWST, we ran SED fitting and derived physical parameters for each galaxy component and classified them as star-forming or quiescent.
We classified the galaxies in different Types based on the star-forming activity in their core and disk. The Type I have a star-forming disk with a red star-forming core, the Type II are quiescent disks with a SF core and the Type III are star-forming disks with a quenched bulge.
The main results of this study are:
* ∼ 70% of the DSFGs in our sample have a red deeply dust attenuated compact star-forming core that can represent up to 80% of the total SFR of the galaxy but only 20-30% of its stellar mass. Contrary to the simulations that predict blue nuggets, these compact red cores are surrounded by large less obscured disks. Most of these cores are measured or predicted to be SMGs. However, telescopes like ALMA or NOEMA would only be sensitive to the most obscured part of the galaxy. This study demonstrates the necessity to combine near-IR imaging to sub-mm data to fully grasp the nature of DSFGs.
* 64% of our galaxies are at least weakly lopsided, and 27% strongly lopsided. The lopsidedness could be caused by asymmetric cold gas accretion and minor mergers feeding preferentially one side of the disk, which would, depending on the orientation of the accretion favor a star-forming or quiescent disk. Lopsidedness could also be triggered by a major merger disrupting the disk, and/or via large scale instabilities even if our study favors accretion. The fact that lopsidedness is so common among our sample means that most DSFGs have a complex SFH and do not calmly evolve without any interaction with their environment.
* 23% of the galaxies of our sample have a quiescent disk but a star-forming core. If one of them is compatible with a blue nugget, the others are not. Their observed outside-in quenching could then find its origins in their strong lopsidedness that favors VDI and rapid transportation of gas towards the center or from large scale instabilities and clump migration accreting the gas from the disk to feed it to the core.
* Most of the galaxies have a disk with patches/clumps of different RGB color that are not radially symmetric. The color variations within the disks are mostly driven by dust attenuation. These variations are another indicator that Main Sequence DSFGs have a complex SFH.
* Interestingly, among the quiescent disks, we find evidence of clump-like structures. These clumps are not (or very weakly) star-forming, they are mostly populated by old stars but seem to be to massive to be compared to the globular clusters we observe in the local universe.
This work demonstrates the power of the JWST in probing for the first time spatially resolved galaxies in the near-IR at Cosmic Noon, where the only available data was the unresolved images from Spitzer/IRAC. This allows reliable studies of quenching and dust attenuation at sub-galactic scales in DSFGs, facilitating the understanding of their morphologies and formation and evolution mechanisms that appear to be more complex than previously thought.
CGG acknowledges support from CNES.
P.G.P.-G. acknowledges support from Spanish Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 through grant PGC2018-093499-B-I00.
();a,
aa
|
http://arxiv.org/abs/2307.05598v1 | 20230710194032 | Shock cooling emission from explosions of red super-giants: II. An analytic model of deviations from blackbody emission | [
"Jonathan Morag",
"Ido Irani",
"Nir Sapir",
"Eli Waxman"
] | astro-ph.HE | [
"astro-ph.HE",
"astro-ph.SR"
] |
firstpage–lastpage
The Synthesis Lab: Empowering Collaborative Learning in Higher Education through Knowledge Synthesis
Bodong Chen
August 12, 2023
====================================================================================================
Light emission in the first hours and days following core-collapse supernovae is dominated by the escape of photons from the expanding shock heated envelope. In a preceding paper, Paper I, we provided a simple analytic description of the time dependent luminosity, L, and color temperature, T_ col, for explosions of red supergiants with convective polytropic envelopes and in the absence of significant circum-stellar medium. It is valid up to H recombination (T≈0.7 eV). The analytic description was calibrated against the results of numerical calculations, approximating radiation transport by diffusion with a "gray" (frequency independent) opacity. Here we present the results of a large set of 1-dimensional numeric multi-group (frequency dependent) photon diffusion calculations, for a wide range of progenitor parameters (mass, radius, core/envelope mass and radius ratios, metalicity) and explosion energies, using opacity tables that we have constructed for this purpose (and are publicly available) including the contributions of bound-bound and bound-free transitions. We provide an analytic description of the small, ≃10% deviations of the spectrum from blackbody at low frequencies, hν< 3T_ col, and an improved (over Paper I) analytic description of the strong suppression of the flux due to line absorption at high frequencies, hν> 3T_ col. We show that the effects of deviations from ionization and excitation LTE and of `expansion opacity' corrections are small, and that the effect of deviations from a polytropic density distribution are also small.
Our analytic results are a useful tool for inferring progenitor properties, explosion velocity, and also relative extinction based on early multi-band shock cooling observations of supernovae.
radiation: dynamics – shock waves – supernovae: general
§ INTRODUCTION
In core collapse supernovae (SNe) explosions, a radiation mediated shock (RMS) traverses outwards through the stellar progenitor, heating and expelling material as it passes. If no significant circumstellar material (CSM) is present around the star, arrival of the shock at the surface produces a hard UV/X-ray ∼10^45 erg s^-1 `shock-breakout' emission, lasting from tens of minutes to an hour. The breakout pulse is then followed in the coming hours and days by thermal UV/optical `shock-cooling' emission, caused by diffusion of photons out of the shock-heated stellar ejecta. Typical luminosities and temperatures during shock-cooling are of the order 10^42-10^44 erg s^-1, and 1-10 eV. As the photons diffuse out, deeper parts of the ejecta are gradually exposed over time <cit.>.
In order to constrain the properties of the progenitor star, it is helpful to have high cadence multi-band observations in the first hours of shock-cooling <cit.>. Among these measurements, ultraviolet observations are especially important, as they are closer to the thermal emission peak, and can be used to determine the emission temperature and the UV extinction self-consistently <cit.>.
Combined with an accurate theoretical model, these measurements can be used to reproduce the progenitor and explosion parameters, including radius, surface composition, explosion energy per unit mass, and the extinction. Analytic models are especially important for solving the "inverse problem" of inferring system parameters and uncertainties from the observed spectral energy distribution (SED).
Emission during shock-cooling is amenable to modeling in the case of `envelope breakout' (i.e.in the absence of CSM with signifcant optical depth), largely because the system is near local thermal equilibrium (LTE) at this time. However, catching supernovae within the first hour presents a practical challenge, and few such observations have been achieved <cit.>. Existing and upcoming observatories, such as the Zwicky Transient Factory (ZTF) <cit.>, the upcoming Vera Rubin Observatory <cit.>, and the expected launch of the wide-field UV space telescope ULTRASAT <cit.> will greatly increase the quantity and quality of early measurements, enabling a systematic study.
The numeric calculations presented here provide a systematic analysis of the deviation of the spectra of the emitted radiation from blackbody, which enables an improvement of the accuracy of the analytic models. We adopt a `multigroup' (MG) numeric approach, where radiative transfer is solved under the diffusion approximation separately for different photons frequencies, including coupling to hydrodynamics. Several codes using different approximations and schemes have been used to address the shock breakout and cooling problem: STELLA <cit.> is a 1 dimensional code that solves for the angle-dependent intensity with a variable Eddington factor, employing a ray-tracing scheme. Their opacity includes free-free, bound-free and atomic lines from <cit.>, and employs the Sobolev approximation, based on <cit.>. STELLA should contain the required physical components for modeling the SED of shock-cooling emission, and this has been done in several works <cit.>. <cit.> numerically solved the planar shock breakout problem under the diffusion approximation and incorporating free-free opacity. They also include inelastic Compton scattering, which is important for shock breakouts at high velocities. Other hydrodynamically coupled multigroup codes are presented in the literature <cit.>, though they have not been used or formulated to solve for shock-cooling emission.
Many other codes focus computational efforts on line modeling and are specialized to describe emission at later phases of the supernova, when the ejecta is freely-coasting. The ARTIS <cit.> and SEDONA <cit.> codes both solve frequency-dependent, time-dependent radiative transfer using Monte Carlo. ARTIS is primarily used to solve type Ia SN problems, while SEDONA has also been used to calculate core-collapse supernova emission <cit.>. The latter contains a module for coupling hydrodynamics to radiation in a Lagrangian sense, though to our knowledge this has not been used in the context of the first day of shock-cooling emission. Other multigroup codes solve for radiative transfer under a `steady-state' approximation <cit.>.
It is not guaranteed that the steady-state approximation is always appropriate at earliest times, though our results suggest that it may be reasonable for modeling if one employs realistic density and temperature profiles.
In a preceding paper, <cit.>, we modeled shock-cooling emission following breakout from a spherical envelope, assuming local thermal equilibrium (LTE). Using earlier analytic results <cit.>, we derived an analytic description of the bolometric luminosity L and color temperature T_ col, that provides a good approximation (to 10% in L and 5% in T_ col) of the results of our 'gray' diffusion simulations for the emission over ∼1 hour to ∼1 week after shock breakout for a wide range of explosion energies and progenitor parameters (radii, core and envelope masses, metallicities). The analytic expression is advantageous relative to those of previous analytic works <cit.> thanks to its calibration using a large set of numeric results incorporating a realistic opacity with free-free, bound-free, and bound-bound components, which have an important effect on emission. We also used a set of MG diffusion simulations to show that the spectrum is well described by a Planck spectrum with an effective color temperature, except at UV frequencies where the flux may be significantly suppressed due to line absorption.
In this work we relax the assumption that the photons are at LTE and account for frequency-dependent emission, absorption and transport both numerically and analytically. We produce hydrodynamically coupled 1-dimensional multigroup simulations for a wide range of explosion energies and progenitor parameters that span the parameter range of Paper I - explosion energies in the range E=10^50-10^52 erg, progenitor masses and radii M=2-40M_⊙, R=3×10^12-2×10^14 cm, envelope to core mass ratios M_ env/M_ c=10-0.3, and metallicities Z=0.1Z_⊙-Z_⊙. Our simulations are based on the work of <cit.>. Their code allows inelastic Compton scattering, which we do not incorporate in the bulk of the simulations, due to its negligible effect during shock-cooling. We include a frequency-dependent opacity κ_ν, for which we use both the publicly available TOPS code <cit.> and one that we developed ourselves and is now available to the community <cit.>. Relative to TOPS, our code is largely open source, and can produce opacity tables at arbitrary density, temperature, and frequency resolution. It is based on experimentally verified atomic line lists <cit.>. We note that our MG simulations cannot be used to make predictions for the exact spectrum including lines, as they have finite frequency resolution and do not include expansion opacity and deviations of excitation and ionization states from LTE (see <ref>). However, they are very useful for predicting the coarser-grained, line averaged SED in shock-cooling following envelope breakout.
This paper is organized as follows. In <ref> we define our notation and summarize the analytic results of , , , and , that we use in this paper. In <ref> we describe our numeric calculations and present convergence and code verification tests. In <ref> we describe our opacity tables, as well as convergence and verification tests for these. In <ref> we derive an analytic description of the deviations of shock cooling emission spectra from blackbody, and in <ref> we compare the analytic description to simulation results. In sections <ref> and <ref>, we address the sensitivity of our results to "expansion opacity" corrections and to deviations of plasma ionization and excitation from those of LTE. A comparison to earlier work, including to STELLA radiation transport calculations for several non-polytropic profiles obtained using MESA stellar evolution calculations <cit.>, is given in <ref>. The agreement of our results with these earlier calculations provides additional support to our code's validity, and to the conclusion of the detailed analysis of SW17, who demonstrated that the shock cooling emission is not sensitive to deviations of the density profile from a polytropic one. Our results are summarized and discussed in <ref>.
§ RMS BREAKOUT AND SHOCK-COOLING EMISSION- SUMMARY OF EARLIER ANALYTIC RESULTS USED IN THIS PAPER
RMS breakout and shock-cooling is extensively discussed in the literature and briefly summarized in . At radii r close to the stellar radius R (δ≡ (R - r)/R≪ 1), the initial density of a polytropic envelope approaches a power-law,
ρ_0 = f_ρρ̅δ^n.
Here ρ̅≡ M/(4π R^3/3) is the average pre-explosion density of the ejecta (exculding the mass of a possible remnant), and n=3/2 for convective RSG envelopes. f_ρ is a numerical factor, of order unity for convective envelopes, that depends on the inner envelope structure <cit.>. The predicted breakout and cooling emission are nearly independent of f_ρ.
As the shock approaches the edge of the star, it accelerates down the steep density profile and the flow approaches the self-similar solutions of <cit.>. The shock velocity diverges in this regime as
v_ sh = v_ s∗δ^-β_1 n,
with β_1=0.19, and with v_ s∗ a constant defined by eq. (<ref>). Based on numerical calculations, <cit.> find
v_ s∗≈ 1.05 f_ρ^-β_1 v_∗, v_∗≡√(E/M),
where M is the mass of the ejecta, E is the energy deposited in the ejecta, and v_∗ is its characteristic expansion velocity. This approximation holds to better than 10% for M_ env/M_ c<1/3, and overestimates v_ s∗ by approx. 20% for M_ env/M_ c=0.1 .
Breakout takes place when the scattering optical depth of the plasma layer lying ahead of the shock equals τ_ es=c/ v_ sh. We denote the shock velocity v_ sh and pre-shock envelope density ρ_0 at this point in terms of breakout parameters ρ_ bo and v_ bo, respectively. We may rewrite eqs. (<ref>) and (<ref>) as
ρ_0=ρ_ bo ( v_ boτ/c)^n/(1+n), v_ sh = v_ bo (v_ boτ/c)^-β_1 n/(1+n).
The location at which breakout "occurs", i.e. where τ=c/ v_ sh, is given by
δ_ bo = (n+1)c /κρ_ bo v_ bo R,
where κ is the opacity.
For RSGs, ρ_ bo and v_ bo are approximately related to the progenitor parameters and explosion energy by <cit.>
ρ_ bo = 1.16 × 10^-9 M_0^0.32 v_∗,8.5^-0.68 R_13^-1.64κ_0.34^-0.68 f_ρ^0.45 g cm^-3,
v_ bo/v_∗ = 3.31 M_0^0.13 v_∗, 8.5^0.13 R_13^-0.26κ_0.34^0.13 f_ρ^-0.09.
Here, R= 10^13R_13 cm, κ=0.34 κ_0.34 cm^2 g^-1, v_∗=v_∗,8.5 10^8.5 cm s^-1, and M=1 M_0 M_⊙. The duration over which the breakout pulse is emitted from the star is approximately given by the shock crossing time of the breakout layer,
δ_ boR/ v_ bo =(n+1)c /κρ_ bo v_ bo^2= (n+1)t_ bo=74.9 ρ_ bo,-9^-1κ_0.34^-1 v_ bo,9^-2 s,
where ρ_ bo=10^-9ρ_ bo,-9 g cm^-3 and v_ bo= 10^9 v_ bo,9 cm s^-1. The observed pulse duration may be longer than this intrinsic duration due to light travel time effects, which spread the pulse over R/c.
δ_ bo is given as a function of progenitor parameters and explosion energy as
δ_ bo=0.02 R_13^0.90 (f_ρ M_0 v_ s*,8.5 κ_0.34)^-0.45,
where v_ s*=v_ s*,8.5 10^8.5.
For later use, we provide here the density profile during the spherical phase, given by eq. (9) of . We recast this equation in terms of r and t using eqs. (3), (4) and (8), and eq. (<ref>) from this paper,
ρ(r,t) = 1.69 × 10^-11 (f_ρ M_0)^0.27 v_s*,8.5^8.73 r_14^-11.73 t_ d^8.73 g cm^-3,
where the radial coordinate is r=r_14 10^14 cm. Alternatively, using eq. <ref>,
ρ(r,t) = 1.82 × 10^-11 R_13^2 v_ bo,9^7.73κ_0.34^-1 r_14^-11.73 t_ d^8.73 g cm^-3.
Similarly, assuming a self-similar diffusion profile <cit.>, we have for the temperature:
T(r,t) = 4.83 R_13^1/4 (f_ρM_0)^0.27 v_ s*,8.5^2.66κ_0.34^0.02 t_ d^2.14 r_14^-3.18 eV,
T(r,t) = 5.02 R_13^0.71 v_ bo,9^2.31ρ_ bo,9^0.08κ_0.34^1/3 t_ d^2.14 r_14^-3.18 eV.
In , we described shock cooling emission by interpolating between the exact planar phase solution () valid at early times (hours), and the later approximate spherical phase solution ().
The combined bolometric luminosity L and emission (color) temperature T_ col are given by
L/L_ br=t̃^-4/3+t̃^-0.172× Aexp(-[at/t_ tr]^α),
T_ col/T_ col,br=min[0.97 t̃^-1/3,t̃^-0.45].
Assuming a blackbody spectral distribution, the emitted luminosity is then given by,
L_ BB=L×π B_ν(T_ col)/σ T_ col^4.
Here {A,a,α}={0.9,2,0.5} , and t̃≡ t / t_ br. t_ tr is roughly the time at which the photons will be able to diffuse out of the envelope in dynamical time. The br (break) subscript marks the values at the transition between the planar and spherical phase. They are given in terms of the model parameters v_ s* ,f_ρM, and R as
t_ br= 0.86 R_13^1.26 v_ s*,8.5^-1.13
(f_ρM_0κ_0.34)^-0.13 hrs,
L_ br=3.69×10^42 R_13^0.78 v_ s*,8.5^2.11
(f_ρM_0)^0.11κ_0.34^-0.89 erg s^-1,
T_ col,br= 8.19 R_13^-0.32 v_ s*,8.5^0.58
(f_ρM_0)^0.03κ_0.34^-0.22 eV.
M_0 denotes mass in units of solar mass. Both the break values and t_ tr can be directly deduced from observations.
Eqs. (<ref>)-(<ref>) are valid at times 3R/c < t < min[ t_0.7 eV , t_ tr/a ], where
3 R / c = 0.67 t_ br,3^-0.1 L_br,42.5^0.55 T_br,5^-2.21 hrs,
t_0.7 eV = 8.01
t_ br,3 T_ br,5^2.22 days.
The former is the time past which we showed light-travel time effects to be unimportant. The latter is the time at which the photosphere temperature is T=0.7 eV, based on , roughly corresponding to Hydrogen recombination. The transparency time, t_ tr, occurs roughly when the dynamical time matches the diffusion time, given by
t_ tr = √(κ M_ env/8 π c v_ s ∗),
= 19.5 √(M_ env,0κ_0.34 v_ s*,8.5^-1) days.
For later use we also define the homologous time t_ hom, which is also the early validity of the formula:
t_ hom = R/5v_ s,* = 0.1 R_13/v_ s*,8.5 days.
During shock-cooling, the luminosity is determined at the diffusion depth, the location from which photons will diffuse outwards in dynamical time <cit.>. The color temperature meanwhile, is roughly determined by the temperature at the thermal depth, the last absorption surface for diffusing photons (the radius from which photons diffuse out of the ejecta without further absorption). In this paper we treat these quantities as frequency-dependent (compare with the approximate prescription, , eq. 30). Specifically the thermal depth r_ col is defined by
τ_⋆,ν(r=r_ col,ν)≡∫_r_ col,ν^∞ρ√(3κ_ abs,ν(κ_ abs,ν+κ_ es))dr'=1,
where the abs, es and ν subscripts indicate absorption, (electron) scattering and frequency dependence, respectively. This integral is often approximated in the literature as √(3τ_ absτ_ es)
or √(3τ_ abs(τ_ abs+τ_ es)). We find that the choice of approximation has a negligible effect on the SED due to the steep ρ(r) dependence,
with the exception of regimes with strong lines, where the observed effect can be tens of percents.
§ DESCRIPTION OF THE NUMERICAL CODE
We numerically solve the radiation hydrodynamics equations of a spherically symmetric flow. We allow the photon distribution to deviate from thermal equilibrium with a multi-group treatment, and handle radiative transfer under the diffusion approximation. For the matter equation of state (EOS), we assume an ideal Hydrogen gas in LTE, including ionization as dictated by the Saha equation. The MG emission/absorption and diffusion opacities (effective opacities for each photon energy group), are based on our tables with a solar mix-like composition.
This section is structured as follows. The equations of the numerical scheme are given in <ref> and <ref>, and the initial and boundary conditions are described in <ref>. The validation of the numeric code and the convergence of the calculations are described in <ref>.
§.§ Radiation-hydrodynamics equations
In Lagrangian coordinates, the velocity v and density ρ evolve in response to the radiation energy density u_ν per unit frequency ν and matter energy density e, as follows:
dr/dt= v,
ρ = ρ_0r_0^2/r^2∂ r_0/∂ r,
d v/dt=-1/ρ1/r^2d/dr(1/3r^2[e+q+∫_0^∞ u_ν d ν])
,
where the 0 subscript denotes initial values. Correspondingly, the energy densities evolve according to,
du_ν/dt=.∂ u_ν/∂ t|_ emis/abs+.∂ u_ν/∂ t|_ compres+.∂ u_ν/∂ t|_ diff,
de/dt= -∫_0^∞.∂ u_ν/∂ t|_ emis/abs dν +.∂ e/∂ t|_ compres.
The emission / absorption, compression, and diffusion terms are given by
.∂ u_ν/∂ t|_ emis/abs=ρκ_ abs,ν c [4π B_ν (T)/c-u_ν],
.∂ u_ν/∂ t|_ compres=-[4/3u_ν -1/3∂(ν u_ν)/∂ν]1/r^2∂(r^2 v)/∂ r,
.∂ e/∂ t|_ compres=-(e+p+q)1/r^2∂(r^2 v)/∂ r,
.∂ u_ν/∂ t|_ diffusion=-1/r^2∂ (r^2 j_ν)/∂ r,
where the frequency-dependent photon energy flux density j is given by
j_ν=-1/ρκ_*(c/3∂ u_ν/∂ r-1/c∂ j_ν/∂ t),
and the Planck energy density per unit frequency is given by
4π B_ν (T)/c=8π/ c^3h ν^3/e^h ν/T-1.
κ_* is the diffusion opacity, given by the Rosseland mean of sum of the Thomson scattering opacity κ_ es and the absorption opacity κ_ abs,ν across the frequency bin (of each photon energy group in the MG calculation), κ_*^-1= (Δν_ bin)^-1∫ dν(κ_ es+κ_ abs,ν)^-1, where κ_ abs,ν is given by our high-resolution opacity tables. The absorption opacity of each group is given by κ_ abs= (Δν_ bin)^-1∫ dνκ_ abs,ν.
In a small fraction of the simulations we also include ineslastic Compton scattering using the Kompaneets equation, as shown below. Inelastic Compton scattering is unimportant during shock-cooling, but can have an important effect on the breakout spectrum, which we include in our tests of the code in <ref>.
.∂ u_ν/∂ t|_ scat=-ρκ_ es c ν/m_ e c^2∂/∂ν[ T ∂/∂ν (ν u_ν) .
+ . (hν - 4 T) u_ν + c^3/8π(u_ν/ν)^2 ],
where m_ e is the electron mass and T is the matter temperature (in units of energy).
Each of the above equations is solved explicitly using operator splitting, with the exception of the diffusion term, where u and j are solved for together implicitly. The flux is initially started as j_0=-1/ρκ_ esc/3∂ u_0/∂ r. The time derivative term in eq. (<ref>) becomes important in optically thin regions. In our problem it
does not have an important effect since the luminosity is determined deep in the ejecta.
Following <cit.>, we rewrite the radiative compression term in eq. (<ref>) as
.∂ u_ν/∂ t|_ compres=-u_ν( 1 -1/3∂log u_ν/∂logν)1/r^2∂(r^2 v)/∂ r.
We include Hydrogen recombination in the EOS, given by
e=(1/γ-1)n(1+Y)T-(1-Y)n I_H, p=n(1+Y)T,
where Y(T) is the ionization fraction, I_H is the Rydberg energy, n is the atomic number density, γ=5/3 is the monotonic adiabatic index. This prescription becomes an ideal gas equation of state when the plasma is fully ionized. Y is solved for iteratively using the Saha equation, assuming the presence of Hydrogen only.
§.§ Numerical scheme
In our numerical scheme, we solve the continuity equations (eqs. <ref>-<ref>) by a standard staggered mesh leap-frog method. Energy evolution in time is solved via operator splitting. The equations are divided into parts, with diffusion (including a flux limiter), radiative processes and compression calculated consecutively as follows.
Frequency dependent diffusion, is solved implicitly using a Newton Raphson (NR) solver, then the matter is compressed explicitly. The output from these is then fed into the radiative processes (emission, absorption, and scattering), which are solved iteratively using two loops of NR solvers that each solve for energy conservation. In the inner loop, we solve implicitly for u_ν, while keeping e constant, while in the outer loop we solve implicitly for e. The inner loop includes several protections from non-physical results in u_ν, including an `overshoot' protection to prevent u_ν from crossing B_ν(T) when attempting to equalize u_ν=B_ν(T) (see details in <ref>). The initial guess for u_ν involves solving radiative transfer explicitly, and if the NR solver fails to find a solution after 30 iterations, the solver also attempts a solution starting from the original u_ν value.
Finally, where needed (e.g. eq. <ref>, <ref>, κ_*, κ_ abs,ν), we extract the temperature and pressure by solving the equation of state, eq. (<ref>) implicitly. The entire set of equations for the evolution of the matter and radiation energies is solved using a predictor-corrector with opacities updated at every iteration prior to the diffusion step.
§.§.§ Time steps
The minimum of the following constraints limits our simulation time step. For grid cells i, the usual Courant upper limit is Δ t_ c=f_ cmin{Δ x_ i/C_ s,i}, where Δ x is the grid spacing and C_s is the speed of sound. Diffusion also limits the time step according to Δ t_ d=f_ dmin{u_ν/∂ j_ν/∂ x} _ i,ν, where the minimum is taken over cells i and bins centered at frequency ν. The factor f_ d, along with all the similar f factors here, is of order unity and smaller than one, and our results are shown to be insensitive to the exact value. Finally, we limit the time step also by limiting the maximal change due to radiative processes of the total energy density of the radiation/plasma,
Δt_r=f_r
u/j_B+j_C j_B+j_C<0,
e/j_B+j_C j_B+j_C>0,
where u=∫_0^∞u_ν dν is the bolometric radiation energy density. The effective plasma and Compton scattering emissivity are given by j_ B=ρ c∫_0^∞κ_ν(B_ν-u_ν) dν and j_ C=4uρκ_ esc(T-T_γ)/m_ ec^2, where κ_ν is the frequency-dependent absorption opacity, B_ν is the blackbody distribution, and T is the matter temperature. The radiation temperature is defined as in <cit.> as
T_γ=1/4u∫_0^∞[hν u_ν+c^3/8π(u_ν/ν)^2]dν
§.§.§ `Protections' on Diffusion and Radiative Transfer
The very high opacity, reaching κ_ν∼10^6 cm^2 g^-1 at some frequencies, may lead to numeric problems in the application of the implicit solution with finite time steps. At infinitely small time steps, u_ν will be kept close to B_ν at such large opacity regions. However, using finite time steps, the implicit result for u_ν can at times `overshoot' B_ν (or proceed in the wrong direction due to strong dependence of κ_ν on temperature). In order to avoid impractically short time steps, we add several limiting `protections' immediately after the inner Newton Raphson solver for u_ν. Namely, if the resulting implicit u_ν lies outside of the range between u_ν of the previous step and B_ν, we override the result to the nearest of these values. We also limit the change during the emission/absorption time step to |Δ u_ abs,nu/u_ν|<f_ abs, where f_ abs is in the range {0.1,0.5}. Its value does not affect our results. Though the latter constraint affects the relative rates of physical processes, absorption still proceeds quickly relative to the other processes in this scheme.
We also add a flux limiter to the simulations. The P1 diffusion approximation doesn't require one in principle. However, at certain frequencies, strong absorption and the strong sensitivity of κ_ν to temperature, can lead to a situation where the photon energy density in a particular frequency group and cell depletes abruptly -for example to match B_ν(T)-. The change may occur faster than the time in which diffusion can respond given finite time resolution, and thus may lead to non-physical flow. Namely, either flux would flow in the wrong direction, or |j|>u_ cellc, where u_ cell is the energy density in the cell from which the flux j is exiting, and c is the speed of light. In lieu of adding another time constraint, we insert a flux limiter as follows,
j_ν→ j_ν, FL=j [1+(|j_ν|/u_ cellc)^m]^-1/m
where m is a positive integer constant.
The derivative for j -defined in between cells- with respect to u, in either one of the adjacent cells, is written as
∂ j_ν, FL/∂ u=α/[1+|γ|^m]^1/m-γ^m-1/[1+|γ|^m]^1/m+1[j_ν/|j_ν|α-|γ|c∂ u/∂ u_ cell],
where α≡∂ j_ν/∂ u and γ≡ j_ν/u_ cellc. This derivative is used in the Newton Raphson solver during the diffusion step.
We test the flux limiter on the gray diffusion runs with various values of m, finding a deviation of less than a percent from the non-flux limited runs when m=8, which we choose in our simulations.
§.§ Initial and boundary conditions
Our numeric calculation involves a succession of three simulations, with each simulation starting later in physical time using a snapshot of the hydrodynamic variables as described by its predecessor. Each successive simulation also contains increasing physical complexity; i.e. first a hydrodynamic-only calculation, then a gray diffusion calculation, and finally a MG simulation. This way, later time stages of interest include all the relevant processes, while allowing the computations to be performed in practical time. All simulations are proceeded through to the latest times for comparison.
Following and , we begin with a simplified progenitor structure, comprising of a uniform density core surrounded by a polytropic envelope at hydrostatic equilibrium. We start a hydrodynamic only simulation where we inject a high thermal energy density in the innermost cells of the core and capture the resulting shock using artificial viscosity. Then a radiative diffusion-hydrodynamics gray simulation is started between 24 and 8 shock crossing times prior to shock breakout, with the exact start time have negligible effect in this range. Both of the simulations are identical to the ones in (and described there in detail), with the important exception that in the gray simulation we increase the initial cell resolution towards the stellar edge[At the edge we use a grid spacing Δ r_ grid∝ (R-r), down to a predetermined scattering optical depth of τ_ es≲ 10^-3. The additional edge resolution has no effect on the results of the gray simulation and allows us to improve edge resolution in the subsequent MG calculation.].
Then we begin a multi-group diffusion simulation, as described in <ref>. The simulation is started between 20 shock crossing times prior to breakout and up to 2 R/c times after breakout, with the exact time at which the simulation is started having negligible effect on the later shock-cooling emission. Typical resolutions for each of the respective stages listed above are 4000-8000, 1600-3200, and 200-1600 cells (MG runs that are started after shock breakout typically have 50-200 cells), with 32-256 photon groups in the MG phases. All calculations are continued until at least after the recombination time t_0.7 eV.
Multigroup simulations that are started prior to breakout have a similar initial grid to the gray diffusion simulations. Namely the grid changes smoothly, with modest resolution in the interior, highest resolution at the starting location of the shock, and steadily decreasing resolution outwards , before approaching a constant for τ≲ c/v_ bo, and finally increasing resolution at the stellar edge, with Δ r ∼ (R-r), down to at least a scattering optical depth of τ_ es∼ 10^-2. MG simulations that were started after shock breakout have a simpler initial cell grid, spaced logarithmically in τ_ es, with the same stellar edge resolution. All MG simulations have photon frequency bins that are constant in time and are spaced logarithmically.
For all simulations we assume a static reflective boundary in the inner surface, and a free boundary at the outer surface that for the diffusion simulations accelerates as ∂_t v_ b = j_ bκ / c, where the subscript b denotes boundary values. The boundary flux is given by j_ b=f_ eddcu_ b, where f_ edd=0.3-0.5 is the Eddington factor. The results are insensitive to the exact value of f_ edd since the flux is determined deep within the plasma, at τ∼ c/v≫1.
§.§ Code validation and numerical convergence
In , we validated our numerical hydrodynamical-only code against the analytic planar stellar breakout solutions of <cit.> and <cit.>, and our gray diffusion code against the analytic planar "Sakurai-Weaver" Anzats solutions of . We also reproduced the bolometric breakout flux expected from a planar stellar breakout, as also described in . In <cit.>, an earlier version of the multi-group code that we use here, underwent several test problems involving radiative diffusion, emission/absorption, and inelastic Compton scattering.
We perform here two additional tests of the multigroup code; the problem of a steady planar radiation mediated shock, and the breakout spectrum in a hydrogen dominated stellar envelope, both including inelastic Compton scattering and only free-free absorption opacity.
We calculate the structure of the steady planar radiation mediated shock at two representative velocities, β= v/c=1% and 10%, i.e. spanning breakout velocities in our parameter range. We find for both cases that density ρ, velocity v, and bolometric photon energy density u, converge to 3%, 0.5%, and ∼1% relative to the analytic result <cit.>. For the β=1% simulation, the photons are in LTE, and u_ν is in excellent agreement with a Planck distribution matching total bolometric luminosity u. For β=10%, the photons carrying most of the energy are in Compton equilibrium. While an exact analytical solution does not exist for the spectrum in this case, we find good agreement between the numeric results for the frequency of the peak of the radiation energy density behind the shock and an analytic estimate using a Wien distribution[The Wien distribution is determined by two parameters, the photon energy density u, which is determined analytically, and the number density of photons n_γ, which we extract from the simulation.], see figure <ref>. At lower frequencies, the energy density transitions to a thermal distribution due to large free-free opacity, producing a visible deviation from the Comptonized spectrum, as is expected.
Next, we compare our results for envelope breakout with the approximate table values from , which again assume a fully Comptonized Wien spectrum. We find reasonable (10's of %) agreement in peak temperature and luminosity (fig. <ref>), which is somewhat remarkable, since the temperature and luminosity profiles in these tables are a function of only the breakout parameters (R,ρ_ bo,β_ bo - see <ref>). For the comparison we extract these parameters from the simulation without performing additional fitting for the SED. There is again a noticeable deviation in the low energy tail due to thermalization. <cit.> performed MG diffusion calculations of the planar envelope breakout phase, and obtained similar results to ours. They find 10's of % agreement with in peak temperature and luminosity, as well as a thermalized low-frequency tail of similar shape.
In , we showed convergence of the hydrodynamic and gray simulations. Our MG calculations are also converged with respect to spatial resolution. Doubling the spatial resolution produces at most a few percent change (and often less than 1%) in L_ν≡∂ L/∂ν. We also verify that we are converged with respect to the resolution of the outermost cell, finding that L_ν varies by less than 1% when the minimum scattering optical depth varies between τ_ es∼ 10^-2 and 10^-3, in agreement with the conclusions of <cit.>.
We note that our frequency resolution is high enough that our SED is insensitive to the number of photon frequency groups, but is coarse relative to the atomic line scale, which we discuss at length in <ref>.
As described in , we include in both the gray and MG diffusion simulations, a non-radiating plasma component coupled with artificial viscosity q. This addition helps stabilize against numerical instabilities associated with the density inversion that occurs at the outer edge of the ejecta.
§ OUR COMPOSITE OPACITY TABLE
Calculating the frequency dependent opacity requires employing several approximations and assumptions. Primary challenges involve solving the many-electron Schrödinger equation and estimating microplasma interactions between species. The assumption that all degrees of freedom are in thermal equilibrium is often, though not always, employed. Due to these approximations, there are large uncertainties in the opacity. For example, a factor of 2 discrepancy in the Rosseland mean exists between TOPS and OP <cit.> in our regime of interest (as shown in ).
We built our own frequency-dependent opacity table, containing free-free, bound-free, and bound-bound components, and assuming local thermal equilibrium. The code that produces these tables is now available to the community on github <cit.> and produces tables at arbitrary density, temperature, and frequency resolution (we use Δν/ν∼ 10^-5-10^-6 in practice).
In we used the Rosseland mean opacity from the high-resolution tables to provide a formula for blackbody emission. Here we bin the tables in frequency to produce our multigroup opacities. For the absorption term (κ_ abs,ν in eq. <ref>) we use the average opacity across each bin, while for the diffusion term (κ_* in eq. <ref>) we take the Rosseland mean across the bin. In this work, we include 10 important atoms up to iron: H, He, C, N, O, Ne, Mg, Si, S, Fe (though the opacity code can handle an arbitrary mixture). We show later that the resulting supernova lightcurves are insensitive to the exact composition in the case of Hydrogen dominated envelopes.
As discussed in , the frequency-dependent TOPS opacities are verified against experiments at temperatures and densities exceeding tens of eV and 10^-6g cm^-3 respectively <cit.>. The Kurucz database of atomic transitions was calibrated against measured line frequencies and oscillator strengths <cit.>, but is incomplete for highly ionized species at high, T>10 eV, temperature. We separately run simulations using both our opacity table and using TOPS.
This section is written as follows. In <ref>, we describe the construction of the opacity, and in <ref> we describe tests and results of the table.
§.§ Opacity - Construction
We solve for the ionization and excitation population numbers self-consistently with the Saha equation, and using electron levels provided by the NIST database[<https://www.nist.gov/pml/atomic-spectra-database>]. For the free-free components and Hydrogen-like bound-free and bound-bound components, we use the equations provided in <cit.>. We verify our free-free calculations against TOPS, finding 7% agreement or better.
For line transitions in Hydrogen-like ions with charge Z, the decay rate is given by A(Z)∼ν^2f=Z^4A(Z=1), where A(Z=1) is the decay rate for the equivalent Hydrogen line, and ν∼ Z^2 is the frequency of the atomic transition. Non Hydrogen-like bound-free absorption is based on the table in <cit.>[This table only includes photoionization from electons in the ground-state. To calculate the electron ground occupation levels, we include all states up to 0.3 eV above the ground state to account for hyperfine splitting.]. In general, the results of the table agree w/TOPS up to a factor of 2, with the exception of a pure Fe mix we tested, where we observe an order of magnitude difference. Bound-bound oscillator strengths, degeneracies and lower energy levels are taken from Kurucz CD 23[Kurucz line lists are expanded semi-regularly, representing ongoing progress. Our simulation results are insensitive to the choice of line list, including CD 23 and the most up-to-date file from Oct. 8th 2017. In cases of mismatch between the reported lower and upper state energy gap and the transition frequency, we use the lower state and the transition frequency. Missing decay rates are estimated analytically where available. For nearly all lines, the natural line width is smaller than the thermal width, which in turn is smaller than our grid resolution.].
Our line sampling method varies with line width relative to the grid resolution Δν_ grid. Lines with full width half max (FWHM) that are much thinner than the grid spacing, Δν, are sampled as a single grid point, with magnitude proportional to ∼ (Δν)^-1 to conserve total flux. Broad lines are sampled as a Voight function ϕ_ν, including native line width and thermal broadening. Intermediate width lines are sampled from the frequency derivative of ∫ϕ_ν dν such that oscillator strength is conserved. We find that we are insensitive to the exact choice of cutoff between each of the sampling regimes in the ranges 1/30<FWHM/Δν_ grid<1/3 and 1<FWHM/Δν_ grid<10, respectively. We therefore choose the cutoffs at FWHM/Δν_ grid=1/10 [To avoid cases where the Lorentz wings can have an affect, we also require the FWHM of the Lorentzian (not including thermal broadening) to be 3×10^3 times thinner than δν_ grid to sample as a delta function. We also don't use the delta function for any of the Hydrogen lines.] and 3.
For quicker calculation in the latter two cases, the broad Voight wings are interpolated at 100 times coarser resolution than the frequency grid, and combined at the end of the calculation[We are insensitive to the location of the cutoff between fine and coarse sampling resolution. Nominally, we place the cutoff 3 FWHM's away from the center.]. Finally, ions and electronic states with electron occupation fractions below 10^-14 are removed (we tested that this omission has no effect).
We also include Red-wing continuum suppression <cit.>, and
the Hummer-Mihalas factor that suppresses higher electron states <cit.>.
Our opacity code allows computation of other ingredients that were not added in this work as they were shown to have negligible influence on the multigroup simulations in our range of parameters, including the Dappen Anderson Mihalas factor <cit.>, and electron collisional broadening. These latter processes are described in the documentation for the opacity.
§.§ Tests of the Opacity Calculation and Results
We test our frequency-dependent opacity code against TOPS and OP for the simplest case of a pure Hydrogen mixture, finding good agreement (≲ 15%) in both Rosseland mean and frequency-dependent opacity (see figs. <ref> and <ref>). As a further sanity check, we also compare our bound-bound opacity for a pure Fe mix with an independent code from <cit.>, finding excellent agreement (not shown). We find that our simulations are converged with respect to the underlying opacity table, as tested in more than 5 separate parameter choices spanning progenitor radius and explosion energies for 128 photon groups. We observe <2% difference in the SED when varying between Δν/ν∼ 10^-5-10^-6 base grid, and <1% when changing the number of {R≡ρ/T^3,T} grid points from {16,66} to {30,120}.
§ DEVIATIONS FROM BLACKBODY - CALIBRATED ANALYTIC MODEL
Here we provide an analytic description of the deviations of emitted spectrum from blackbody.
Our approximation formulae are derived piecewise, based on the strength of the absorption opacity, κ_ abs,ν, relative to the scattering opacity, κ_ es. At frequencies where κ_ abs,ν>κ_ es, the emitted flux may be approximated as a blackbody with a frequency-dependent thermal depth (surface of last absorption) r_ col,ν, and corresponding frequency-dependent color temperature T_ col,ν=T(r_ col,ν), given by
L_ν, BB=4π r_ col,ν^2 B_ν(T_ col,ν).
At frequencies where the absorption opacity is smaller than the scattering opacity, κ_ abs,ν<κ_ es, we base our approximation on the flux f_ν emitted by a semi-infinite planar slab of temperature T in the two-stream approximation <cit.>,
f_ν=4π/√(3)√(ϵ_ν)/1+√(ϵ_ν)B_ν(T),
where
ϵ_ν=κ_ abs,ν/(κ_ abs,ν+κ_ es).
We therefore suggest
L_ν,ϵ=(4π)^2/√(3)r_col,ν^2√(ϵ_ν)/1+√(ϵ_ν)B_ν(T_col,ν),
as an approximation for the escaping spectral luminosity in this regime <cit.>.
We first derive an expression describing the emission at the regime of relatively low absorption opacity, κ_ abs,ν<κ_ es, which occurs primarily at intermediate frequencies near and below the Planck peak (e.g. hν∼ 1-3 eV in fig. <ref>). Absorption in this regime is dominated by free-free transition with a small bound-free contribution. Neglecting the bound-free contribution we approximate eq. (<ref>), that defines the frequency-dependent thermal depth, as
∫_r_ col,ν^∞ρ√(3κ_ ff,νκ_es)dr' =1.
Here we have neglected κ_ abs,ν with respect to κ_ abs,ν and used
κ_ abs,ν→κ_ ff,ν=
4.13×10^-31g_ ffρ T^-1/2(hν)^-3(1-exp(-hν/T)) cm^2 g^-1,
where the density ρ is in cgs, temperature T is in ergs. We approximate the gaunt factor g_ ff=√(3)/πK_0(hν/T)∼0.717(hν/T)^-0.27, with K_0 being the zeroth modified Bessel function of the second kind.
Solving eq. (<ref>) using the analytic / spherical phase density profiles (eqs. <ref> - <ref>) we obtain the radius, temperature and opacity at the thermal depth,
r_ col,ν = 1.29 × 10^14 R_13^-0.01
(f_ρM_0)^0.09 v_ s*,8.5^0.78κ_0.34^0.03 t_ d^0.80ν_eV^-0.08 cm,
T_ col,ν = 2.13 R_13^0.28
(f_ρM_0)^-0.02 v_s*,8.5^0.17ν_ eV^0.25κ_0.34^-0.08
t_ d^-0.42 eV,
κ_ ff,ν =
0.02 (f_ρ M_0)^-0.05
R_13^-0.23 v_ s*,8.5^-0.66κ_0.34^-0.29
t_ d^-0.19ν_ eV^-1.66 cm^2 g^-1.
We find that modifying the expression for r_ col,ν to
r_ col,ν = R + 1.29 × 10^14 R_13^-0.01
(f_ρM_0)^0.09 v_ s*,8.5^0.78κ_0.34^0.03 t_ d^0.80ν_eV^-0.08 cm
while keeping the expressions for T_ col,ν and κ_ ff,ν unchanged provides a good description of the spectrum also at the planar phase (and at the transition from planar to spherical evolution),
In break notation we have:
r_ col,ν = R + 2.18 × 10^13 L_ br,42.5^0.48 T_ br,5^-1.97κ_0.34^-0.07t̃^0.80ν_ eV^-0.08 cm,
T_ col,ν = 5.47 L_ br,42.5^0.05 T_ br,5^0.92κ_0.34^0.22t̃^-0.42ν_ eV^0.25 eV,
κ_ ff = 0.03 L_ br,42.5^-0.37 T_ br,5^0.56κ_0.34^-0.47t̃^-0.19ν_ eV^-1.66 cm^2 g^-1,
where t̃=t/t_ br and R is given in terms of the break parameters by
R = 2.41×10^13 t_ br,3^-0.1 L_br,42.5^0.55 T_br,5^-2.21 cm.
The thermal depth values can then be inserted into eq. (<ref>) with ϵ_ν=κ_ ff,ν/(κ_ ff,ν+κ_ es) in order to describe the emission in the low absorption frequency range.
For frequency regions with strong absorption, where κ_ abs,ν<κ_ es, we return to eq. (<ref>). The thermal depth at these frequencies is located at the outer edge of the ejecta, where the density decreases sharply and the temperature, determined by the free-streaming photons, is nearly uniform. We therefore approximate r_ col,ν≈ const. (ν) and T_ col,ν≈ const. (ν) for these frequencies, and describe the emission as a gray blackbody L_BB, eq. (<ref>). At low frequencies where the free-free opacity dominates, we find numerically that the luminosity is well approximated by L_BB(0.85 T_ col). Meanwhile, at frequencies near and above the Planck peak, where atomic transitions dominate, we use both the simulations and a separate analytic estimate (see <ref>) to improve upon the approximate L_ BB (0.74 T_ col) description of the UV suppression of , replacing the suppression factor 0.74 with a function of (R,t) lying in the range [0.6,1].
The combined freq-dept formula is thus
L_ν = [L_ BB (0.85 T_ col)^-m + L_ν,ϵ^-m]^-1/m hν<3.5 T_ col
1.2 × L_ BB(0.85 R_13^0.13 t_d^-0.13× T_ col) hν>3.5 T_ col,
where m=5, and L_ν,ϵ is again given by eqs. (<ref>) and (<ref>) with the choice κ_ abs,nu→κ_ ff,ν. The 1.2 factor accounts for modest UV excess we observe in our results at the planck peak due to the presence of strong lines. The frequency slope in the Raleigh-Jeans regime is similar, but slightly lower than the blackbody value L∼ν^2.
Eq. (<ref>) can be further simplified to be given in terms of only L and T_ col, with a minor decrease in the approximation's accuracy,
L_ν =
π/σL/T_ col^4[ (B_ν(0.85 T_ col)/(0.85)^4)^-m + .
. ( 8/√(3) x^-0.155 T_ col,5^-0.1√(ϵ_ a)/1+√(ϵ_ a) B_ν(1.63 x^0.247 T_ col) )^-m]^-1/m hν<3.5 T_ col
1.2× L_ BB(1.11 L_42.5^0.03 T_5^0.18× T_ col) hν>3.5T_ col,
where x=hν/T_ col, T_ col = 5 T_ col,5 eV, and ϵ_ a = 0.0055 x^-1.664 T_ col,5^-1.0996. L and T_ col are given by eqs. (<ref>) and (<ref>).
§ NUMERIC RESULTS
In figs. <ref> and <ref> we plot shock-cooling numeric results from over a dozen multigroup simulations. Deviations from our gray blackbody formula (eqs. <ref>-<ref>) are small. In the Raleigh Jeans regime, the SED slope is slightly shallower than L ∼ν^2. As frequency increases in this regime, L_ν passes from 10's of percents above the Planck distribution to 10's of percents or more below it. Near the Planck peak at hν∼ 3.5 eV, the SED can be 10's of % in excess of the prediction of our blackbody formula (50% in extreme cases, further discussed in <ref>). In the Wien tail, the flux is suppressed due to line absorption.
The SED obtained numerically is also compared to our frequency dependent formula (eq. <ref>) yielding good agreement, with an RMS error of Δ L_ν/L_ν≲20% for hν<3 T_ col (and several 10's of % or more in the Wein tail, reflecting a very small inaccuracy in the radiation temperature). The frequency-dependent formula is generally closer to the simulation results than the gray formula prediction throughout. Both formulas shown in the figure use breakout parameters (ρ_ bo, β_ bo, R) that are derived by comparing the breakout bolometric luminosity to (as was done in ), without additional fitting.
We find in many simulations that the ratio of bolometric luminosity between the multigroup and gray simulations is approximately given by (κ_ es/κ_ Ross), where the Rosseland mean opacity κ_ Ross is evaluated at the diffusion depth, where L is determined (recall that the gray simulation only include Thomson opacity). This factor is less important during the planar phase, but generally reduces L in the multigroup simulation by 10's of % during the spherical phase, primarily due to Wien suppression.
We test the sensitivity of our results to the choice of opacity table using simulations that span the progenitor radius and explosion energy parameter range. When we use TOPS table instead of our own, we find that near recombination time, T_ col in the presence of the TOPS-derived opacity can be lower by up to 10's of % (usually <5% - see example in fig. <ref>). This result is in general agreement with our gray analysis in , where we concluded that for T_ col<4 eV, it is preferable to use our opacity table due the presence of lab-confirmed lines. For higher temperatures, early during shock-cooling, we find negligible (few percent) SED difference between the two opacity tables, since most of the observable frequencies (hν≲10 eV) are in the Rayleigh Jeans regime, and are less affected by the presence of lines.
We also test the sensitivity to composition in our simulations, finding a negligible variation when metallicity varies from Z=0.1 solar to solar metallicity (see fig. <ref>), in agreement with results. We conclude that the SED in Hydrogen dominated envelopes is insensitive to metallicity.
§ EXPANSION OPACITY, FINITE FREQUENCY RESOLUTION, DEVIATIONS FROM LTE
§.§ Expansion opacity and finite frequency resolution
Our numeric calculation method assumed that the opacity, κ_ν, does not vary significantly across the extent Δ r of a single spatial resolution element. The large velocity gradients in the flow we are considering, combined with the presence of strong absorption lines, may lead to significant variations of κ_ν across Δ r due to the space dependent Doppler shift. This effect may have a significant impact on the calculation of photon transport. For the diffusion calculation, we use a Rosseland average of the opacity, κ_R,i=1/κ^-1 where the average is over the frequency band ν_i-ν_i+1, implying a photon mean free path of l_R=1/(κ_R,iρ). The Rosseland mean is dominated by the frequencies at which the opacity is low, i.e. where lines are absent, and hence the presence of strong lines does not affect l_R significantly. In the presence of a large velocity gradient, the photon mean free path may become significantly shorter- as the photon propagates through the varying velocity of fluid, the frequencies of the lines shift and the photon will be absorbed when it reaches a position where the shifted frequency of a strong line coincides with the photon's frequency. In other words, the effective Doppler broadening "closes" the low κ_ν "windows" in frequency space, that allow a low value of κ_R and large l_R. The absorption mean free path is given in this case by
l_ exp≈c/ vΔν/νr,
where Δν is the frequency separation between strong lines, and we have used ∂ v/∂ r=v/r as appropriate for the homologous expansion phase. A line is considered "strong" if it leads to τ>1 when integrated over the photon propagation path taking into account the Doppler shift- for line opacity κ_ν=κ_lν_lδ(ν-ν_l), τ_ν=∫ dr ρκ'=κ_lρ ct(ν_l/ν) (where κ' is the Doppler shifted opacity). Thus, Δν is the frequency distance between lines with
κ_lρ ct>1.
l_ exp can be used to define an effective "expansion opacity", taking into account the finite velocity difference across Δ r due to the expansion of the plasma. The ratio between the Rosseland and expansion mean free paths is
l_ exp/l_R=(κ_Rρ r)c/ vΔν/ν
=(κ_Rρ r)c/ v(νdN/dν)^-1≈ 10τ_Rc/ v(νdN/dν)^-1,
where dN/dν is the (strong) line density and we have used ρ∝ r^-10 (as appropriate for the shock cooling expanding plasma) yielding 10τ_R=κ_Rρ r.
In the region where the escaping flux is determined, τ_R≈ c/ v, we have l_ exp/l_R≈10(c/ v)^2(Δν/ν)≈ 10^3.5(Δν/ν). We therefore expect l_ exp/l_R≫1 in this region, and thus only a negligible effect of the "expansion opacity" on the escaping flux. This is demonstrated to be the case in Fig. <ref>, showing l_R/l_ exp for various photon energies near recombination time as a function of τ_ es, the electron scattering optical depth. As demonstrated in the figure, the suppression of the mean free path due to the "expansion opacity" effect at this time may affect the spectrum at photon energies >5 eV, since l_ exp/l_R<1 may be obtained at the thermalization depth of such high energy photons. We arrive at the same conclusion from examination of simulations spanning different progenitor radii and explosion energies.
In order to test the possible impact of the expansion opacity on the high energy spectrum, we compare (see fig. <ref>) the flux obtained in the numeric simulation, which does not include the effects of expansion opacity, with the flux obtained using the analytic approximations of eqs. (<ref>) and (<ref>) with r_ col,ν (eq. <ref>) calculated with the density and temperature profiles obtained in the simulation. In the latter we use a high resolution, Δν/ν∼10^-5 opacity table and take into account the Doppler shifts of the lines[T_ col,ν, and ϵ_col,ν are determined at r=r_ col,ν, and we approximate ϵ_col,ν= τ_ abs,ν / (τ_ abs,ν + τ_ es), where the optical depths again include Doppler shifting and are evaluated up to the thermal depth, (τ_*,ν=1).].
The SED is given by
L_ν, Dopp =
Eq. (<ref>) τ_ es ( τ_*,ν=1) ≤ f_ cut
Eq. (<ref>) τ_ es ( τ_*,ν=1) > f_ cut,
where f_ cut determines the scattering optical depth τ_ es at the thermal depth below which we neglect the effect of scattering. The results are not sensitive to the choice of f_ cut in the range 0.3-3. We show results for f_ cut=1.
The analytic approximation, with r_ col calculated using a course frequency grid and neglecting expansion opacity effects, as done in the numeric simulations, reproduces well the spectrum obtained in the simulations.
The modifications of r_ col, and the implied flux modification in the analytic approximation, due to the inclusion of the Doppler shifts of lines using a high resolution frequency grid, yield therefore an estimate of the magnitude of this effect. The good agreement between the simulations results and the analytic estimate obtained as described above implies that the effect of expansion opacity is small, typically of order 10%.
The comparison of the analytic and numeric results described above, reveals several frequency bins at intermediate photon energies, 5-8 eV, in which the flux obtained in the numeric simulation exceeds that which is obtained by the analytic estimate by a factor of a few. This is due to the presence of strong and relatively isolated lines, which lead to a large average absorption opacity κ_ abs,i of the photon bin, which in turn leads to a blackbody photon spectrum across the frequency bin with temperature corresponding to the plasma temperature. Since the radiation energy density at intermediate photon energies is below the plasma temperature at radii where the optical depth is small (see <ref> and fig. <ref>), the flux obtained in bin i is significantly larger than the flux obtained at other frequencies. This result is an artifact of the numeric calculation using finite frequency resolution- the emitted flux would match the blackbody spectrum (of the plasma temperature) only across the (very) narrow line width, hardly affecting the total flux in the finite frequency range of bin i. In order to remove this effect, we have identified and removed a few such isolated lines from the opacity table used in the simulations.
The 5 removed lines are listed in table <ref>. Much of the remaining discrepancy between the simulations and the analytic estimate is likely due to the fact that the flux obtained in the numeric simulation exceeds that of the analytic results due to an overestimate of the effect of lines.
§.§ Deviation from LTE Ionization and Excitation
The density of the plasma emitting the radiation during the hours to days of shock breakout and cooling is relatively high, and the radiation is close to thermal equilibrium with the plasma - see fig. <ref>. This situation is quite different from that prevailing later, on weeks time scale, where the density is low and the radiation is far from thermal equilibrium, causing the ionization fraction and excitation level distribution to deviate largely from Boltzmann.
The relatively large density implies that the time scale for all relevant processes (electron-electron and electron-ion collisions, photo-ionization and excitation, electron impact excitation), with the exception of electron impact ionization, are much shorter than the dynamical time (see fig. <ref>). This, combined with the fact that the photon spectrum at energies exceeding the Planck peak (e.g. hν≳ 3 eV during recombination), is close to thermal out to very low τ_ es, implies that the ionization fraction and the excitation level distribution of the low energy excited states are both close to LTE. The fact that at low optical depth, τ_ es≲ 1, the radiation energy density falls below that of LTE at low photon energy, implies that the level distribution of the higher energy excited states may deviate from LTE at the outer edge of the ejecta. However, we note that the distribution of excited energy levels are strongly dominated by UV transitions to and from the highly populated ground state, hence deviations from LTE occupation of the higher energy states are expected to be mild. In addition, the effect of lines on the SED in this energy range is small, and hence we do not expect a significant effect due to deviations from LTE.
§ COMPARISON TO EARLIER WORKS
<cit.> (herafter ) produced an analytic model including deviations from blackbody due to free-free opacity. They arrive at an SED that disagrees with our simulations by an order of magnitude or more in the infrared and up to a factor of two near the Planck peak. In <cit.>, the formula is compared to simulations produced by the STELLA code. They conclude, similarly, that the model's infrared behavior does not describe the SED for hν<3T_ col, and suggest using a blackbody formula for this range. In fig. <ref>, we compare the formula, including the <cit.> corrections, with the results of our MG simulations. We find that their analytic model does not reproduce well the simulation results, due the different values obtained in their analytic approximation for L and T_ col, which in turn is largely due to their opacity approximation neglecting bound bound transitions (see Paper I).
Next, we compare our numeric results to those obtained using STELLA <cit.> radiation transport calculations <cit.> for several non-polytropic profiles obtained using MESA stellar evolution
calculations <cit.>. In each case, we approximate the progenitor density profile used in earlier simulations by a simple polytrope. We show two such examples in fig. <ref> and summarize the profile parameters we use in table <ref>. In figs. <ref> and <ref> we then compare the resulting emission from our multigroup simulations to the published results, without performing any further fitting.
The comparison to the results of <cit.> is shown in fig. <ref>. We find a good agreement with our results when we include in our calculations only bound-free (neglecting bound-bound) line opacity. This is due most likely to the fact that <cit.> treated bound-bound absorption as scattering. While this approximation may be valid at late times, when the density of the plasma is low and the radiation energy density is far below that of thermal equilibrium, this is not a good approximation at the early stage of shock cooling emission (see discussion in <ref>).
The comparison to the results of <cit.> is shown in fig. <ref> (top). We find a good agreement between the results of the different calculations.
Finally, fig. <ref> (bottom) shows a comparison to the results of <cit.>. There is good agreement between the results at the latest observed times (t=3 days after breakout). At early time, there is good agreement at longer wavelengths, λ>3000Å, where the free-free emission dominates, but there is a clear discrepancy at higher frequencies. The very high temperature of the emission peak obtained in <cit.> at these times, T∼70 eV, appears to be non-physical, since for the corresponding progenitor parameters it is associated with the temperature of the envelope at an optical depth τ∼ 1000. <cit.> arrive at the same conclusion, noting that that particular model features a very weak line absorption opacity, and as a result, unreasonably high emission temperatures.
The agreement of our results with these earlier calculations provides additional support to our code’s validity (in particular to the applicability of the diffusion approximation), and to the conclusion of the detailed analysis of SW17, that the shock cooling emission is not sensitive to deviations of the density profile from a polytropic one.
§ DISCUSSION AND SUMMARY
We derived an analytic description of the deviations of shock cooling spectra from blackbody for red supergiant explosions. The approximation is given by eqs. (<ref>) and (<ref>). The definitions of (and equations for) all variables appearing in eqs. (<ref>) and (<ref>) are given in the Appendix. The analytic description holds from post breakout (t>3R/c), up to H recombination (T≈0.7 eV), or until the photon diffusion time through the envelope becomes shorter than the dynamical time, see eq. (<ref>). The analytic expressions were calibrated against a large set of numeric MG calculations, and provide an excellent approximation to the numeric results- see <ref>, figures <ref> and <ref>. In accordance with SW17 and Paper I, we find that the results are not sensitive to metalicity and to the ratio of core to stellar radii, in the range R_ c/R=10^-1-10^-3. In <ref> and <ref> we showed the effects of deviations from ionization and excitation LTE and of `expansion opacity' corrections are small.
Our analytic formula depends on the same four parameters as the previous one in , {R, v_ s∗,f_ρ M,M_ env}. Of these parameters, the SED is most sensitive to the progenitor radius R, and the ejecta velocity v_ s∗, and insensitive to the parameters f_ρ M and M_ env, where M and M_ env are the total mass of the ejecta and mass of the envelope, and f_ρ is a dimensionless factor of order unity that depends on the inner envelope structure (see eq. <ref>). Therefore, the former two parameters are the ones most readily extracted from observations. We note also that deducing parameter values is hindered by the difficulty of determining T_ col at early times, when the maximum observable photon frequencies (hν≲ 10 eV) may be located below the thermal peak. For this regime, the frequency-dependent deviations from blackbody may prove to be an important discriminator between models of the Raleigh-Jeans part of the spectrum.
Our results are compared in <ref> to earlier numerical results including STELLA radiation transport calculation results for several non-polytropic profiles <cit.> obtained using MESA stellar evolution calculations. The agreement of our results with these earlier calculations provides additional support to our code's validity (in particular to the applicability of the diffusion approximation), and to the conclusion of the detailed analysis of SW17, that the shock cooling emission is not sensitive to deviations of the density profile from a polytropic one.
Based on our analysis of <ref>, we find that the emergence of Balmer lines does not coincide with the recombination time but likely occurs earlier (approximately at T∼2-3 eV). While the recombined H fraction at this time is very low, and the effect of the lines on the SED should be small, the lines may be visible in a spectrum. The appearance of the first Balmer lines should not be associated with the recombination time t_0.7 eV, when the temperature drops to 0.7 eV.
Our frequency-dependent formula agrees with the simulations' results with an RMS error of ≲20% (with the exception of the Wien tail, where deviations are larger but reflect a very small inaccuracy in radiation temperature). This error is somewhat larger than the uncertainty in our numeric results, ≈ 10%, as quantified by the difference between the simulations' results and the high frequency-resolution calculation (eq. <ref>). Therefore, in order to best incorporate the uncertainty of our analytic model when comparing to observations, we suggest using a covariance matrix (as function of (ν,t)) of the residuals between the frequency dependent formula (eq. <ref> or <ref>) and our published simulations (see Data Availability).
§ ACKNOWLEDGEMENTS
We thank Barack Zackay for his contribution to our numerical code, as well as Gilad Sadeh for insightful discussion. EW's research is partially supported by ISF, GIF and IMOS grants.
§ DATA AVAILABILITY
Numerical codes used in this paper will be provided upon reasonable request to the corresponding author. Our opacity table code is available online for public use at <cit.>. Numeric results from our simulations are available at <https://www.dropbox.com/s/ub5zg1rngodjb1d/RSG_Solar_Metallicity.mat?dl=0>.
mnras
§ SUMMARY OF MODEL EQUATIONS
The basis for the analytic approximations of the spectra given in this paper is the analytic approximation of Paper I for the bolomteric luminosity
L and the color temperature T_ col (eqs. <ref>-<ref>),
L/L_ br=t̃^-4/3+t̃^-0.172× Aexp(-[at/t_ tr]^α),
T_ col/T_ col,br=min[0.97 t̃^-1/3,t̃^-0.45].
Here {A,a,α} ={0.9,2,0.5}, t̃=t/t_ br, and we define t=0 as the time at which the breakout flux peaks. The break parameters (with br subscript) are given as a function of progenitor radius R, ejecta velocity v_s*, and total ejecta mass M, by (eqs. <ref>-<ref>)
t_ br= 0.86 R_13^1.26 v_ s*,8.5^-1.13
(f_ρM_0κ_0.34)^-0.13 hrs,
L_ br=3.69×10^42 R_13^0.78 v_ s*,8.5^2.11
(f_ρM_0)^0.11κ_0.34^-0.89 erg s^-1,
T_ col,br= 8.19 R_13^-0.32 v_ s*,8.5^0.58
(f_ρM_0)^0.03κ_0.34^-0.22 eV.
Here, R= 10^13 R_13 cm, κ=0.34 κ_0.34 cm^2 g^-1, v_ s*=v_ s*,8.5 10^8.5 cm s^-1, M_0 denotes mass in units of solar mass, and f_ρ≃1 depends on the inner structure of the envelope (see eq. <ref>). v_ s∗ is related to the characteristic ejecta velocity v_∗ by (eq. <ref>)
v_ s∗≈ 1.05 f_ρ^-0.19v_∗, v_∗≡√(E/M),
where E is the energy deposited in the ejecta.
Our analytic approximation for the luminosity and spectra of the shock cooling emission, taking into account deviations from a blackbody spectrum, is (eq. <ref>)
L_ν = [L_ BB (0.85 T_ col)^-m + L_ν,ϵ^-m]^-1/m hν<3.5 T_ col
1.2 × L_ BB(0.85 R_13^0.13 t_d^-0.13× T_ col) hν>3.5 T_ col,
with m=5 and (eqs.<ref>, <ref>)
L_ BB=L×π B_ν(T_ col)/σ T_ col^4,
L_ν,ϵ=(4π)^2/√(3)r_col,ν^2√(ϵ_ν)/1+√(ϵ_ν)B_ν(T_col,ν), ϵ_ν=κ_ ff,ν/κ_ ff,ν+κ_ es
and (eqs. <ref>-<ref>)
r_ col,ν = R + 2.18 × 10^13 L_ br,42.5^0.48 T_ br,5^-1.97κ_0.34^-0.07t̃^0.80ν_ eV^-0.08 cm,
T_ col,ν = 5.47 L_ br,42.5^0.05 T_ br,5^0.92κ_0.34^0.22t̃^-0.42ν_ eV^0.25 eV,
κ_ ff = 0.03 L_ br,42.5^-0.37 T_ br,5^0.56κ_0.34^-0.47t̃^-0.19ν_ eV^-1.66 cm^2 g^-1.
Here L_ br=L_ br,42.5 10^42.5 erg s^-1, T_col=5 T_ col,5 eV, and ν=ν_ eV eV, and R in terms of the break parameters is (eq. <ref>)
R = 2.41×10^13 t_ br,3^-0.1 L_br,42.5^0.55 T_br,5^-2.21 cm.
A simpler approximation, that depends only on the L and T_ col and is slightly less accurate, is (eq. <ref>)
L_ν =
π/σL/T_ col^4[ (B_ν(0.85 T_ col)/(0.85)^4)^-m + .
. ( 8/√(3) x^-0.155 T_5^-0.1√(ϵ_ a)/1+√(ϵ_ a) B_ν(1.63 x^0.247 T_ col) )^-m]^-1/m hν<3.5 T_ col
1.2× L_ν, BB(1.11 L_42.5^0.03 T_5^0.18× T_ col) hν>3.5T_ col
where x=hν/T_ col, T_ col = 5 T_5 eV, and ϵ_ a = 0.0055 x^-1.664 T_ col,5^-1.0996.
Our analytic approximations are valid at (eq. <ref>-<ref>)
3 R / c = 17 R_13 min < t < min[t_ 0.7 eV, t_ tr/a],
where
t_0.7 eV = 6.86 R_13^0.56 v_ s*,8.5^0.16κ_0.34^-0.61 (f_ρM_0)^-0.06 days,
t_ tr = 19.5 √(κ_0.34M_ env,0/v_ s*,8.5) days.
|
http://arxiv.org/abs/2307.04991v1 | 20230711030458 | Periodic Trajectories and Topology of the Integrable Boltzmann System | [
"Sean Gasiorek",
"Milena Radnović"
] | math.DS | [
"math.DS",
"37J35, 37J39, 37J46, 37C79, 37C83, 70G40"
] |
Adversarial Training Over Long-Tailed Distribution
Guanlin Li
Nanyang Technological University, S-Lab
[email protected]
Guowen Xu
City University of Hong Kong
[email protected]
Tianwei Zhang
Nanyang Technological University
[email protected]
August 12, 2023
============================================================================================================================================================================================================================
We consider the Boltzmann system corresponding to the motion of a billiard with a linear boundary under the influence of a gravitational field. We derive analytic conditions of Cayley's type for periodicity of its trajectories and provide geometric descriptions of caustics.
The topology of the phase space is discussed using Fomenko graphs.
Keywords: Integrable Boltzmann system, Kepler problem, billiards, periodic trajectories, Poncelet theorem, Fomenko graphs
MSC2020: 37J35, 37J39, 37J46, 37C79, 37C83, 70G40
§ INTRODUCTION
The Boltzmann system <cit.> consists of a massive particle moving in a gravitational field with a linear boundary (i.e. the wall) between the particle and the centre of gravity.
The reflections off the boundary are absolutely elastic, meaning that the kinetic energy remains unchanged by them, and they obey the billiard reflection law, i.e. the angles of incidence and reflection are congruent to each other.
It was recently shown in <cit.> that this system is integrable, since it has, in addition to the energy, a second integral of motion. That additional integral can be geometrically seen in the fact that, for each trajectory, there is a fixed circle such that all arcs of the trajectory are Kepler conics with one focus at the centre of gravity and the second focus on that fixed circle.
In <cit.>, this system is analysed further, and it is proved that the Boltzmann map is equivalent to a shift on a given elliptic curve, which then implies a Poncelet-style closure theorem in this setting:
if a given trajectory of the Boltzmann system is n-periodic, then each trajectory consisting of arcs with foci on the same circle as the initial n-periodic trajectory is also n-periodic.
This Poncelet-style closure result is the initial point for this paper, from which we find the analytic conditions for periodicity of the trajectories of the Boltzmann system. Those conditions and examples are presented in Section <ref>.
In Section <ref>, we discuss the geometry of Boltzmann trajectories.
We show the existence of caustics and the focal reflection property in Theorems <ref> and <ref>.
In Section <ref>, the phase space of the Boltzmann system is analysed.
We describe in detail the dynamics on the singular level sets in Theorem <ref> and, using the Fomenko graphs and invariants, we provide topological description of the isoenergy manifolds in Theorem <ref>.
Before we proceed to those discussions, we will briefly recall, following <cit.>, the construction and recent results of the Boltzmann system.
§.§ Integrability of the Boltzmann billiard
Here we review of notions and results from <cit.>, which we will immediately use in Section <ref> to derive analytical conditions for periodicity of the Bolzmann system.
In the classical 2-body Kepler problem with an inverse square central force law, solutions to the reduced problem in some inertial reference frame are conics with one focus at the origin. The position r and linear momentum p of the reduced body relate to the angular momentum L by
L = r×p,
which is an integral of motion and defines the plane of motion.
The total energy E and Laplace–Runge–Lenz vector A = p×L-𝐫|𝐫| are also integrals of motion. The conic is an ellipse if E <0, a parabola if E=0, and one branch of a hyperbola if E>0.
Following the notation from <cit.>, we consider the Boltzmann system in the plane with coordinates x_1, x_2 with the centre of gravity at the origin and wall at x_2 =1.
Vector 𝐀 will then also be in that plane and we denote its components by A_1, A_2, while 𝐋 is orthogonal to the plane and we denote its third component by L.
It was shown in <cit.> that
D := L^2 - 2A_2
is an integral of motion for the Boltzmann system.
In the coordinates (x_1,x_2), the Kepler conic is given by
: x_1^2 + x_2^2 = (D+2A_2 - A_1 x_1 - A_2 x_2)^2.
One focus of is at the origin and the other one is F_2:=(A_1/E, A_2/E) that lies on a circle of radius R/|E| with
R^2 := 1+2DE + 4E^2 centred at the point (0,2).
The point (0,2), the centre of the circle , is symmetric to the origin Ø(0,0) with respect to the wall.
We will show in Section <ref> that this point will play other significant roles in the dynamics of the Bolzmann system, see Theorems <ref> and <ref>.
The pair (E,D) determines the level set X(D,E) of the system.
According to <cit.>, the level set X(D,E) has a compactification that is a smooth projective curve of genus 1 whenever
the following conditions hold:
D^2≠4, 1+2ED+4E^2≠0, D+2E≠0.
Such level sets correspond to the non-degenerate Liouville tori.
If some of the inequalities eq:singular-conditions are not satisfied, the level set X(D,E) will be singular.
We note that in Section <ref>, where our goal is to find analytic conditions for periodicity, only non-singular level sets are of interest, so we will assume there that the conditions eq:singular-conditions hold.
On the other hand, the singular level sets will be analysed in more detail in Section <ref>.
In the non-degenerate case, the elliptic curve corresponding to X(D,E) is:
y^2=(1-s^2)(1-k^2s^2),
with
k^2=D+4E-2R/D+4E+2R.
That elliptic curve can also be seen in the affine space with coordinates (x,A_1,A_2) as:
A_1^2+A_2^2-4EA_2=1+2DE,
x^2+1=(A_2+D-A_1x)^2,
where (x,1) is a reflection point on the wall and 𝐀=(A_1,A_2) is the corresponding Laplace-Runge-Lenz vector.
The Boltzmann map is given in those coordinates as a composition of two involutions i and j:
i(x,A_1,A_2)=(x',A_1,A_2),
j(x,A_1,A_2)=(x,A_1',A_2'),
with
x'=-2(A_2+D)A_1/1-A_1^2,
A_1'=(x^2-1)A_1-2xA_2+4xE/x^2+1,
A_2'=-2xA_1-(x^2-1)A_2+4x^2E/x^2+1.
The involution i maps one intersection point of the Kepler ellipse to the other one.
On the other hand, the involution j preserves the intersection point, but maps the vector 𝐀 before the reflection to the vector 𝐀' after the reflection, i.e. switches to the next Kepler ellipse arc of the Boltzmann trajectory.
Both involutions i, j have fixed points <cit.>, thus they represent point reflections on the Jacobian of the elliptic curve, and
the Boltzmann map, as their composition j∘ i, is equivalent to following shift of the Jacobian of that curve:
u↦ u+∫_-1^s_0ds/y,
where
s_0=D+2E+R/D+2E-R,
which is the shift by the divisor Q_+-P_+, where points P_+, Q_+ are given by:
P_+(-1,0),
Q_+(
D+2E+RD+2E-R,
-4i(D+2E)R(D+2E-R)^2√((D + 2E)(D + 4E + 2R))).
Thus the n-periodicity of any trajectory on X(D,E) is equivalent to:
n(Q_+-P_+)∼0.
Since that condition does not depend on the initial point of motion, but only on the constants D, E, we will have the Poncelet property: n-periodicity of one trajectory of the Boltzmann system implies that all trajectories with the same constants D, E of motion will also be n-periodic <cit.>.
§ PERIODIC TRAJECTORIES OF THE BOLTZMANN SYSTEM
In this section, we will first make a very brief review of results connected with closure theorems of Poncelet type and the corresponding Cayley-type conditions, in particular in the context of billiards.
After that, we provide in Theorem <ref> the analytic conditions for periodicity of the Boltzmann billiard, and then illustrate it by several examples.
We note that closure theorems of Poncelet type and the corresponding analytic conditions originate in classical works of XIXth century mathematicians.
Namely, the classical Poncelet porism states that the existence of a closed polygon inscribed in one conic and circumscribed about another one implies the existence of infinitely many such polygons;
moreover, each point of the circumscribed conic is a vertex of one of them and all of those polygons have the same number of sides <cit.>.
While Poncelet's approach was synthetic geometric, Jacobi gave alternative proof using addition formulas for elliptic functions, see Jacobi, Schoenberg.
Explicit analytic conditions for closure were derived by Cayley <cit.>.
Modern algebro-geometric approach to Poncelet theorem and those analytic conditions can be found in GH1977,GH1978.
The interest in the Poncelet theorem, its generalizations and its applications has been renewed in the last few decades, and there is a large body of works regarding that.
A recent detailed account on the history of Poncelet theorem can be found in DC2016a,DC2016b.
More about analytic conditions for the Poncelet theorem and the generalizations, with a review of modern development in the theory of billiards can be found in DR2011,DR2014.
Among various generalisations of the Poncelet theorem, those in distinct geometries or those where a potential field is present were developed, see for example Veselov1990,DJR2003,GKT2007,KT2009,GR2021 and AF2006,Fed2001.
For even more results, see references therein.
The conditions for n-periodicity of the Boltzmann billiard are given in the following theorem.
The trajectories of the Boltzmann system with integrals D and E satisfying eq:singular-conditions are n-periodic if and only if
B_3B_4B_m+1B_4B_5B_m+2B_m+1B_m+2B_2m-1=0
with
n=2m ≥ 4,
or
B_2B_3B_m+1B_3B_4B_m+2B_m+1B_m+2B_2m=0
with
n=2m+1≥3,
where B_0, B_1, B_2, B_3, …are the coefficients in the Taylor expansion of:
√((2(D+2E)ξ-R)(4R(D+2E)^2ξ^2 + 2(D+2E)(D^2+2DE-2)ξ +R))
around ξ=0.
The algebro-geometric condition for n-periodicity is n(P_+-Q_+)∼0, with points P_+, Q_+ given by (<ref>) on the curve (<ref>) <cit.>.
We note that P_+ is a branching point of that curve.
To simplify the calculations, we make the coordinate transformation (s,y)→(ξ,η) such that ξ(P_+)=∞, ξ(Q_+)=0:
ξ=1/s+1-D+2 E-R/2 D+4 E,
η=y/(s+1)^2·(D+2E)√((D + 2E)(D + 4E + 2R)).
In these new coordinates, the elliptic curve (<ref>) becomes
:
η^2 =(2(D+2E)ξ-R)(4R(D+2E)^2ξ^2 + 2(D+2E)(D^2+2DE-2)ξ +R).
We now derive the Cayley-type conditions similarly as in <cit.>.
The divisor condition n(Q_+ - P_+)∼ 0 is equivalent to the existence of a meromorphic function on with a unique pole at P_+ and unique zero at Q_+ such that the pole and zero are both of multiplicity n.
We denote by Ł(nP_+) the linear space of all meromorphic functions on which have a pole of order at most n at P_+ and are holomorphic elsewhere.
A basis of Ł(nP_+) is:
{1, ξ, ξ^2, …, ξ^m, η, ηξ, …, ηξ^m-2},
for n=2m;
{1, ξ, ξ^2, …, ξ^m, η, ηξ, …, ηξ^m-1},
for n=2m+1.
It can be derived, in the same way as it is done in <cit.>, that there is a nontrivial linear combination of these functions with a zero of order n at ξ=0 if and only if the stated determinant conditions hold.
Now, we use the analytic conditions from Theorem <ref> to construct examples of periodic trajectories.
[Period 3]
The Cayley-type condition for a 3-periodic trajectory is B_2=0.
The coefficient B_2 is calculated from the Taylor expansion as stated in Theorem <ref>:
B_2 = -(D+2E)^2(4(D^2-4)E^2 + 4D(D^2-3)E+D^4-2D^2-3)/2|1+2DE + 4E^2|.
By assumption eq:singular-conditions, D+2E ≠ 0 and 1+2DE + 4E^2 ≠ 0, so B_2 =0 is equivalent to
4(D^2-4)E^2 + 4D(D^2-3)E+D^4-2D^2-3=0,
which is precisely the condition stated in <cit.>.
In Figure <ref>, two 3-periodic trajectories are shown.
[Period 4]
Theorem <ref> states that the analytic Cayley-type condition for 4-periodic trajectories is B_3=0, which is equivalent to:
(D^2+2DE-1)((D+2E)^2(D^2-4)-1)=0.
Examples are shown in Figure <ref>.
[Period 5]
The condition for a 5-periodic trajectory is B_2 B_4 - B_3^2=0, which is equivalent to:
0 = D^12-6 D^10+3 D^8+60 D^6-169 D^4+42 D^2+5
+64 (D^2-4)^3 E^6
+64 (D^2-4) (3 (D^2-7) D^2+52) D E^5
+16 (D^2-4)(15 D^6-90 D^4+251 D^2+4) E^4
+32 (D^2-4) (5 D^6-25 D^4+71 D^2+13) D E^3
+4 (386 D^6-452 D^4-537 D^2+15 (D^2-8) D^8+52) E^2
+4 (3 D^10-21 D^8+46 D^6+22 D^4-257 D^2+47) D E.
In Figure <ref>, two such trajectories are shown.
[Period 6]
The analytic condition B_3 B_5 - B_4^2=0 for 6-periodic trajectories is equivalent to:
0 = [ 4(D^2-4)E^2 + 4D(D^2-3)E+D^4-2D^2-3 ] [(D^2+2DE-1)^2-4(D+2E)^2 ]
×[-1+(D^2-4)(D+2E)^2 ((3D^2-4)(D+2E)^2+16E(D+2E)+6) ].
The first factor in the above expression is the condition for 3-periodic trajectories, so we find the solutions from the other two factors. Two such examples are shown in Figure <ref>.
The trajectories in the right-hand sides of Figures <ref> and <ref> meet the wall at a right angle at its leftmost and rightmost points. These perpendicular trajectories are fixed in the involution j of <cit.> and occur when A_2 = 2E/1± R.
§ GEOMETRIC PROPERTIES OF THE BOLTZMANN SYSTEM
In planar elliptical billiards, each trajectory has a caustic, that is a curve which is touching all segments of that trajectory.
Moreover, the focal property also holds: if a segment of a given trajectory contains a focus of the boundary, then then the next segment will containing the other focus.
In this section, we will prove that the trajectories of the Boltzmann system have caustics and that the focal property can also be formulated and proved in this case.
The first step is to establish that for any Kepler conic , there are particular confocal conics that are tangent to . See Figure <ref>, left.
The Kepler conic given by (<ref>) is touching two unique conics with foci Ø(0,0) and (0,2), which are given by:
_±: x_1^2/(R±1/2E)^2 -1 + (x_2-1)^2/(R±1/2E)^2 =1.
Moreover, we have:
* the points of tangency of and _± are
B_± = (A_1α_±, 2+(A_2-2E)α_±), with
α_±=
(4E^2-(R±1)^2) (R(R±1)-2E(A_2-2E))/2ER^2(4E^2-(R±1)^2)-8A_1^2E^3;
* the slope of the joint tangent line to and _± at B_± is:
m_± = A_1(R±1)(2E(A_2 - 2E)-(R±1)R)/2A_1^2 E (R±1) - R(A_2 - 2E)(4E^2- (R±1)^2);
* the points B_±, and the non-origin focus F_2 of are collinear and they lie on the following line:
Ł: (2E-A_2)x_1 + A_1 x_2 = 2 A_1.
Follows by a straightforward calculation.
Now, Proposition <ref> directly implies the existence of caustics for the Bolzmann system.
For each fixed pair (E,D) satisfying the conditions eq:singular-conditions there are two unique conics _+ and _- with foci Ø(0,0) and (0,2) such that all arcs of each trajectory on the level set X(E,D) are touching those two conics.
The existence of caustics is illustrated in Figure <ref>, were 100 iterations of the Boltzmann map for (E,D) = (-7/24,7/4) are shown.
The arcs of the trajectory have a hyperbolic caustic above the wall and the an elliptical caustic with tangencies both above and below the wall.
Next, we will show that the point which is symmetric to the centre with respect to the wall has the focusing property for the trajectories of the Boltzmann system, see Figure <ref>.
Suppose that an arc of a given trajectory of the Bolzmann systems contains the point (0,2).
Then all arcs of that trajectory will also contain point . Furthermore, the trajectory asymptotically converges to the vertical axis x_1=0.
The first part of this theorem follows from Theorem <ref>, when the caustic _- given by eq:GeneralCaustic is degenerate, i.e. R-1=2E, which implies that D=2.
The corresponding Kepler ellipses have major axis 2a=-1/E.
Denote by x_0 a point on the wall which belongs to one such ellipse _0,
and let x_1, x_2, … and _1, _2, …be the subsequent reflection points and arcs of the Boltzmann trajectory.
The origin Ø is the focus of each elliptic arc _i and we denote by F_i the other focus.
By the caustic property, the trajectory starts at x_0, travels along the first Kepler ellipse _0, passes through the point of the degenerate caustic _1 and intersects the wall at x_1. The “string construction" applied to ellipses _0 and _1 implies -1/E = |Ox_1| + |x_1 F_0| and -1/E = |O x_1| + |x_1 F_1| respectively.
Those equalities give |x_1 F_0| = |x_1 F_1|, so that we may think of F_0, F_1 as the intersection of and a circle centred at x_1 with radius R_01. By the position of x_1, it is necessarily the case that the x_2-coordinate of F_1 is larger than the x_2-coordinate of F_0.
Repeating this process with _1 and _2 produces the next focus F_2 with larger x_2-coordinate than that of F_1. We find the x_2-coordinates of the F_i are monotonically increasing and are bounded above by the top of the circle at coordinates (0,2-1/E). As the F_i approach this point, the defining components of each Kepler ellipse (x,A_1,A_2) approach (0,0,-1), respectively, which corresponds to the vertical axis x_1=0.
§ THE PHASE SPACE
In this section, we will analyse the phase space of the Boltzmann system.
While the discussion in <cit.> assumes complex values of D, E, we will consider only real values to correspond to the real motion in the Kepler problem.
Moreover, we will consider only the part of the phase space containing bounded trajectories, i.e. when the arcs of the trajectories are ellipses.
We note that the last assumption implies that each trajectory is bounded and has infinitely many reflections.
Boundedness is important for us, since then the level sets and the isoenergy manifolds in the phase space will be compact, which allows us to use the Fomenko invariants for the topological characterization.
Subsection <ref> contains a brief review of the bifurcation set for the Boltzmann system, based on <cit.>.
Then, in Subsection <ref>, we provide the analysis of the motion on the singular level sets.
In Subsection <ref>, we present the topological description of the compact isoenergy manifolds for the Boltzmann system, using Fomenko graphs.
§.§ The bifurcation set
The bifurcation set for the Boltzmann system can be represented in (E,D)-plane, with restrictions on D and E to produce real motion when the arcs of trajectories are ellipses or degenerate to straight segments.
In particular, these restrictions determine an infinite region in the plane bounded by the curves 1+2DE + 4E^2=0, E=0, D+2E=0, D=2, as shown in the right-hand side of Figure <ref> <cit.>. We call this region R.
As explained in Theorem <ref>, the caustics _± are only dependent upon the values of (E,D).
The curve _- is an ellipse for all (E,D) ∈R. However, the curve _+ is a hyperbola for (E,D) ∈R with D<2, an ellipse for (E,D) ∈R with D>2, and degenerate consisting of the two points Ø (0,0) and (0,2) when (E,D) ∈R with D=2 and E>-1/2.
The left-hand side of Figure <ref>, shows an arc of a trajectory corresponding the level set X(E,D).
§.§ Singular level sets
The singular level sets of the Boltzmann system are placed on the boundary of the region R and within R on the line D=2. The next theorem describes the dynamics on those level sets.
The singular level sets in the phase space of the Boltzmann system consist of:
* A single closed orbit corresponding to the limiting motion on the wall along the minor axis of the ellipse _+, for each (E,D) such that D+2E=0 and 0<D<2;
* A single closed orbit, corresponding to a 2-periodic trajectory on an ellipse whose minor axis is placed along the wall, for each (E,D) such that 1+2DE+4E^2=0 and D>2;
* A single closed orbit corresponding to a periodic trajectory lying on the x_2-axis and bounded by the point (0,-1/E) when D=2 and -1<E<-1/2;
* A closed orbit and a separatrix when D=2 and -1/2<E<0. The closed orbit corresponds to a periodic trajectory lying on x_2-axis and bounded by the point (0,2) and the wall, and the separatrix contains the trajectories with elliptic arcs that contain the point (0,2).
The singular level sets correspond to the values E, D which do not satisfy the inequalities eq:singular-conditions.
We consider separately each of the three possible cases.
*First, we assume D+2E=0.
By setting x_2 =1, we can find the x_1-coordinates of the reflection points of the particle with the wall:
x_1^± = -A_1(A_2+D) ± L √(D+2E)/1-A_1^2,
for A_1^2 ≠ 1.
From there, the Kepler conics will be hyperbolas tangent to the wall from above when E>0; or parabolas and ellipses tangent to the wall from below for E ≤ 0 if and only if Δ =0.
Thus, the limiting motion for the Boltzmann system, whenever E≤0, will be along the minor axis of the limiting caustic _+, see Figure <ref>.
*Second, we assume 1+2DE+4E^2=0.
Under this assumption, the Kepler conics are ellipses orthogonal to the wall at both intersection points, i.e. their minor axes lie on the wall x_2 =1 and major axes lie on the vertical coordinate axis, x_1=0. See Figure <ref>.
We can derive this condition using the involutions i, j, see (<ref>). First, a necessary condition for such a 2-periodic ellipse is x_1^+ = -x_1^-, or equivalently, i(x_1,A_1, A_2) = (-x_1, A_1, A_2). This means
-2(A_2+D)A_1/1-A_1^2 -x_1 = -x_1 A_1 =0 or A_2 = -D,
which correspond to conics whose intercepts with the wall are symmetric about x_1=0. To further correspond to a 2-periodic trajectory, we seek fixed points of j:
j(x_1,A_1,A_2) = (x_1,A_1,A_2) A_1 = x_1(2E-A_2).
Satisfying equations (<ref>) and (<ref>) leads to several possibilities.
Case 1: Suppose A_1=0 and x_1 ≠ 0. Then A_2 = 2E, and the second focus of the conic (<ref>) is F_2 = (0,2), and the equation of the conic simplifies to
x_1^2/-D/2E-2 + (x_2-1)^2/1/4E^2=1.
By equation (3) of <cit.>, these conditions also mean R^2=0 1+2DE + 4E^2 =0. This matches Felder's 2-periodic condition and is consistent with the geometric description that the second focus F_2 in the Boltzmann system lies on a circle of radius R/|E| centred at (0,2). In turn, equation (<ref>) becomes
x_1^2/1/4E^2-1 + (x_2-1)^2/1/4E^2=1.
This equation represents an ellipse in the (x_1,x_2) plane for -1/2 < E < 1/2 and E ≠ 0. Moreover, in the (E,D)-plane, the curve 1+2DE+4E^2=0 is a hyperbola with asymptotes D+2E=0 and E=0, and branches lying in the second and fourth quadrants. Further assuming D+2E>0 to ensure two distinct intersection points of the ellipse with the wall, this forces the pair (E,D) to lie on the branch in the second quadrant. Therefore we have a 1-parameter family of ellipses corresponding to 2-periodic trajectories given by equation (<ref>) for -1/2<E<0. This family of ellipses approaches a degenerate ellipse (or segment connecting the foci) as E → -1/2^+ and approaches an ellipse of unbounded major and minor axes as E → 0^-.
Case 2: Suppose A_1=0 and x_1=0. The Kepler conic equation (<ref>) reduces to
x_1^2+x_2^2 = (2A_2+D-A_2x_2)^2
and must pass through the point (0,1), which implies A_2 = -D ± 1. However, this reduces equation (3) of <cit.> to
(D+2E)(D±2)=0,
with D having the opposite sign of A_2. By assuming D+2E>0, both possibilities turn equation (<ref>) into x_1=0. Thus the only such conic is the degenerate conic x_1=0. Dynamically, this can be seen as the particle repeatedly bouncing directly up and down with no component of motion to the left or right.
Case 3: Suppose A_2 = -D. Since L^2=D+2A_2 = -D, we have D <0, which does not belong to the region R.
*Third, assume D^2=4.
Since D is positive within the bifurcation set R, we have D=2.
In the Boltzmann system, the Kepler conic is an ellipse whose foci are F_1 = (0,0) and F_2 = (A_1/E,A_2/E), and whose major axis has length 1/|E|. The second focus F_2 lies on a fixed circle of radius R/|E| centred at (0,2). The degenerate Kepler ellipses will occur when, as F_2 varies along , the length of the minor axis approaches 0, which occurs if and only if F_2 is a distance 1/|E| from the origin. Thus we seek solutions to the system
x_1^2 + (x_2-2)^2 = 1+2DE+4E^2/E^2 and x_1^2+x_2^2 = 1/E^2.
The solutions are (x_1,x_2) = (±√(4-D^2)/2E,-D/2E). Thus the line x_2=-D/2E is a line which can intersect in zero, one, or two points; depending on the location of F_2 relative to this line, the Kepler conic (<ref>) will be an ellipse, hyperbola, or degenerate. See Figure <ref>.
The condition D=2 corresponds to the case when the line x_2 = -D/2E is tangent to the circle at the point (0, -D/2E). As such, all points except the point of tangency are allowable locations for F_2 and the Kepler conic is an ellipse.
§.§ Topological description of isoenergy manifolds
In this section we give a topological description of the isoenergy manifolds for the Boltzmann system using the topological invariants developed by Fomenko and his school, see BMF1990, BF2004, BBM2010 and references therein.
Those invariants can be used for 3-dimensional submanifolds of the phase space of integrable systems with two degrees of freedom.
The Liouville folitation of such submanifolds is represented by a graph, which is obtained by shrinking each leaf of the foliation to a point.
Thus, the smooth families of Liouville tori will create edges which connect together at vertices corresponding to the singular leaves.
Each type of the singular leaf corresponds to a letter-atom.
To complete the topological description, each edge and some subgraphs are marked with rational and natural numbers.
A detailed account of those invariants, together with theoretical bakcground and examples can be found in <cit.>.
The Fomenko graphs were extensively used for studying the topology of integrable billiards: elliptical ones DR2009, DR2010, within the domains bounded by confocal parabolas <cit.>, with Hooke's potential <cit.>, in the Minkowski plane and on ellipsoids and the hyperboloid in the Minkowski space DR2017,DGR2022, non-convex billiards <cit.>, billiards with slipping <cit.>, and broader classes of systems and their bifurcations SRK2005,VK2018, FV2019, PRK2018,PRK2021.
For the larger body of works on the topic, see also the references therein.
The subsets of the phase space for the Boltzmann system corresponding to fixed negative values of E are compact 3-dimensional manifolds which are represented by the Fomenko graphs as shown in Figure <ref>.
In order to determine the Fomenko graphs and the corresponding numerical invariants, we will analyse behaviour near the singular level sets.
*First, we consider the case -1<E<-1/2.
For any pair (E,D), such that -1<E<-1/2 and -2E<D<2, the corresponding level set is a single Liouville torus, thus one edge of the Fomenko graph corresponds to those tori when E is fixed and D varies.
According to Theorem <ref>,
each level set corresponding to D∈{2E,2} consists of a single closed orbit, i.e. the A-atom of the Fomenko graph.
Consider the nature of the Boltzmann trajectories near each of the boundary components. Near D+2E=0, the two branches of the caustic _- are near the wall from above and below, keeping the arcs of the Kepler ellipses trapped vertically between the wall and _-, and horizontally within _+. These trajectories limit to the motion along the wall, between the vertices of the minor axis of _+, see Figure <ref>.
Near the upper bound D=2, the length of the minor axis of _+ shrinks to 0.
The Kepler conics are bounded vertically between the wall and _-, and horizontally between the shrinking arcs of _+.
These trajectories limit to a simple up-and-down 2-periodic trajectory between the wall at (0,1) and the lowest point of , which has coordinates (0,2-R). See Figure <ref>.
In order to calculate the numerical invariants, we need to choose two admissible bases on a Liouville torus corresponding to a point on the edge of the Fomenko graph.
Each of those two bases is chosen accordingly to one of the singular level sets corresponding to the endpoints of the edge of the graph, and then the numerical invariants are calculated from the matrix of the transformation which maps one basis to the other one.
For details, see <cit.>.
In this case, one admissible basis, taken according to the singular level set with D+2E=0, can be chosen so that it consists of a preimage of a segment orthogonal on the wall and the segment placed on the wall.
The other admissible basis, taken according to the singular level set with D=2, can be chosen to consist of the preimages of the same segments, but in the reversed order.
*Second, we analyse the case when -1/2< E <0.
For any pair (E,D), such that -1/2<E<0 and D+2E>0, D≠2, 1+2DE+4E^2>0, the corresponding level set is a single Liouville torus, thus the Fomenko graph has two edges: each one connecting the singular level set corresponding to D=2 with the two remaining singular level sets.
According to Theorem <ref>,
that level set corresponds to the A^*-atom, while the two other level sets correspond to the A-atoms of the Fomenko graph.
Near the lower bound D+2E=0, the behaviour is the same as described in the previous case and shown in Figure <ref>, thus the admissible basis can be chosen as in the previous case.
Near the boundary D=2, the caustic _- narrows around the x_2-axis, as shown in the left of Figure <ref>.
Near the upper boundary 1+2DE + 4E^2=0, the radius of the circle shrinks to 0, and the trajectories are trapped between the inner elliptic caustic _- and the outer elliptic caustic _+. These trajectories limit to the 2-periodic trajectory which aligns with the upper half of the outer caustic _+, as shown in Figure <ref>.
This discussion shows that in this case, the Boltzmann system will be Liouville equivalent to the billiard within half-ellipse, as the Fomenko graph shown in the right-hand side of Figure <ref> is identical, see for example DR2009,Fokicheva2014.
§.§ Acknowledgment
The research of M. R. was supported
by the Australian Research Council, Discovery Project No. DP190101838 Billiards within confocal quadrics and beyond, and by the Science Fund of Serbia grant Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics, MEGIC, Grant No. 7744592.
amsalpha
*
|
http://arxiv.org/abs/2307.04053v1 | 20230708220300 | How is Fatherhood Framed Online in Singapore? | [
"Tran Hien Van",
"Abhay Goyal",
"Muhammad Siddique",
"Lam Yin Cheung",
"Nimay Parekh",
"Jonathan Y Huang",
"Keri McCrickerd",
"Edson C Tandoc Jr.",
"Gerard Chung",
"Navin Kumar"
] | cs.CL | [
"cs.CL"
] |
How is Fatherhood Framed Online in Singapore?
Sen Lu, Abhronil Sengupta
School of Electrical Engineering and Computer Science
The Pennsylvania State University
University Park, PA 16802, USA
Email: {senlu, sengupta}@psu.edu
============================================================================================================================================================================================
The proliferation of discussion about fatherhood in Singapore attests to its significance, indicating the need for an exploration of how fatherhood is framed, aiding policy-making around fatherhood in Singapore. Sound and holistic policy around fatherhood in Singapore may reduce stigma and apprehension around being a parent, critical to improving the nation's flagging birth rate. We analyzed 15,705 articles and 56,221 posts to study how fatherhood is framed in Singapore across a range of online platforms (news outlets, parenting forums, Twitter). We used NLP techniques to understand these differences. While fatherhood was framed in a range of ways on the Singaporean online environment, it did not seem that fathers were framed as central to the Singaporean family unit. A strength of our work is how the different techniques we have applied validate each other.
Keywords: fatherhood, singapore, social media
§ INTRODUCTION
Fatherhood is now an unprecedentedly visible cultural phenomenon in Singapore. This increased attention is related to the inaugural nationwide fatherhood movement, Dads for Life, the continual development of parenting magazines and the recent emergence of fatherhood blogs within the Singapore internet sphere. In recent times, various fatherhood-related initiatives in Singapore have collaborated with government agencies, business corporations, and community organizations on initiatives to create awareness of the importance of the father’s role, develop commitment to good fathering, and encourage fathers to spend time with their children. In Singapore, the introduction of paternity leave and encouragement for fathers to play a bigger role in childcare and child-raising suggest that the government is sympathetic to the pursuit of gender equality. However, there is a gap between the perception of the importance of fathers and the actual involvement of fathers in their children’s lives. In addition, the role of fathers continues to be recognized primarily as that of a breadwinner. Yet fathers want to do more and experience parenthood as a very fulfilling experience, to which they are highly committed <cit.>. The proliferation of discussion about fatherhood in Singapore attests to its significance as a commercial, ideological, and cultural subject, indicating the need for an exploration of how fatherhood is framed, aiding policy-making around fatherhood in Singapore. While there has been research around how fatherhood is framed in the Singapore context, there is limited analysis of how fatherhood is framed on social media, news outlets, or online forums. Such platforms are where opinions or news on fatherhood are forwarded, people get parenting information, or get quick answers to fatherhood questions. Studying how fatherhood is framed in the online Singaporean context is central to crafting progressive and effective policy around parenting in Singapore, as well as managing the media landscape. Sound and holistic policy around fatherhood in Singapore may reduce stigma and apprehension around being a parent, critical to improving the nation's flagging birth rate. Policies developed in Singapore around fatherhood may then be implemented in nearby East Asian countries, which have similarly low birth rates, to mitigate a rapidly aging society and a shrinking taxpayer base. In this paper, we demonstrate how fatherhood in Singapore is framed on multiple online platforms (news outlets, parenting forums, Twitter). Our main research question (RQ) is as follows: How is fatherhood in Singapore framed on various online platforms? Our findings suggested that while fatherhood was framed in a multiplicity of forms online, it did not seem that fathers were core to the family.
§ RELATED WORK
Fatherhood Framing Online
Work on fatherhood in Singapore is limited. Recent work proposed the concept of Confucian masculinity to explain how the depiction of active fatherhood reinforced the ubiquitous normal family that upholds patriarchal ideology and perpetuates patriarchal power, obscuring the contradictions of class, race, and sexuality that exist in Singapore <cit.>. Other work examined the fatherhood discourses in new dad ads; feature articles from Today’s Parents, a parenting magazine; articles from Life Dads, a government electronic newsletter on fatherhood; and blog entries from three fatherhood blogs <cit.>. The study employed critical discourse analysis, and proposed a Hegemonic Fatherhood Discourse Schema to postulate that the new father/man and traditional father/man ideology is the hegemonic fatherhood in Singapore, ultimately serving the interests of the Singapore state. While past work detailed framing around fatherhood in Singapore, previous research did not compare framing across online platforms, or provide an overview of fatherhood framing to develop policy or informational tools. While there was limited fatherhood research in the Singapore context, there was relatively more research on fatherhood framing online in other contexts. For example, recent work <cit.> used discussion threads from two Web-based parenting communities, r/Daddit and r/PreDaddit from Reddit. Results demonstrated that men used web-based communities to share the joys and challenges of the fatherhood experience.
§ DATA AND METHOD
Data We first selected three content experts who had published at least ten peer-reviewed articles in the last three years around fatherhood. We ensured the content experts were either from Singapore or conducted research on fatherhood/parenthood in Singapore. Given the wide disciplinary focus of fatherhood research, we sought to select a range of experts across disciplines. We recruited one expert from each of these disciplines: Public policy, social work, computational social science. Selecting experts from a range of fields allows results to be contextualized to fields where fatherhood research is concentrated, allowing for findings to be drawn on by stakeholders in public policy, social work, and computational social science. The context experts separately developed lists of online platforms most relevant to fatherhood in Singapore. Each expert developed a list of ten platforms independently, and we selected only platforms common to all three experts' lists. For each online platform, experts also provided up to 10 examples, where applicable, of websites, or forums, and we selected examples common to all experts' lists. The final list of platforms is as follows: Singapore news outlets (Straits Times, Channel NewsAsia, TODAYonline), parenting forums (singaporemotherhood.com, singaporeparents.com.sg/forum, forums.hardwarezone.com.sg/threads/welcome-to-hwzs-parenting-kids-early-learning-forum.5684416, mummysg.com/forums), Twitter (filtering only posts related to Singapore). Examples of platforms not selected: Facebook, Instagram, Reddit, LinkedIn. We were not able to collect Facebook and Instagram data as there was limited support for CrowdTangle, the main mode of Facebook/Instagram data collection. Similarly, the pushshift.io Reddit API had limited support and Reddit data collected was incomplete. LinkedIn had limited fatherhood posts and posts were mostly centered on non-family content. To capture fatherhood-related text on these platforms, we used queries based on a related systematic review e.g., father* OR dad* OR patern* OR paternal OR paternity OR stepdad* OR stepfather* OR step-dad* OR Step-father* OR papa. We used only English-language keywords as most of discussion in the Singapore internet environment is in English. English is also the major language of communication in Singapore. For forums, we used automated scraping techniques (Beautiful Soup) to obtain forum posts from 2010 to 2023, with the same set of keywords. We ran a search for querying the keywords in the title of the forum post or replies to the forum post. We collected all posts that contained these keywords within the forum posts and replies. Regarding Twitter, we used the Twitter API and the indicated keywords to collect tweets from 2011 to 2023. Finally, for news articles, we used Nexis to obtain news archives from 1992 to 2023. To prepare the data for analysis, English stop words such as the, a, an were removed, along with abbreviations, and terms were stemmed using Porter’s stemming algorithm. Stemming converts words with the same stem or root (e.g., innovative and innovator) to a single word type (e.g., innovate). We organized data into four streams for analysis: Twitter (tweets), news (news articles), forums (forum posts).
Sentiment
Sentiment analysis can aid us in comprehending how sentiment around fatherhood is expressed in the online arena. As an example, forums may be more likely to have lower sentiment compared to news. DistilBERT was used for sentiment analysis. DistilBERT was used separately on data from each platform. The model assigns sentiment based on each article or post. Sentiment is from a -1 to 1 scale, where values <0 are negative sentiment, >0 are positive sentiment, and close to 0 are neutral. To stay within the admitted input size of the model, the text length (title + body text) was clipped to to 512 tokens.
Emotion Recognition
Emotion recognition can help us understand how emotions are expressed across various platforms, indicating differences in how fatherhood is framed in Singapore. For example, forums may be more likely to contain anger compared to news. We used DistilBERT for emotion recognition. The model was applied separately on data from each platform. The model assigns emotions (anger, fear, joy, love, sadness, surprise) based on each article or post. To stay within the admitted input size of the model, we clipped the length of the text (title + body text) to 512 tokens.
We provided an overview of the data in Table <ref>. Two reviewers independently examined 10% of the articles or posts within each dataset to confirm salience with our research question. The reviewers then discussed their findings and highlighted items deemed relevant across both lists. We noted the following relevance proportions: News outlets (82%), Twitter (90%), Parenting forums (78%).
§ RESULTS
Overview
We first explored sample posts across platforms. News outlets generally mentioned fatherhood in the context of providing demographic data about interviewees, with excerpts such as So the 40-year-old eye specialist and father of three had to wrap up his work at the hospital quickly, or when interviewees were referring to their fathers with no specific reference to fatherhood e.g., Mr Lee, whose father founded the clan association, rents out its third floor to a small media firm. Broadly, news outlets did not seem to focus on the experience of fatherhood, with the bulk of articles mentioning fathers as a demographic indicator. Twitter posts focused on people recounting incidents, often humorous or heart-warming, with their fathers e.g., My dad was telling me something serious and he hit his leg against the table and I burst out laughing so he had no choice but to laugh, Dad brought back homemade fresh horfun (noodles) from the temple. It's delicious. Twitter seemed to have a greater focus on fathers playing a core function in the Singapore family unit. Posts from forums were very diverse topically. Several posts were about hiring a helper for a young child: My husband is totally against the idea of employing a helper, as he does not like a stranger living with us; I am a father of a newborn baby girl. I recently engaged a confinement lady by the name of Auntie Judy. Such posts suggest the significant role domestic helpers play in the Singaporean family, and how a portion of a father's role is perhaps to oversee the hiring of the domestic helper. Other posts were about suspected infidelity e.g., So my Wife of 2 years has been cheating on me with another male colleague, perhaps indicative of the strain parenting is related to within some Singaporean families.
We then provided word clouds in Figure <ref> as an overview of the data. Across all datasets, words such as time, work, now were prominent, perhaps indicative of how work and likely limited time are central to fatherhood in Singapore. Most common trigrams for news articles centered on leaders of Singapore, who were father and son: Lee Kwan Yew and Lee Hsien Loong. This may indicate that the mainstream news media discussion around fatherhood had little to do with fathers' role in a family, but simply around familial relationships within major news stories. In 1992 - 2003, common trigrams in the news were engineer success story and pressure parent counting. From 2004 - 2019, common trigrams were two baby boy, first new baby, and first time parent. From 2020 - 2022, common trigrams were generation grit family, and grit family love. Broadly, news trigrams may detail how the initial focus was on children bringing pride and wealth to their families, with a transition toward celebrating new births. In more recent years, forums tended to focus on how the family unit could overcome struggles. The most common trigrams in Twitter focused on celebrating fathers through specific events such as Father's Day and birthdays: happy father's day, happy birthday daddy. Such phrases indicated that Twitter may be used to celebrate fathers, but only in relation to pre-defined events, instead of fathers being celebrated for time put toward caregiving etc. Common trigrams in 2011 - 2020 were love u dad, dad love love. 2021 onwards, popular trigrams were feel fulfilling husband, and last nite daddy. Twitter data demonstrated a shift from declaring love for one's father, to fathers indicating how they were fulfilled in their role. Unlike other datasets, there appears to be a shift towards a more active form of fatherhood in Singapore, where fathers describe pride in their role. Trigrams in forums centered on perceived marital infidelity, such as wife unfaithful husband, and assisted reproductive technologies, such as ivf mommy toben, and cousin egg donor. Forums seemed to be platforms where people sought support around spousal infidelity and assisted reproductive technologies, rather than discuss fathers' role in the family unit. The most common trigrams in forums changed over time, with phrases such as gave birth daughter, and first time dad in 2010 - 2019, but with phrases such as happen file divorce, and judged urged divorcing in 2020. In 2021, common trigrams were conceiving single women, while in 2022, trigrams such as crave physical intimacy, and physicial intimacy normal were popular. Forums, while initially around celebrating birth, may have become places where people sought information around divorce, assisted reproductive technologies, and physical intimacy. Broadly, descriptive data indicated shifting framing around fatherhood, but a limited focus on fathers as core to the Singapore family.
Sentiment
We presented sentiment analysis results across each platform in Table <ref>. News and Twitter had higher proportions of positive sentiment (53.7% and 57.0% respectively) compared to forums (27.2%). Forums had the highest proportion of negative sentiment (65.9%), compared to news and Twitter (43.8% and 33.8% respectively). We then presented sentiment analysis results over time for each platform in Figure <ref>. News data exhibited several fluctuations but had the greatest rise in positive sentiment post-2009. The nationwide fatherhood movement, Dads for Life, started in 2009, may explain the increase in positive sentiment. Examples of news article content with positive sentiment were as follows: A group of prominent figures from various organisations and businesses have banded together to start up the Fathers Action Network. The network aims to kick-start a movement called Dads for Life to get fathers more involved with their families, especially in their childrens' lives. This follows a fatherhood perception survey conducted in April and May this year by a Ministry. Most felt that being a father and raising children is one of the most fulfilling experiences a man can have.; Work is work and family is family. Our ultimate goal is still our family. Work is just a means to get the money so we should be very clear about it. And that is the sort of spirit that the Dads for Life movement wants to inspire. After 2017, positive sentiment declined over time, and was overtaken by negative sentiment. Forums had broadly negative sentiment 2015 onward, reaching a peak in 2017, followed by a steady decline. Twitter exhibited mostly positive sentiment 2013 onward with a steady decline after. We suggest that the high proportion of positive sentiment in the news may be related to governmental initiatives and the high proportion of negative sentiment in forums may be related to a more frank discussion of the stresses of parenting.
Emotion Recognition
We presented emotion recognition results across each platform in Table <ref>. News had the highest proportion of joyous (61.3%) and loving (34.2%) posts, perhaps reflecting governmental initiatives around fatherhood. While Twitter and forums had similar levels of joyous posts (56.6% and 44.2% respectively), they were still not as high as news. Similarly, loving posts on Twitter and forums (2.4% and 4.1% respectively) were far lower than news outlets. We suggest that the emotion in the news reflects pro-fatherhood governmental initiatives, but these do not always filter successfully to other media. We then presented emotion recognition results over time for each platform in Figure <ref>. News data exhibited several fluctuations but had the steepest rise post-2009. Dads for Life, started in 2009, may explain the uptick in news articles, especially around joy. Examples of news article content that were coded as joy: It's a happy Father's Day for SAFRA, as it is set to receive funds from the "Dads for Life" movement to pump up father-friendly activities for its members over the next two years.; He will be running alongside his daughter in the Dads For Life 800m Father and Child Challenge, a new category in the annual SAFRA Singapore Bay Run and Army Half-Marathon. Mr Shariff, who was born without part of his left leg, said: I signed us up because I want to show her how running can make her happy. Both Twitter and forum posts saw a sudden spike post-2013 onward, mostly around joy. We suggest that the shift in emotion may be due to a delayed reaction to Dads for Life. Broadly, we forward that the 2009 Dads for Life movement and other similar policies may have catalyzed emotional reactions around fatherhood in the Singapore online arena. However, the rises in emotion were not sustained and seemed to decline by 2023, perhaps indicative that new policy levers may need to be rolled out.
§ DISCUSSION
Our RQ was to explore how fatherhood in Singapore is framed on various online platforms. A strength of our work is how the different techniques we applied validate each other as well as reveal differences across platforms. While fatherhood was framed in a range of ways on the Singaporean online environment, it did not seem that fathers were framed as central to the Singaporean family unit. Results also indicated that governmental initiatives may have some effect on altering the framing of fatherhood, but are not lasting in effect. The concordance in our results suggests the veracity of our findings and we hope that results can add to research and policy around fatherhood in Singapore. Our evidence adds to previous research, where we provided data on how governmental initiatives may initially buttress framing around fatherhood, but needs to be sustained to provide broad and lasting support for fathers. Key to how fatherhood is framed in Singapore is the inclusion of fathers' viewpoints when writing news articles on fatherhood. Where possible, fathers themselves should be consulted on articles about fatherhood. For example, a panel staffed by fathers can comment on fatherhood-related online news articles, providing suggestions on how articles can more accurately represent fathers' concerns <cit.>. Our findings relied on the validity of data collected with our search terms. We used a range of established techniques to search for all articles/posts relevant to fatherhood, and our data contained text aligned with how fatherhood is framed. We were thus confident in the comprehensiveness of our data. We only used English-language text but will include other languages in future work. Given the token limits for the emotion recognition technique, we were not able to use emotion recognition for the entirety of longer news articles. We note that the recall of the search string was not tested. We note that our data may not be generalizable to how fatherhood is framed globally. Our goal was not to identify who was doing the framing around fatherhood e.g., family members or government. Future studies will seek to identify which stakeholders were likely involved in the framing.
splncs04
|
http://arxiv.org/abs/2307.04689v1 | 20230710164304 | Trapping and imaging single dysprosium atoms in optical tweezer arrays | [
"Damien Bloch",
"Britton Hofer",
"Sam R. Cohen",
"Antoine Browaeys",
"Igor Ferrier-Barbut"
] | physics.atom-ph | [
"physics.atom-ph",
"cond-mat.quant-gas",
"quant-ph"
] |
[email protected]
Present address: Department of Physics, Stanford University, Stanford, California 94305, USA
[email protected]
Université Paris-Saclay, Institut d'Optique Graduate School, CNRS,
Laboratoire Charles Fabry, 91127, Palaiseau, France
We report the preparation and observation of single atoms of dysprosium in arrays of optical tweezers with a wavelength of 532, imaged on the intercombination line at 626. We use the anisotropic light shift specific to lanthanides and in particular a large difference in tensor and vector polarizabilities between the ground and excited states to tune the differential light shift and produce tweezers in near-magic or magic polarization. This allows us to find a regime where single atoms can be produced and imaged.
Using the tweezer array toolbox to manipulate lanthanides will open new research directions for quantum physics studies by taking advantage of their rich spectrum, large spin and magnetic dipole moment.
Trapping and imaging single dysprosium atoms in optical tweezer arrays
Igor Ferrier-Barbut
August 12, 2023
======================================================================
Trapping and cooling of single atoms in tweezer arrays <cit.> has allowed tremendous progress in quantum science and metrology <cit.>.
These techniques were first used on alkali atoms <cit.>, before being extended to alkaline-earth species <cit.> and molecules <cit.>.
In parallel to this progress, experiments with quantum gases of lanthanides have explored dipolar physics <cit.> and topology <cit.> among other examples.
Controlling lanthanides in single-atom tweezers will permit leveraging their specific properties.
Their anisotropic light-matter interaction <cit.> results in a broad tunability of trapping potentials useful to produce sub-wavelength interatomic distances <cit.> or for quantum-enhanced sensing <cit.>.
Dimers with a large magnetic dipole moment <cit.> or atoms with an electric dipole <cit.> might be produced to study quantum magnetism <cit.> in tweezer arrays.
Finally, their many transitions from the ground state, spanning a broad range of wavelengths and linewidths makes them an interesting platform for studies of collective light-matter interactions <cit.>.
In this Letter we demonstrate single-atom trapping of dysprosium in optical tweezers, imaging on the narrow intercombination line by making use of the strong anisotropic light shift of Dy.
The rich spectra of optical transitions of lanthanides have been used to operate efficient laser cooling and produce degenerate quantum gases <cit.>.
Transitions from the 6s^2 electrons are similar to those of two-electron atoms such as Yb and Sr, and the methods developed to produce single atoms of these species can be adapted to lanthanides.
Here we rely on the intercombination line between G=4f^106s^2 ^5I_8 and E=4f^10(^5I_8)6s6p(^3P^∘_1) (8,1)^∘_9 of Dy, generally used for magneto-optical traps <cit.>, to produce and image single Dy atoms.
This transition has a wavelength λ = 626 and a linewidth Γ=2π×135.
Another advantage of lanthanides is their non-vanishing vector and tensor polarizabilities.
The tensor polarizability was recently used to demonstrate magic trapping for the Dy intercombination transition at a trap wavelength of 1070nm <cit.>.
We rely in this work both on the tensor and vector polarizabilities <cit.> to obtain magic trapping at 532nm.
We generate 5×5 tweezer arrays with 5 spacing at a wavelength of 532 [The trapping laser is a Coherent Verdi V10 with measured wavelength 532.208.] using a 2D acousto-optic deflector (AOD) driven by a multitone signal <cit.>.
The tweezer light is sent through a 0.5 NA microscope objective (Mitutoyo G Plan Apo 50X) placed outside a glass cell, resulting in a tweezer waist w_0≈500 [The waist is defined as the 1/e^2 radius of a gaussian beam that fits best the expected radial intensity profile.].
Each trap has a power of 2, yielding a potential depth of about 150.
Our setup is schematized in Fig. <ref>(a) and more details will be published in <cit.>.
We use the ^162Dy isotope in this work.
The experiment begins with a 2D magneto-optical trap (MOT) on the broad transition of Dy at 421, as in <cit.>, to cool and redirect atoms towards a glass cell.
In the glass cell, we capture the atoms with a two color core-shell MOT <cit.> and eventually transfer them to a MOT using only the narrow intercombination line.
Following the MOT loading stage, the atoms are pumped in the lowest Zeeman state |g⟩=|G, J=8, m_J=-8⟩ by ramping the intensity to I=0.1 I_ sat, with I_ sat=72^2, and detuning to Δ=-(2π) 1.5 <cit.>.
The tweezers are overlapped for 100 on the MOT. After this, each trap is filled with more than one atom on average.
Dysprosium has a large Zeeman manifold in both the ground state (J=8) and excited state (J'=9).
This strongly influences imaging and cooling since the scattering rate on a narrow transition depends on the atom's internal state.
We apply a magnetic field of 7G to isolate a closed σ^- transition between |g⟩ and |e⟩ = |E, J'=9, m_J'=-9⟩.
This leaves the π (m_J=-8 ↔ m_J'=-8) and σ^+ (m_J=-8 ↔ m_J'=-7) transitions strongly off-resonance, respectively detuned by about 13 and 25 (resp. 95 Γ and 190 Γ). It ensures negligible photon scattering rates for these transitions
and the atoms are then imaged solely on the cycling σ^- transition.
To obtain single atoms we induce light-assisted collisions that eject pairs of atoms from the multiply-loaded tweezers <cit.>.
We observe that such collisions take place in a few milliseconds when shinning red-detuned light. The collision pulse lasts for 10 and has the same parameters as used for imaging specified below.
After this, the tweezers are randomly loaded with zero or one atom, with a filling fraction close to 50%.
Next, to image single atoms, we need to precisely tune the trapping potential.
Indeed for such a narrow linewidth,
high fidelity single-atom imaging requires magic trapping where |g⟩ and |e⟩ have the same polarizability <cit.>.
Whether or not such a condition exists for a given species depends in general on the trapping wavelength.
In contrast with other species, the strong anisotropy of the polarizability of lanthanides allows one to tune the differential polarizability between |g⟩ and |e⟩ by changing the tweezer polarization <cit.>.
This can lead to magic trapping in broad ranges of wavelengths.
Measurements of the scalar, vector and tensor polarizabilities for both the ground state G and excited state E at 532 will be reported in <cit.>.
We use the large vector polarizability of the excited state and create an elliptic polarization of the tweezers, with Jones vector (ϵ_x,ϵ_y)=(cosθ,i sinθ) in the plane perpendicular to the magnetic field. Fig. <ref>(e) shows the shift of the transition measured with fluorescence spectroscopy as a function of trap power for different ellipticities θ. We find an ellipticity θ≃ +6^∘ for which the transition |g⟩↔|e⟩ is magic [see fig. <ref>(c, d)].
This magic trapping condition allows us to image single atoms in the tweezers.
Fluorescence is induced by a single non retro-reflected beam with propagation axis having components along both the radial and axial directions of the tweezers <cit.>, which is necessary to cool efficiently while imaging.
It is red-detuned by Δ = - 1.0 Γ
and has an intensity I=0.8 I_ sat. The duration of the imaging pulse is typically 30.
The light scattered by the atoms is collected onto a CMOS camera (Hamamatsu C15550-20UP) through the same microscope objective used to focus the traps.
For a single shot image as in Fig. <ref>(a), we count the number of collected photons in a small circular area around each trap.
We repeat the experiment, reloading the MOT and the tweezers for every shot, and we record the histogram of the collected fluorescence as shown in Fig. <ref>(c).
The histograms exhibit two peaks characteristic of the single-atom regime: one peak corresponding to zero atoms and the other peak, with about 50 photons detected, corresponding to a single atom in the trap.
These histograms are shifted and broadened by background light.
This light is due to the tweezers beam at 532 going through the microscope and causing the glass to fluoresce at longer wavelengths, including the imaging wavelength of 626.
To mitigate this effect, two angle-tunable dichroic filters, one short-pass and one long-pass (Semrock TSP01-628 and TLP01-628), are placed on the path before the camera to transmit only a narrow wavelength band around 626.
This reduces the light reaching the camera to about 20 photons per pixel per second for 50 of 532 light going through the microscope.
This remaining background can be seen in figure <ref>(b).
To determine the presence of a single atom in a given picture, we compare the number of photons collected to a given threshold.
If the fluorescence is higher than the threshold, we label the trap as containing an atom, otherwise we label it as empty.
In the following, we characterize the fidelity and induced losses of our imaging. The fidelity represents the probability to correctly label the initial presence of an atom in a trap. In addition, losses might be induced by the imaging sequence through which a ground state atom initially present in the trap is not detected in a subsequent imaging pulse. Both infidelity and imaging-induced losses will limit the ability to image and re-arrange large atomic arrays <cit.>.
The experimental fluorescence histograms are well modeled as the sum of three distributions.
The first peak is centered on the number of background photons N_0, with area the empty-trap probability P_0≃50.
A second peak represents events where an atom is present for the full duration of the imaging.
It is centered on N_0 + N_1 where N_1 is the number of photons scattered by the atom.
Its area is P_1× P_ survival where P_1=1-P_0 is the probability to have initially one atom in the trap and P_ survival is the probability that the atom survives imaging.
The third contribution is a flat distribution that bridges the two peaks, visible in Fig. <ref>(d), that corresponds to the events where atoms are lost while they are being imaged <cit.>.
Its area is P_1 × P_ loss, with P_ loss = 1 - P_ survival.
We give more details on the exact modelization of the distributions in <cit.>.
Adjusting this model to the observed histograms, we extract the parameters N_0, N_1, P_0, P_ loss and estimate the best threshold to maximize the imaging fidelity F (see <cit.>).
All quantities above depend in general on every imaging parameter such as exposure, imaging intensity and detuning, as well as tweezer power.
We optimized them to have the highest imaging fidelity.
For example, we show in Fig. <ref>(a) F and P_ loss for several exposure times.
At short duration, the fidelity is low because an atom does not scatter enough photons to be clearly distinguished from the background.
The fidelity increases with exposure, eventually reaching a maximum after a few tens of milliseconds.
However, the loss probability increases linearly with time.
The imaging duration we choose is then a compromise between high fidelity and low losses.
In typical conditions, we image the atoms in 30, which is resilient to small fluctuations of parameters and we reach F=99.1(0.2) and P_ loss=6.1(0.8) [This means that out of the 6% of atoms that are lost during the imaging, most (roughly 5 in 6) are correctly labeled before being lost.].
To identify the origin of the losses, we measured the influence of the imaging parameters on P_ loss.
We took a first picture to detect the atoms, then applied an imaging pulse for 30 varying the imaging parameters and finally measured the probability for the atom to have survived this pulse by taking a last image.
As shown in Fig. <ref>(b), we observe that P_ loss increases linearly with imaging power.
We also measure the average number of detected photons before an atom is lost N_ ph, loss = -N_ detected / ln(P_ survival) [This comes from P_ survival(t)=e^-t/τ_ loss=e^-N_ detected/N_ ph, loss], where N_ detected is the number of detected photons during the pulse.
For I≲ I_ sat, N_ ph, loss is approximately constant. It decreases for higher intensities [gray area in Fig. <ref>(b)] due to less efficient Doppler cooling <cit.>.
We also find that N_ ph, loss is constant when varying the detuning for Δ≲ -1 Γ.
Thus our observations suggest that the probability to lose an atom is directly proportional to the time it spends in the excited state |e⟩.
This could be caused by a decay from |e⟩ to dark or non-trapped states.
However, the intercombination transition is closed and we have also checked that the atoms are not pumped to other Zeeman states of the ground manifold.
These losses are thus likely due to further excitation by the trapping light from |e⟩ to a highly excited state in Dy's dense spectrum.
We indeed observe that the leakage to non-imaged states increases with trap power: Fig. <ref>(c) shows N_ ph, loss for fixed imaging parameters as a function of trap power at 532. A steep decrease is observed showing that a stronger trap means a higher loss probability per imaging photon.
We thus conclude that losses are due to a two-photon event: an atom in |e⟩ absorbs a trap photon, sending it to a highly excited state from which it then decays to non-imaged states.
There indeed exists a state with a dipole-allowed transition with |e⟩ (4f^105d6p, J=10 at 34776.04) lying only about 400 away from the sum of the two laser frequencies <cit.>.
These losses are the main factor limiting imaging fidelity, and using a tunable trapping laser to increase the detuning from this state should allow to mitigate them.
We expect this to be necessary for other lanthanides because of their dense spectrum.
We further observe that dark atoms can decay back to |g⟩ from metastable states. Indeed, a trap initially containing an atom and that became dark sometimes spontaneously becomes bright again although the MOT is turned off.
This can be seen on Fig. <ref>(a) where we plot the fluorescence of a single trap continuously imaged and observe discrete jumps from bright to dark and vice-versa.
Starting from initially empty traps, we do not observe the appearance of atoms, ruling out reloading from residual background pressure.
Similar observations were reported with Yb in <cit.>, identified as the excitation of the atom to metastable states and spontaneous decay to the ground state.
To measure the average time it takes for the atoms to come back, we apply a pulse of imaging light for 1.5.
After this pulse, about 70 % of the atoms are no longer imaged.
We plot in Fig. <ref>(b) the fraction of these dark atoms that subsequently reappear as a function of the wait time.
We thus observe that 35 of them come back after a typical time τ=0.48(0.08). From these measurements we extract a branching ratio of about 65 % of decay towards trapped metastable states versus non-trapped ones <cit.>. We leave for future research the exact identification of these states.
We finally measured the temperature and lifetime of atoms in the tweezers. The lifetime in particular is important in views of sorting atoms to form large ordered arrays <cit.>.
For this, we used the release and recapture method, see <cit.>.
Directly after imaging, we measured a temperature of 6.3(0.2), slightly higher than the Doppler temperature for the intercombination transition (T_D= 3.2). Next, in shallow tweezers (depth U_0=150, P_ trap = 2), we observed a heating rate of 1.7(0.2), that limits the lifetime in the absence of cooling to about 10.
This heating rate is compatible with the off-resonant scattering of trap photons in the ground state.
Indeed from the calculated imaginary part of the polarizability at 532 <cit.>, we expect a heating rate of a few microkelvins per second. We mitigated this heating by applying cooling light (intensity I=5× 10^-3 I_ sat, detuning Δ=-1.3 Γ), and observed a lifetime of 300(30), limited by the two-photon losses studied above (see <cit.>).
In conclusion, we have demonstrated single-atom trapping and high-fidelity imaging of Dy on the intercombination line in tweezers rendered magic by fine tuning the tweezer polarization.
Single-atom trapping of lanthanides opens exciting opportunities.
For instance it can be used to obtain subwavelength distances using the anisotropic polarizability <cit.> or also by directly loading an accordion lattice.
This could be used to create atomic waveguides <cit.>, or to prepare directly extended Bose-Hubbard models <cit.> from optical tweezers.
We acknowledge fruitful discussions with Maxence Lepers and Jeff Thompson, experimental assistance by Florence Nogrette and critical reading of the manuscript by Thierry Lahaye and Giovanni Ferioli. We note that another setup based on a similar 421-nm 2D-MOT loading a 626-nm 3D-MOT of Dy atoms has been developed in the group of L. Chomaz <cit.>. We have widely benefited from exchanges between our groups.
This project has received funding by the Agence Nationale de la Recherche (JCJC grant DEAR, ANR-22-PETQ-0004 France 2030, project QuBitAF) and by the European Union (ERC StG CORSAIR, 101039361, ERC AdG ATARAXIA 101018511).
Supplemental Material
§ TRAP HOMOGENEITY
We homogenize the traps by imaging the array after the AOD (AA Opto Electronic DTSXY-400-532-002) but before the microscope objective by reflecting a fraction of the incoming beam off of a beam sampler.
The beam sampler's angle with respect to the propagation of the trapping light from the AOD to the microscope is minimized so as not to distort the image of the traps.
We verified that the reflected intensities are proportional to the transmitted ones so that when the imaged intensities are homogeneous the transmitted ones are homogeneous as well.
The RF signal used to create the array of traps is generated by an arbitrary waveform generator (Spectrum M4i.6621-x8 AWG) followed by an amplifier before being sent to the AOD.
The AWG produces a set of sine waves at equally spaced frequencies.
When all the tones are in phase the maximum voltage amplitude scales linearly with the number of traps and quickly saturates the amplifier.
To avoid this we optimize the phases to minimize the signal's envelope, following the same protocol as in <cit.>.
We finally image the trap intensities and feedback altering each of these tones' amplitude sequentially to minimize the trap intensities' variance. We finally obtain a standard deviation of trap intensities of about 2 %. When measuring the |g⟩↔|e⟩ transition frequency in non-magic conditions, we did not observe a significant inhomogeneity of the traps.
In our magic-polarization tweezers, the polarization homogeneity over all traps is important. We find that the acousto-optic deflector can lead to polarization inhomogeneity of the order of a degree or more for linear polarization. To prevent this polarization inhomogeneity, we placed a polarizer directly after the AOD. It turns a polarization inhomogeneity into power inhomogeneity, which is corrected by the feedback discussed above.
§ IMAGING FIDELITY
To estimate the imaging fidelity F, we assume that the histogram of the number of collected photons follows a simple model:
If no atom is present in the trap, we collect on average N_0 photons due to the background light. The probability that n photons reach the camera is then given by P^P_N_0(n), where where
P^P_λ(n)=λ^n e^-λ/n!
is the Poisson distribution with mean λ.
In addition to the shot noise, the distribution is also broadened by the Gaussian camera readout noise and the probability to have a count x is ∑_n=0^+∞ P^P_N_0(n) g_n, σ(x) where g_n, σ is a Gaussian distribution of mean n and standard deviation σ=1.6. (This value is given by the read noise of the camera for a single pixel multiplied by the square root of the number of pixels over which the fluorescence is integrated.)
Similarly, if an atom is present throughout the total duration of the imaging, it scatters N_1 photons on average and the probability to have a count x is ∑_n=0^+∞ P^P_N_0 + N_1(n) g_n, σ(x).
The last possibility corresponds to an atom being lost during the imaging, after scattering a random number of photons M between 0 and N_1. The probability to collect n photons is then
P^L_N_0, N_1(n) = 1/N_1∫_0^N_1P^P_N_0 + M (n) dM
This corresponds to a smooth flat distribution that bridges the peaks for the zero and one atom cases.
Combining all three possibilities as shown on Fig. <ref>, the probability that n photons reach the camera is
P^N(n) = P_0 P^P_N_0(n)
+ (1-P_0)[(1-P_ loss) P^P_N_0 + N_1(n) + P_ loss P^L_N_0, N_1(n) ]
where P_0≃50 is the probability that the trap is initially empty and P_ loss is the probability to lose an atom while imaging it.
Taking into account the Gaussian noise of the camera, the probability to measure a count x is then
P^X(x) = ∑_n=0^+∞ P^N(n) g_n, σ(x)
The mean of this distribution is
⟨ X ⟩ = N_0 + N_1(1-P_0)(1-P_ loss /2 )
and its variance is
Var(X) = σ ^ 2 + N_0
+ N_1 (1 - P_0)×
[
N_1 (P_0 (1 - P_ loss/2) ^ 2 + P_ loss/3 (1 - 3 P_ loss/4 ))+ (1 - P_ loss/2)]
This distribution is characterized by the set of parameters P_0, P_ loss, N_0, N_1 and σ that we need to estimate to compute the fidelity as a function of the imaging parameters for example on figure <ref>(a).
We record the histogram in the case were the tweezers are not loaded, which correspond to setting P_0=1. In this case ⟨ X ⟩ = N_0 and Var(X) = N_0 + σ ^ 2 so we can extract N_0 and σ.
To measure P_ loss we record three pictures and measure the probability that the middle picture removes an atom. The last two parameters P_0 and N_1 can be extracted from the mean and variance of the distribution when the tweezers are loaded normally.
Once the parameters have been estimated, one can compute the fidelity of the imaging f(s) as a function of the threshold s used to classify the presence of an atom.
The fidelity f(s) is defined as the probability to correctly label the initial presence of an atom in the trap.
There are three events that contribute to the imaging fidelity : no atom is present in the trap and the fluorescence collected is below the threshold; the atom is lost during imaging but scatters more photons than the threshold, or the atom is kept during the full imaging and scatter enough photons such that the fluorescence is higher than the threshold.
The probabilities of these events are respectively the blue, red and green areas on figure <ref>.
The sum of these three contributions gives us the imaging fidelity for a given threshold :
f(s) = P_0 P^P_N_0(X<s)
+ (1-P_0)[(1-P_ loss) P^P_N_0 + N_1(X>s) + P_ loss P^L_N_0, N_1(X>s) ]
We then compute the optimal fidelity F and the best threshold s_0 by maximizing f. This method is used to calculate the fidelity in the rest of the text.
It is worth remarking that even if the zero and one atom peaks (blue and green curves on fig. <ref>) have negligible overlap, the imaging fidelity does not reach 100 because a fraction of the red curve is below the threshold.
This corresponds to the case where atoms are lost before having scattered enough photons to be distinguished from the background and this eventually limits our imaging fidelity to F=99.1(0.2).
§ BRANCHING RATIO
Here we describe how we extracted the branching ratio of trapped to non-trapped metastable states: When imaging, we have measured a loss probability from |g⟩ of about 6 % in 30, which corresponds to a rate of γ_ g≃2.
These atoms leak in two channels: towards trapped metastable states (which we call here |t⟩) with a rate αγ_ g, and towards non-trapped states with a rate (1-α)γ_ g with α the branching ratio.
The |t⟩ atoms can then decay back to |g⟩, with a rate γ_ t-g. Under the application of imaging light, the atom numbers in |g⟩ and |t⟩ then follow the rate equations
ṅ_ g=-γ_ gn_ g+n_ tγ_ t-g
ṅ_ t=αγ_ gn_ g-n_ tγ_ t-g
First, we extract γ_ t-g. For this we applied a 1.5 imaging pulse and then removed the atoms remaining in |g⟩. After this, the atoms that are in |t⟩ will eventually decay back to |g⟩ with a rate γ_ t-g. In Fig. <ref>(b) show the fraction of atoms that have disappeared during the pulse, that re-appear in |g⟩ as a function of wait time. By fitting the data with an exponential saturation, we extract a decay rate γ_ t-g≈2. Such long lifetimes are similar to that observed with blue MOTs <cit.>.
Finally, in Fig. <ref> we apply an imaging pulse of variable time, and plot the fraction of atoms remaining in |g⟩ after the pulse. Fitting then the data of Fig. <ref> with the rate equations using the measured rates γ_ g, γ_ t-g and leaving α as a free parameter, we obtain a good agreement with the data for α = 0.65, (dashed line in Fig. <ref>).
§ TEMPERATURE MEASUREMENTS
To measure the temperature of the atoms in the tweezers, we use the release and recapture technique <cit.>.
We suddenly turn off the tweezers for a few tens of microseconds and then switch them back on.
The fraction of recaptured atoms depends on the initial temperature of the atoms, the lower the more likely for atoms to be trapped again, and we extract the temperature by comparing with numerical simulations.
A typical temperature measurement is shown in <ref>(a).
Just after the imaging step described in the main text, the temperature of the atoms is T_0=6.3(0.2).
In the absence of cooling light, the atoms slowly heat up in the tweezers.
For a tweezer power P_ trap = 2.1 (trap depth of 150), we measure a heating rate of 1.7(0.2) (see figure <ref>(b)). This heating rate is compatible with expectations from the imaginary part of the polarizability <cit.>. We note that it is dominated by contributions from the broad transitions near 400 rather than by the close narrow transition at 530.3.
§ TRAPPED ATOMS LIFETIME
Figure <ref>,
represents the fraction of remaining atoms after a given wait time, under continuous cooling at Δ = -1.3 Γ, I=10^-3 I_ sat, from which we extract an exponential lifetime of 300(30), dashed line. This lifetime is limited by cooling-induced losses. By measuring lifetimes at different cooling powers, we could extrapolate the vacuum lifetime to be larger than 500 seconds.
|
http://arxiv.org/abs/2307.05404v1 | 20230711161211 | Gravitational waves induced by scalar-tensor mixing | [
"Pritha Bari",
"Nicola Bartolo",
"Guillem Domènech",
"Sabino Matarrese"
] | astro-ph.CO | [
"astro-ph.CO",
"gr-qc"
] |
[email protected]
[email protected]
[email protected]
[email protected]
^a Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, via Marzolo 8, I-35131 Padova, Italy
^b INFN, Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy
^c INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
^d Institute for Theoretical Physics, Leibniz University Hannover, Appelstraße 2, 30167 Hannover, Germany.
^e Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L'Aquila, Italy
This paper explores the physics of second-order gravitational waves (GWs) induced by scalar-tensor perturbation interactions in the radiation-dominated Universe. We investigate the distinctive signatures of these GWs and their detectability compared to scalar-induced GWs. Unlike scalar-scalar induced GWs, scalar-tensor induced GWs do not present resonances or a logarithmic running in the low frequency tail in the case of peaked primordial spectra. But, interestingly, they partly inherit any primordial parity violation of tensor modes. We find that chirality in primordial GWs can lead to distinguishing effects in scalar-tensor induced GWs in the ultraviolet (UV) region. We also address a potential divergence in our GWs and explore possible solutions. This study contributes to our understanding of GWs in the early Universe and their implications for cosmology and GWs detection.
Gravitational waves induced by scalar-tensor mixing
Sabino Matarrese^a,b,c,e
August 12, 2023
===================================================
§ INTRODUCTION
With the groundbreaking detection of gravitational waves (GWs) from a binary black hole merger <cit.>, modern cosmology has witnessed a significant breakthrough, leading to an increased focus within the scientific community on designing more precise GWs observations and developing robust theoretical predictions. Among the diverse range of GWs sources <cit.>, it is plausible that a cosmic background of GWs permeates the universe. This GWs background represents a potential smoking gun of inflation <cit.> and encapsulates invaluable information about the early Universe <cit.>,
as GWs barely interact with intervening matter. The search for the cosmic GWs background is a central focus in cosmology, pursued through avenues such as B-mode polarization of the cosmic microwave background (CMB) and direct detection using interferometers <cit.>.
Gravitational waves are metric tensor perturbations that can arise from vacuum fluctuations during inflation, particularly in the context of single-field slow-roll inflation. However, within the broader framework of inflationary models, classical production mechanisms for GWs have also been explored <cit.>. Regarding non-inflationary mechanisms, a strong GWs signal can be produced by topological defects <cit.>, or phase-transitions <cit.>. GWs are also induced by primordial fluctuations after inflation. While at linear order in cosmological perturbation theory scalar and tensor fluctuations decouple, it is no longer the case at higher orders. For instance, the product of two spatial gradients of scalar fluctuations has a non-vanishing transverse and traceless projection that sources tensor fluctuations. This has led to extensive research, with a significant focus on scalar perturbations as the seeds for second-order tensor perturbations, primarily because scalar perturbations dominate at the linear level. Such a topic has indeed a long history, starting from <cit.> and has been later developed in, e.g. <cit.>. Recent results from PTA collaborations <cit.> have identified a stochastic GWs background. This intriguing signal has sparked various interpretations, including inspirals of supermassive black hole binaries <cit.>, phase transitions <cit.>, and cosmic strings <cit.>. Among these possibilities, scalar-induced gravitational waves (SIGWs) have garnered attention and are being considered in many analyses <cit.>. These “scalar-scalar induced GWs” have also been used as probes for primordial black holes <cit.>. Analytic integral solutions for second-order GWs, induced by various quadratic combinations of cosmological perturbations, during both matter and radiation-dominated epochs, are provided in <cit.>.
Curvature (scalar) perturbations are tightly constrained by the latest Planck data on cosmic microwave background (CMB) scales, with an amplitude of A_S=2.1× 10^-9 <cit.>. On the other hand, the recent joint analysis using Planck2018, BICEP2/Keck2015-2018, and LIGO-Virgo-KAGRA data has placed a tight constraint on r (<0.028) <cit.>, where r=A_T/A_S is the so-called tensor-to-scalar perturbation ratio and A_T is the amplitude of the tensor power-spectrum at CMB scales. However, these constraints would not be applicable to smaller scales without large extrapolations, leaving the amplitude of both scalar and tensor fluctuations mainly unconstrained. In fact, on small scales, there exist various well-motivated mechanisms in the literature that can generate such large perturbations, both scalar perturbations as mentioned in the induced gravitational waves scenario related to primordial black holes formation, and tensor perturbations in the context of induced matter perturbations <cit.>.
While the general expectation is that scalar-scalar induced GWs dominate the secondary GWs signal, the product of scalar-tensor and tensor-tensor also source GWs at second order. Scalar-tensor interactions may also play an important role in the wave-optics limit of the GW background <cit.> (see e.g. Refs. <cit.> for wave-optics effects in astrophysics). It is thus important to systematically study the physics of these additional GWs signals and investigate their distinct signatures and any chance at detecting them.
In this paper, we focus on gravitational waves induced by interactions between scalar and tensor perturbations in the radiation-dominated Universe, by treating such scalar-tensor interactions as a source to GWs in the early Universe.
Understanding these interactions in cosmology might be important for a general description of cosmic GWs propagating through an inhomogenous Friedmann-Lemaître-Robertson-Walker (FLRW) universe.
Scalar-tensor induced GWs have been explored before in Refs. <cit.>. In particular, Ref. <cit.> pointed out that, for Dirac delta primordial scalar and tensor spectra, scalar-tensor induced GWs may dominate the high-frequency regime of the total induced GWs. However, there is no general answer regarding the detectability of this signal compared to scalar-induced GWs. While GWs induced by scalar-tensor interactions are novel and intriguing in its own right, it is important to address this latter point. We improve and clarify previous studies on scalar-tensor induced GWs in several ways:
* We derive general formulae allowing for parity violation of primordial tensor modes. We show how such parity violation is partly inherited by scalar-tensor induced GWs and discuss their distinct signatures. We also take into account finite width of primordial spectra.
* We identify possible (unphysical) infra-red divergences in the scalar-tensor induced GWs which can be traced to the fact that a constant scalar mode is naively allowed to source tensors. We propose a (rough) regularization for such divergences motivated by the existence of a locally inertial frame.
* We find that for peaked primordial spectra, the low frequency tail of scalar-tensor induced GWs does not have a logarithmic running, in contrast to scalar-scalar induced GWs. There is also no resonant peak.
This paper is structured as follows. The next section presents the evolution equation for second-order induced GWs when the primordial perturbations are scalar and tensor. Section <ref> discusses the general form of the power-spectrum of scalar-tensor induced GWs and explores properties of the kernel. In Section <ref>, an example is provided with peaked scalar and tensor perturbations, considering both chiral and non-chiral waves in the primordial tensor sector. Section <ref> explores potential strategies to circumvent the divergence in the ultraviolet (UV) region. Finally, Section <ref> presents the conclusions and summarizes the findings of the study.
In our analysis, we assume that scalar perturbations are more pronounced on small scales than tensor perturbations, thus neglecting tensor-tensor interactions. Under the condition A_T < A_S, we investigate whether the scalar-tensor contribution to GWs can be distinguished from scalar-induced GWs. We identify a distinct signature of this interaction related to the chirality of GWs. Specifically, while scalar-induced GWs do not possess chirality, we demonstrate that chiral GWs, after interacting with scalar perturbations, exhibit different behaviours in the ultraviolet (UV) region for left- and right-handed waves. This characteristic can be valuable in distinguishing them from scalar-induced GWs.
We assume c=ℏ=M_ pl=1 throughout this paper.
§ TENSOR MODES INDUCED BY SCALAR-TENSOR INTERACTIONS
We consider a perturbed flat FLRW space-time, in the Poisson gauge, in which the metric is described by
ds^2= -e^2Φdt^2+a^2 e^-2Ψ(e^γ)_ijdx^idx^j,
where t is the coordinate time, a(t) the scale factor, Φ and Ψ are scalar perturbations and γ_ij tensor perturbations. We neglect vector perturbations for simplicity. To get an evolution equation of the tensor perturbations sourced by a mixing of first order scalar and tensor ones, we focus on the trace-less part of the ij-th Einstein equation. We start with a general approach: we use the ADM formulation and compute the full non-linear equations, which we present in Appendix <ref>. We then expand the non-linear equations only at linear order in γ_ij, keeping the full dependence on Ψ and Φ. In doing so, we obtain
γ̈_ij-Φ̇e^Φγ̇_ij+3(H-Ψ̇)γ̇_ij-e^2(Φ+Ψ)a^-2∇^2 γ_ij= 2a^-2e^2Φ TT^ab_ij[e^2Ψ(𝒮^ss_ab+𝒮^sst_ab)+𝒮^m_ab],
where 𝒮^ss_ij and 𝒮^sst_ij are respectively given by
𝒮^ss_ij=Φ_,ij+Φ_,iΦ_,j-Ψ_,iΨ_,j-Ψ_,ij+Φ_,iΨ_,j+Ψ_,iΦ_,j
and
𝒮^sst_ij =-[δ^klγ_ki(Φ-Ψ)_,jl+1/2δ^klγ_ki_,j(Φ-Ψ)_,l+1/2δ^klγ_kj,i(Φ-Ψ)_,l.
.+δ^klγ_ikΨ_,jΦ_,l+δ^klγ_ikΦ_,jΨ_,l+δ^klγ_ikΦ_,jΦ_,l-δ^klγ_ikΨ_,jΨ_,l] ,
and the contribution from the energy-momentum tensor of a perfect fluid matter, namely
𝒮^m_ij=∂_i V∂_j V ,
where V is the scalar component of the linear perturbation in the spatial velocity of the perfect fluid. In Eq. (<ref>), TT^ab_ij is the transverse-traceless projector which can be found, e.g., in <cit.>, and latin indices are raised and lowered with the spatial background metric, which at leading order is δ_ij. We present all the details in Appendix <ref>.
Let us focus on the leading order terms in scalar-tensor interactions. In this case, Eq. (<ref>) becomes, in conformal time dη=dt/a,
γ”_ij+2 Hγ'_ij-∇^2 γ_ij=4Φ∇^2 γ_ij+4Φ'γ'_ij ,
where '=d/dη and we used that in the absence of anisotropic stress we have Φ=Ψ.
In what follows we will treat the right hand side of (<ref>) as a source term. To do so, we will follow a perturbative expansion and split
γ_ij=γ^(0)_ij+γ^(1)_ij+... ,
where γ^(0)_ij is the solution to the homogeneous equation and γ^(1)_ij are the scalar-tensor induced GWs. Note that inside such source term in (<ref>) there is a bare Φ, namely without gradients or time derivatives. We will later show that this term is problematic for sufficiently flat primordial scalar spectrum and leads to unphysical divergences.
Now, we decompose scalar and tensor perturbations into their Fourier modes, we respectively have
γ_ij(x,η) = 1/(2π)^3∫ d^3k e^ik.xγ_k, σ (η) ϵ^σ_ij(k̂) ,
Φ(x,η) =1/(2π)^3∫ d^3k e^ik.xΦ_k(η) ,
where ϵ^σ_ij(k̂) are the (transverse-traceless) polarization tensors. To be compatible with the reality condition of the Fourier expansion (<ref>), we work with left and right handed polarization tensors where σ= R, L represents the polarization index. We also choose the normalization given by
ϵ^σ*_ij(k̂)ϵ^σ'ij(k̂)=2δ_σσ' .
In Fourier space Eq. (<ref>) becomes
γ_k,λ”+2ℋγ_k,λ'+k^2γ_k,λ=𝒮_st,λ(k,η) ,
where we defined
𝒮_st, λ(k,η) =-2∑_σ∫d^3k_1/(2π)^3 Φ^ p_k-k_1γ^ p, σ_k_1ϵ^σ_ij(k̂_1) ϵ^ij*_λ(k̂)
×[k_1^2 T_γ(k_1η) T_Φ(c_s|k-k_1|η)-T'_γ(k_1η) T'_Φ(c_s|k-k_1|η)] ,
c_s is the sound speed of scalar fluctuations, which for radiation domination is c_s=1/√(3), and we abused notation and dropped the superscript “1” in γ^(1)_ k in the left hand side of (<ref>) and used the homogeneous solutions for γ^(0)_ k and Φ_ k inside the integrand. We split such homogeneous solutions into a primordial (initial) value and a transfer function, namely
γ^(0)_k, σ(η)=γ^ p_k, σ T_γ(kη)
=γ^ p_k, σ√(π/2kη) J_1/2(kη) ,
and
Φ_k(η) =Φ^ p_k T_Φ(c_skη)=Φ^ p_k 2^3/2 Γ[5/2](c_skη)^-3/2J_3/2(c_skη) ,
where the superscript “p” refers to primordial and T_γ(kη) and T_Φ(c_skη) are respectively the transfer functions for the homogeneous solution to tensor and scalar modes in radiation domination. J_α(x) is the Bessel function of the first kind of order α. We note that T_γ(kη) is the same for both polarizations, unless there are parity violating terms in the gravity sector after inflation.
Applying Green's method, the solution to Eq. (<ref>) reads
γ_k, λ(η) =γ^(0)_k, λ(η)+ γ^(1)_k, λ(η)+ ... ,
with
γ^(1)_k, λ(η) =∫_0^η dη̃ S_st, λ(k,η) G(η, η̃) ,
where S_st,λ(k,η) is given by (<ref>), ... refers to higher order solutions and G(η, η̃) is the Green's function for the tensor modes, namely
G(η, η̃) =y_1(η̃)y_2(η)-y_2(η̃)y_1(η)/y_1(η̃)y'_2(η̃)-y_2(η̃)y'_1(η̃) .
In Eq. (<ref>) y_1 and y_2 being the two homogeneous solutions for γ^(0)_k. Concretely, if we take y_1 to be given by the “growing mode” (<ref>), y_2 is given by “decaying mode” which reads as in (<ref>) but with Y_1/2(x), the Bessel function of the second kind, instead of J_1/2(x). Note that the first term of Eq. (<ref>) is the usual first-order (primordial) GWs, whereas the second one is for the modulated (scalar-tensor induced) GWs. In the next section we present analytical formulae for the kernel and the power-spectrum of scalar-tensor induced GWs.
§ SCALAR-TENSOR INDUCED GW SPECTRUM AND KERNEL FUNCTION
Let us derive a general formula to calculate the spectrum of scalar-tensor induced GWs. We aim to compute the two point function of tensor modes, namely from Eq. (<ref>)
⟨γ_k, λγ_k', λ'⟩= ⟨γ^(0)_k, λγ^(0)_k', λ'⟩+⟨γ^(1)_k, λγ^(1)_k', λ'⟩+... ,
where ... denotes two point functions involving higher order solutions of γ_k, λ, e.g. γ^(2)_k, λ. In this respect, it is important to note that we are neglecting the contribution ⟨γ^(0)_k, λγ^(2)_k', λ'⟩ which naively would be of the same order as ⟨γ^(1)_k, λγ^(1)_k', λ'⟩. However, computing γ^(2)_k, λ involves solving second order equations for Φ, using the solution for γ^(1)_k, λ and computing the third order components in the source term of γ_k, λ. This is out of the scope of this paper. Here we focus on fully understanding the solution γ^(1)_k, λ and the corresponding spectral density.
We write the two point correlation of scalar-tensor induced GWs in terms of a dimensionless power-spectrum which we call Δ^2_γ_1(k). Namely, we have that
⟨γ^(1)_k, λ(η)γ^(1)_k', λ'(η)⟩= (2π)^3 δ_λλ'δ^3(k+k') 2π^2/k^3Δ^2_γ_1, λ(k) ,
where it should be noted that both δ_λλ' and δ^3(k+k') follow from the contraction of the polarisation tensors in Eq. (<ref>) and keeping in mind that the two-point function of the primordial Φ and γ^ p_k, σ are written as
⟨Φ^ p_k Φ^ p_k'⟩ = (2π)^3 δ^3(k+k') 2π^2/k^3Δ^2_Φ(k) ,
⟨γ^ p_k, σγ^ p_k', σ'⟩ = (2π)^3 δ^3(k+k') δ_σσ'2π^2/k^3Δ^2_γ_0, σ(k) .
Doing so, we arrive at
Δ^2_γ_1, λ(k) = k^3/π∑_σ∫
d^3k_1Δ^2_Φ(|k-k_1|) Δ^2_γ_0, σ(k_1)/k_1^3
|k-k_1|^3ϵ^ij,σ(k̂_1) ϵ_ij^λ*(k̂)ϵ^mn, σ(-k̂_1) ϵ^λ *_mn(-k̂)
×( ∫_0^η dη̃ G(η, η̃) [k_1^2 T_γ(k_1η̃ ) T_Φ(c_s|k-k_1|η̃ )-T'_γ(k_1η̃ ) T'_Φ(c_s|k-k_1|η̃ )])^2 .
To perform the integrals, it is convenient to work with the variables given by
v=k_1/k , u=|k-k_1|/k and x=kη .
Further using the properties of the polarization tensors, namely ϵ^(λ)*_ij(k̂)=ϵ^(λ)_ij(-k̂), and ϵ^λ_ij(-k̂)=ϵ^-λ_ij(k̂), we derive a compact expression for the right and left polarizations of the induced GWs respectively given by
Δ^2_γ_1, R/L(k)
=1/32∫_0^∞ dv ∫_|v-1|^v+1 du/v^6u^2 Δ^2_Φ(uk) ℐ^2(x,u,v)
×[ ((v+1)^2-u^2)^4 Δ^2_γ_0, R/L(vk)+((v-1)^2-u^2)^4 Δ^2_γ_0, L/R(v k)] ,
where we used a slash “/” in the subscript of Δ^2_γ_1, λ to differentiate between the case of right and left polarization, and we defined
ℐ(x,u,v)= π/2√(x)∫_0^x dx̃ x̃^3/2 (J_1/2(x̃)Y_1/2(x)-J_1/2(x)Y_1/2(x̃))
×[v^2 T_γ(vx̃) T_Φ(c_sux̃)-d/dx̃T_γ(vx̃) d/dx̃T_Φ(c_sux̃)] .
An overline in ℐ in Eq. (<ref>) denotes oscillation average. By averaging over multiple wavelengths, we can improve the accuracy of parameter estimation, by mitigating the effects of the rapid oscillations. We also note that, in what follows, we formally take the upper limit of the time integral (<ref>) to infinity as the GW frequencies of interest enter the horizon well inside the radiation dominated universe, that is kτ_ eq≫ 1 where τ_ eq is the (conformal) time of radiation-matter equality. The contribution from large values of the conformal time is therefore negligible. In <ref> we present the full time dependence of the kernel. In this case, integrating and taking the oscillation average, we can define the following quantity
ℐ_∞^2(u,v) ≡ x^2×ℐ^2(x→∞,u,v)
=9/2^7(v/c_s u)^2[π^2(1-s^2)^2 Θ(1-s^2)+(2s+(1-s^2)log|1+s/1-s|)^2] ,
where the subscript ∞ refers to the limit x→∞, we multiplied ℐ^2(x→∞,u,v) by x^2 to subtract the typical decay of sub-horizon tensor modes, i.e., γ∝ 1/a, so that ℐ_∞^2(u,v) is time independent and for convenience we defined
s=v^2+c_s^2 u^2-1/2c_suv .
The oscillation average is taken because GW detectors measure the time average of the GW background. Note that, in contrast to scalar-scalar induced GWs, in Eq. (<ref>) only the variable u is multiplied by c_s as it corresponds to the scalar mode, but not v which is related to the momentum of the tensor mode.
The spectral density of scalar-tensor induced GWs is then given by
Ω^st-ind_GW, R/L, c(k) = 1/12(k/ℋ)^2 Δ^2_γ_1, R/L(k)
=1/384∫_0^∞ dv ∫_|v-1|^v+1 du/v^6u^2 Δ^2_Φ(uk) ℐ_∞^2(u,v)
×[ ((v+1)^2-u^2)^4 Δ^2_γ_0, R/L(vk)+((v-1)^2-u^2)^4 Δ^2_γ_0, L/R(vk)] ,
where we used that in the radiation dominated universe H=1/η. The subscript “c” in (<ref>) denotes evaluation at a time where GWs are deep inside the horizon so that they behave as radiation. It is interesting to note from Eq. (<ref>) that if there is primordial parity violation of GWs, such parity violation is inherited by the scalar-tensor induced GWs and it is smeared between the polarizations. For completeness, we also give the formula for the total spectral density which is the sum of both polarizations, namely
Ω^st-ind_GW, c(k) = ∑_λ= R, LΩ^st-ind_GW, λ(k,η)
=1/12∫_0^∞ dv ∫_|v-1|^v+1 du/v^2u^2 Δ^2_Φ(uk) ℐ_∞^2(u,v)(Δ^2_γ_0, R(vk)+Δ^2_γ_0, L(vk))
×[ (1+v^2-u^2)^2/v^2+ (1+(1+v^2-u^2/2v)^2)^2 ] ,
which is of course parity symmetric. In the case of no primordial parity violation we simply take Δ^2_γ_0, R=Δ^2_γ_0, L=Δ^2_γ_0 in Eq. (<ref>). To evaluate the amplitude of the GW spectral density today, we use <cit.>
Ω^st-ind_ GW, R/L, 0h^2 =1.62× 10^-5(Ω_ rad, 0h^2/4.18× 10^-5)(g_ρ(T_c)/106.75)(g_s(T_c)/106.75)^-4/3Ω^st-ind_GW, R/L, c .
Note that if one wants to use the curvature perturbation R, one has that
Δ_Φ^2=(3(1+w)/5+3w)^2Δ_ R^2=4/9Δ_ R^2 ,
where in the last step we use that for radiation domination w=1/3.
This completes the general derivation of the kernel and the spectral density of scalar-tensor induced GWs for general primordial parity of tensor modes. This is one of the new results of our work.
§.§ General behaviour of the kernel and differences with scalar-scalar induced GWs
We proceed to show the general properties of the kernel (<ref>) and its differences with the case of scalar-scalar induced GWs. First, let us examine the infrared (k → 0) behaviour of our GWs. In this limit, u ∼ v ∼ 1/k ≫ 1. Hence, s takes the value
s =1+c_s^2/2c_s .
In contrast, the scalar-induced gravitational waves (SIGWs) exhibit a different behaviour, as s approaches 1 (see Appendix <ref>). In that scenario, a logarithmic running arises, which is not observed in scalar-tensor-induced GWs due to the fact that s never reaches the value of 1 in IR. This logarithmic running has been regarded as a distinctive characteristic of SIGWs, setting them apart from primordial GWs in the infrared (IR) region. Notably, our findings demonstrate that such a feature is absent in our case.
In the opposite limit (k →∞), i.e. the UV one, v→ 1 and u→ 0, which corresponds to the large wavelength limit for the scalars, we observe that the variable s approaches u. As a result, the kernel exhibits the behaviour
ℐ_∞^2(u → 0,v → 1)∼1/u^2 .
Consequently, the integral in Eq. (<ref>) becomes proportional to 1/u^4. The other k→∞ limit, corresponding to a long wavelength tensor modes, leads to v→ 1 and u→ 0. In this case we have s → 1/v, which does not lead to any divergence.
In Eq. (<ref>), another notable feature occurs when s=± 1. In the case of SIGWs, this scenario can lead to a logarithmic resonance. However, in our case, the presence of the term (1-s^2) prevents such resonance from occurring. In fact, for s=± 1, or v=± (1-c_s u), we have
ℐ_∞^2 =9/2^5(1-1/c_s u)^2 .
This diverges in the u → 0 limit.
We would like to emphasize that the kernel remains finite throughout the entire integration region being considered (except for the strict limit u → 0). However, the fact that the integrand grows unboundedly for small u can lead to artificial enhancements in the GWs spectrum. In particular, if both primordial scalar and tensor spectra are flat, the integral does not converge. If the tensor spectra is flat and primordial peaked, say at u=k_p/k, then the GW spectrum grows unboundedly in the UV where k≫ k_p. If the scalar spectrum is flat and the tensor spectrum peaked at v=k_p/k, then the GW spectrum diverges at k=k_p where v=1 and u=0. For these reasons, we will not consider the aforementioned cases and focus only on peaked spectra for which no such divergences occur. We later focus on identifying the source of the divergence and propose a regularization in <ref>.
§ SCALAR-TENSOR INDUCED GWS FROM PEAKED SOURCES
Now, we proceed to demonstrate the effect using a specific choice of input scalar and tensor perturbations: peaked sources. We do so for simplicity and because enhancements of primordial scalar and tensor fluctuations during inflation often lead to peaked primordial spectra <cit.>. We first consider Dirac delta primordial spectra and later discuss the effects of a finite width.
Let us take a Dirac delta source located at k_s/t,*, for both scalar and tensor primordial power-spectra,
Δ^2_Φ(k)=A_Φ δ(lnk/k_s,*) and Δ^2_γ_0, R/L(k)=A_γ_0, R/L δ(lnk/k_t,*) .
Note that in general we may have k_s,*≠ k_t,*. This kind of peaked scalar sources can be relevant for primordial black hole formation <cit.>. Below, we illustrate the impact of this particular choice on the final spectra in two scenarios: when the primordial GWs exhibit chirality and when they do not.
§.§ Non-chiral primordial GWs
We first consider the case when A_γ_0, R=A_γ_0, L=A_γ_0. Then, we consider two different cases:
* Peaks at the same location: The simplest possibility, also considered in <cit.>, is that both peaks are at the same location, namely k_s,*= k_t,*=k_*. In that case, we have
Δ^2_γ_1(k)= A_Φ A_γ_0 (k/k_*)^2[1+k^4/16k_*^4+ 3k^2/2k_*^2]ℐ^2_u=v=k_*/k Θ(2k_*-k) .
Fig. <ref>, left panel, shows the GWs energy density for this case. We report a small difference with Fig. 2 of <cit.>: the high frequency part of the spectrum in <cit.> presents some wiggles, while we find no such feature in Eq. (<ref>). Unfortunately, comparison is not so straightforward because we are not considering tensor-tensor induced GWs and we are using a different prescription for the metric perturbations (Eq. (<ref>)). It is interesting to note though that the precise form of the scalar-tensor and tensor-tensor mixings (<ref>) depends on how one expands the metric (i.e. exponential or linear in γ_ij and Φ). Ultimately, this is related to the gauge issue of induced GWs <cit.>.
* Peaks at different locations: In general it is possible that k_s,* and k_t,* are two independent parameters and so k_s,*≠ k_t,*. If so, we find
Δ^2_γ_1(k) = A_Φ A_γ_0 k^2/k_s,*k_t,*[(k^2+k_t,*^2-k_s,*^2)^2/k^2k_t,*^2+ (4k^2k_t,*^2+(k^2+k_t,*^2-k_s,*^2)^2)^2/16k^4k_t,*^4]
×ℐ^2_v=k_t,*/k, u=k_s,*/k Θ(k_s,*-|k_t,*-k|) Θ(k_t,*+k-k_s,*) .
Naturally, the range of wavenumber of the induced waves increases with a decreasing separation of the two different peaks, which can be seen from the two Heaviside thetas, and the left panel of Fig. <ref>. In other words, for Dirac delta separate peaks, the scalar-tensor induced GWs have an IR and UV cut-off.
By looking at both panels of Fig. <ref> , we see that they exhibit an enhancement in the induced GWs between the two dips that are determined by the first Heaviside theta function in Eq. (<ref>), even though the scalar and tensor source peaks are located at different positions.
This enhancement is related to the fact that we have a very large value of the integrand at small u. We have been able to obtain a solution in case of monochromatic primordial perturbations only because in this case the momenta acquire a single value. For other shapes of source primordial spectra, the integral should be regularised.
It is interesting to note that, contrary to the scalar-scalar induced GWs, the scalar-tensor GWs spectrum for Dirac delta spectra have a finite amplitude at the cut-off k=2k_*. This may look suspicious at first because one expects the GW spectrum to be continuous. However, the sharp cut-off is due to the “unphysical” Dirac delta. Once we consider a log-normal peak the GW spectrum exponentially vanishes near the cut-off, as we shall show later.
§.§ Chiral primordial GWs
For primordial GWs, which have a delta-function peak in only one of the polarizations at the same wave-number as the primordial scalars (Eq. (<ref>)), the present-day total spectral density of the induced GWs can be obtained from Eq. (<ref>)
Ω^st-ind_GW, R/L, c(k,η) = 1/768 A_Φ A_γ_0, R/L (k/k_*)^6[1+32(k_*/k)^4+48(k_*/k)^2]
×ℐ^2_u=v=k_*/k Θ(2k_*-k) .
The left panel of Fig. <ref> displays the spectral density of induced gravitational waves resulting from the interaction between peaked scalar and peaked chiral gravitational waves. In this and the right panel, it can be observed that when the primordial chiral gravitational waves induce gravitational waves of the same polarization, the peak of the spectrum, which is situated in the UV region, is more pronounced than that of the opposite polarization. In the IR region, however, we have an unpolarized induced wave. This could be attributed to the choice of peaked sources. Since the IR region is located far away from the peak of the GWs signal, there is effectively no detectable difference in the behaviour of the polarizations. Although only the case with the right-handed primordial gravitational waves are displayed, the same applies to the left-handed ones. The trend is also manifested in the right panel of Fig. <ref>, which exhibits the same scenarios but with different peak locations of scalar and tensor perturbations.
§.§ Scalar-tensor induced GWs from log-normal peaks
We now consider a more realistic situation where the peaks in the primordial spectra have a finite width. We do so by considering a log-normal spectrum, namely
Δ^2_Φ(k)=A_s/√(2π)σexp[-ln^2(k/k_s,*)/2σ^2] and Δ^2_γ_1, R/L(k)=A_γ_0, R/L/√(2π)σexp[-ln^2(k/k_t,*)/2σ^2],
where we will consider for simplicity that they have the same logarithmic width σ but in principle they can differ. We consider again the two possibilities: same peak location and different peak location. Interestingly, we find that in both cases the amplitude of the scalar-tensor induced GW spectrum is not very sensitive to the width of the primordial spectra. The spectral shape of course changes: it broadens for broader peaks.
For the log-normal we compute the scalar-tensor induced GWs numerically. We show our results in Figs. <ref> and <ref>. For simplicity, we only considered only right polarization for the primordial tensor modes. However, one can consider the case of non-chiral primordial tensor modes by summing both lines in Figs. <ref> and <ref> corresponding to scalar-tensor induced GWs and multiplying by 2. Since we are plotting in logarithmic scale, the change in amplitude is not significant.
In Fig. <ref> we show the case of same peak position on the left and different peak position on the right. See how as one gets closer to the peak all lines are similar to the Dirac delta case. Also, note that for a finite width primordial spectra there is no sharp cut-offs in the scalar-tensor induced GWs, as expected. Since we are considering finite width of the primordial scalar and tensor spectra, we can also show and compare all contributions (primordial tensor, scalar-scalar and scalar-tensor induced GWs). We do so in Fig. <ref> where we show our results for σ=0.01 (left) and σ=0.1 (right). See how in both cases the scalar-tensor induced GWs dominate the spectrum near the cut-off, even for σ=0.1. In this way, we extend the results of <cit.> and show that even in the case of not too sharp and not too broad primordial spectrum, the scalar-tensor induced GW eventually have the potential to show-up in the high frequency part of the spectrum.
§.§ Future prospects for scalar-tensor induced GWs
After demonstrating two examples of scalar-tensor-induced spectra without encountering divergence issues, our attention now shifts to the detectability of these spectra. As stated earlier, we make the assumption that A_γ_0<A_Φ, which enables us to neglect the tensor-tensor contribution and results in an effect that is subdominant compared to scalar-scalar induced GWs.
In the right panel of Fig. <ref>, it is evident that there exists only a limited range of scales
(approximately k/k_* ∈ [1.34, 2]), where the scalar-induced GWs do not surpass the scalar-modulated ones. Detecting the modulated waves amidst the dominance of the former requires identifying a characteristic that can distinguish our effect. As observed in previous section, non-chiral primordial waves lack such a property. However, when scalar modulation affects chiral primordial GWs, a disparity in the energy density between left and right circularly polarized waves becomes apparent. This distinction offers a potential avenue for detecting and studying the modulated waves.
Fig. <ref> provides a comprehensive comparison between the primordial GWs, the realistic log-normal case, and the sensitivities of various probes. It is evident from the plot that while scalar-tensor induced GWs can dominate over SIGWs in the high-frequency range, this dominance is limited to a small range of scales. We observe the same behaviour for σ=0.1 and σ=0.01 in Fig. <ref>. On the other hand, the behaviour of the different-parity induced waves presents a distinguishing characteristic that sets them apart from SIGWs, particularly in the UV scales. This parity-violating behaviour of the scalar-tensor induced waves can be observed in Fig. <ref>, where it extends beyond the peak of the primordial tensor spectrum. While this effect is not very significant for σ∼ 0.1, it becomes important for σ≲ 0.01. We would like to clarify that the figure shown in Fig. <ref> provides a quantitative description of the power-spectrum shape of the induced GWs, rather than the detectability of their chiral properties. It is important to note that planar detectors typically do not have the capability to directly detect the chirality of GWs, unless specific methods are employed, such as leveraging the motion of the solar system with respect to the cosmic reference frame (as discussed in <cit.>, see also the refs. therein). However, studies have shown that by cross-correlating the output of multiple detectors, such as LISA and Taiji, it is possible to detect and study the parity violation in the stochastic gravitational wave background <cit.>. While we mention the potential of using this chirality to distinguish our induced GWs, the actual detectability of the same requires further detailed analysis beyond the scope of this work. In the low-frequency range, however, the induced waves consistently remain unpolarized, as depicted in the corresponding figures.
Upon examining Fig. 10 in <cit.> and Fig. 3 in <cit.>, we observe a similar behaviour in the induced GWs as depicted in our Fig. <ref>. It is noteworthy that these studies investigate GWs production mechanisms involving chiral dark photons, which are entirely distinct from our approach. Based on these observations, a hypothesis emerges, suggesting that there is a distinct polarization behaviour present in the UV region of induced GWs. It says that the polarized primordial component makes the peak of the induced GWs of the same polarization more enhanced compared to that of the oppositely polarized GWs, while the IR region remains unpolarized. However, the thorough investigation and verification of this hypothesis are left for future endeavours.
In Fig. <ref>, we present the recent results from the NANOGrav <cit.>, which may have detected the stochastic gravitational wave background using pulsar timing arrays (PTAs). The SGWB observed by PTAs can be considered as the IR tail of the SIGWs <cit.>. We leave a detailed analysis of our scalar-tensor induced GWs signal with the new PTA data <cit.> for future work. Here, we demonstrate an example where the peak of the SIGWs lies in the 0.1μ Hz range, which is currently beyond the sensitivity range of existing detectors. However, there has been proposals for a future detectors in this frequency range <cit.>. If such a detector, preferably with better sensitivity and ability to detect chirality, is realized, it would be capable of detecting the peak, where it can be distinguishable from our signal based on the latter's chirality properties.
§ ORIGIN OF DIVERGENCES AND REGULARISATION
In <ref> we have anticipated that the momentum integral in scalar-tensor induced GWs contains a divergence for vanishing scalar mode momentum. The divergence in the integrand appears as 1/u^4∼ (k/q)^4 for q≪ k, where here k and q respectively are the internal tensor and scalar mode momentum (note that the because q≪ k we have that the internal tensor mode momentum is |k-q|∼ k so there is not much difference between the external and internal mode in this limit). It should also be noted that we did not encounter such issues in <ref>, as we considered scalar-tensor induced GWs from peaked primordial spectra. But, as soon as we consider a relatively flat scalar primordial spectra, the divergence shows up. For flat scalar and tensor primordial spectra the integral divergences for all k. In this section we identify the source of such unphysical divergence and propose a practical and well-motivated way to regularize it. We leave a deeper study of the regularization procedure for future work.
We proceed as follows. We first derive the general time-dependent kernel to show that divergence is not completely associated with the limit x=kτ→∞ and that artificial enhancements appear for scalar modes with wavelengths larger than that of tensor modes. We will see that this is independent on whether scalar modes are superhorizon or subhorizon. In fact, we show that removing the contribution from superhorizon scalar modes does not solve the problem. We then provide an argument based on a local inertial frame to regularize the integral.
§.§ General time dependent kernel
We start by rewriting the kernel (<ref>) in a more practical form for the general integrations. We rewrite the Bessel functions, J_α(x), in terms of spherical Bessel functions, j_α(x), as
I(x,u,v)=v^2/x( I_y(x,v,u)sin x + I_j(x,v,u)cos x ) ,
where we defined
I_j,y(x,v,u)≡∫_x_i^x dx̃ x̃^2{
j_0(x̃)
y_0(x̃)
}(j_0(vx̃)j_0(c_sux̃)-j_2(vx̃)j_2(c_sux̃)) .
In this way, we have that the first term in Eq. (<ref>) corresponds to the “growing mode” of a free tensor mode in a FLRW background and the second one to the “decaying mode”. We will now compute the functional form of I_j(x) and I_y(x) for general x. Without loss of generality we will take x_i→ 0, which does not influence the present discussion. After several integrations, similar to those in Ref. <cit.>, we obtain that
I_j(x,v,u)= 3(1-s^2)/8c_suvG_0[ Si [x]]+3(1-s^2)/8c_suvxG_1[cos x]
+3/8(c_suvx)^2G_2[sin x]+3/16(c_suvx)^3G_3[cos x]+9/16(c_suv)^3x^4G_4[sin x] ,
and
I_y(x,v,u)= -3/8c_suv(2s+(1-s^2)(log|1+s/1-s|+F_0[ Ci |x|]))+3(1-s^2)/8c_suvxF_1[sin x]
-3/8(c_suvx)^2F_2[cos x]-3/16(c_suvx)^3F_3[sin x]-9/16(c_suv)^3x^4F_4[cos x] .
In the expressions above we have defined
F_0 [ Ci |x|]
= Ci|(1 + c_s u - v) x| +
Ci|(1 - c_s u + v) x| - Ci|(1 + c_s u + v) x|- Ci|(1 - c_s u - v) x| ,
and
F_1[sin x]
= sin[(1 + c_s u - v) x]/1+c_su-v+sin[(1 - c_s u + v) x]/1-c_su+v -sin[(1 + c_s u + v) x]/1+c_su+v-sin[(1 - c_s u - v) x]/1-c_su-v .
Then, G_0[ Si[x]] is obtained by replacing Ci|x| by Si[x] in F_0[ Ci |x|]. Similarly, G_1[cos x] is obtained by replacing sin x by cos x in F_1[sin x]. The other functions F_2,F_3,F_4 and G_2,G_3,G_4 are suppressed when c_suvx≫1. For the interested reader, we write them explicitly in Appendix <ref>. It is straightforward to check that we recover Eq. (<ref>) in the limit of x→∞ and we take the oscillation average.[The function F_0[ Ci |x|] with Ci[x] vanish when x→∞. The other function gives
G_0[ Si[x→∞]]=πΘ(1-|c_su-v|)-πΘ(1-(c_su+v)) .
It is only non-vanishing when |c_su-v|<1<c_su+v which corresponds to s^2<1.
]
§.§.§ Long wavelength scalar modes
With the full time dependence of the kernel, we can study two limits involving a subhorizon induced tensor mode (x≫1) and a long wavelength scalar mode (u≪1): (i) the scalar mode is superhorizon at the time x, i.e. ux≪1, and (ii) the scalar mode is subhorizon at time x, namely ux≫ 1. Note that the limit u≪1 corresponds to the regime where u∼0, v∼ 1 and s∼ 0 in the kernel. For case (i), where ux≪ 1, we find that
I_j(x,ux≪1)≈x-cos xsin x/2+ O(u^2,u^2x^2) , I_y(x,ux≪1)≈ -sin^2x/2+ O(u^2,u^2x^2) .
Note that the integral I_j in Eq. (<ref>) is diverging for x→∞. This is one of the sources of the divergence in the naive oscillation average presented in Eq. (<ref>). Nevertheless, even if we consider the total time dependence of the kernel before doing the oscillation average, the situation does not improve. For instance, the kernel Eq. (<ref>) in this limit leads us to
I(x,ux≪1,x≫1)≈1/2(cos x-sin x/x) .
Looking at Eq. (<ref>), we see that in addition to the regular “growing” mode (sin x/x) there is a non-decaying mode (cos x). As we shall shortly show, after the mode u enters the Hubble horizon, the integrals I_j,y(x) approach a constant. However, the constant will be larger for smaller u modes, since these modes had more time to grow. This can be seen in case (ii), where ux≫1, in which the integrals I_j and I_y respectively yield
I_j(u≪1,ux≫1)≈3π/8c_su-3c_sπ u/32 , I_y(u≪1,ux≫1)≈ -3/4+c_s^2u^2/16 .
We see that despite the fact that after the mode u enters the Hubble horizon, the kernel (<ref>) behaves as a regular kernel for tensor modes in the sense that there is a growing and decaying mode, coefficient of the decaying mode, that is I_j given in Eq. (<ref>) diverges for small u. This is the same divergence that we found in Eq. (<ref>) for I_∞^2(u≪1, v∼1)∼ 1/u^2. However, the total integrand diverges as 1/u^4. We will come back to the remaining 1/u^2 at the end of the section.
In passing, we mention that we find no such behaviour when the wavenumber of the primordial tensor mode vanishes, i.e. when v→ 0, u→ 1 and s→ -∞. Explicitly, in the limit where x≫1, ux≫ 1, vx≫1, we find that
I_j(v≪1,x≫1)≈ 0 , I_y(v≪1,x≫1)≈1/1 - c_s^2 + (5 - c_s^2) v^2/5 (1- c_s^2)^3 .
The integrand of the momentum integral for scalar-tensor induced GWs, Eq. (<ref>), vanishes for v→0.[The c_s=1 case has to be treated separately but one can check that it is regular as well in the limit v→ 0.] Thus, the only issue are scalar modes with wavelengths longer than the tensor mode.
§.§.§ Removing superhorizon scalar mode contribution
To illustrate the fact that the issue is more subtle, let us show that even if we remove the strange non-decaying mode in Eq. (<ref>) by subtracting the contribution from superhorizon scalar modes, the divergence is not cured. The problematic term in the equations of motion for induced tensor modes Eq. (<ref>) is the term that has Φ in front of ∇^2γ_ij. For analytical purposes, we remove the contribution of superhorizon Φ coming from the mentioned term by introducing the following correction terms in Eq. (<ref>), which is to be subtracted to the coefficients of the growing and decaying modes I_j,y
C_j,y(x,v)≡∫_x_i^x dx̃ x̃^2{
j_0(x̃)
y_0(x̃)
}
j_0(vx̃) ,
where we used that T_Φ(x≪1)∼ 1. We will later cut the integral near sound horizon crossing, that is at α c_sux=1 where α is a free parameter. After integration, we find that
C_j(x)=cos (x) sin (v x)-v sin (x) cos (v x)/v (v^2-1) ,
and
C_y(x)=sin (x) sin (v x)+v cos (x) cos (v x)/v (v^2-1)-1/v^2-1 .
We can see that the correction terms C_j and C_y cancel the leading order terms of the non-decaying modes we found in Eq. (<ref>). To see this, we take the limit v∼ 1 in Eqs. (<ref>) and (<ref>), which respectively yields
C_j(ux≪1,v∼ 1)≈x-cos xsin x/2≈ I_j(x,ux≪1) ,
and
C_y(ux≪1,v∼ 1)≈ -sin^2x/2≈ I_y(x,ux≪1) .
We can now follow the corrected kernel until the scalar mode enters the sound Hubble horizon, i.e. we evaluate C_j/y at α c_sux=1 where α∼ O(1) free parameter to be fixed shortly. The corrected kernel is then given by
I(ux≫1)=v^2/x([ I_y(x)-C_y(x=1/(α c_su))]sin x +[ I_j(x)-C_j(x=1/(α c_su)]cos x ) .
In the limit where the divergence appeared, that is for ux≫1, u≪1 and v∼ 1, we find that
C_j(ux≫1,v∼ 1)≈1/2 α c_s u-1/4sin(2/α c_s
u) ,
and
C_y(ux≫1,v∼ 1)≈-1/2sin ^2(1/α c_s u) .
By requiring that the divergence in I_j is cancelled by that in C_j we fix α to
α=4/3π≈0.42 .
This value of α means that we subtract a constant Φ a little bit after sound Hubble horizon crossing, i.e. at uc_sx≈ 2.3.
The results of this section show that the divergence in the kernel can be made finite in the limit x→∞ (as in Eq. (<ref>)) by subtracting the superhozion contribution of a constant Φ. Nevertheless, the coefficient of the kernel become constant which implies that the integrand for scalar-tensor induced GWs Eq. (<ref>) still diverges as 1/u^2 for u≪1. This means that the divergences for long wavelength scalar modes are not entirely due to their superhorizon contribution but in fact some subhorizon as well. In the next subsection we identify the full source of the divergence.
§.§ Source of the divergence and regularization
To understand the source of the divergence, let us split the gravitational potential Φ into a short and a long wavelength contribution, namely
Φ=Φ_ short+Φ_ long ,
where ∂_kΦ_ long≪∂_k γ_ij. In other words, if L_Φ is the characteristic length of Φ_ long and L_γ the characteristic length of γ_ij, we have that ∂_kΦ_ long× L_γ∼ O(ϵ) Φ_ long where we defined ϵ=L_γ/L_Φ. In that case, the equations of motion for scalar-tensor induced tensor modes (<ref>) read
γ”_ij+2 Hγ'_ij-∇^2 γ_ij=4(Φ_ short+Φ_ long)∇^2 γ_ij+4(Φ_ short'+Φ_ long')γ'_ij .
For the time derivatives we have that Φ_ long'≪Φ_ short' so we can neglect Φ_ long'. However, in front of ∇^2 γ_ij we have directly Φ_ short+Φ_ long. In radiation domination, the amplitude of Φ_ short has decayed more than that of Φ_ long and therefore, it appears that the amplitude Φ_ long could be the dominant contribution and have a big impact on γ_ij^(1). For instance, if we consider that Φ_ long= constant, we see that the right hand side of Eq. (<ref>) has now a term proportional only to ∇^2γ_ij. If we think of it as a “source”, then γ_ij is resonating with itself and leading to divergences. We will see this when computing the general kernel.
However, our intuition from the equivalence principle tells us that long wavelength modes cannot affect significantly the physics of short wavelength modes. We can use the fact that ∂_kΦ_ long≪∂_k γ_ij to consider that Φ_ long≈ constant from the point of view of γ_ij. We then rescale the spatial coordinates as
dx_i → (1-2Φ_ long) dx_i so that ∇^2→ (1+4Φ_ long)∇^2 .
This coordinate transformation cancels the constant factor of Φ_ long∇^2 γ_ij in the right hand side of Eq. (<ref>) and we are left with
γ”_ij+2 Hγ'_ij-∇^2 γ_ij≈4Φ_ short∇^2 γ_ij+4Φ_ short'γ'_ij .
Thus, we conclude that long wavelength scalar modes cannot affect the local generation of tensor modes. Although in the strict sense Φ_ long is not a constant, it shows that if we view the scalar-tensor mixing in the right hand side of Eq. (<ref>) as a local source, then we must remove the contribution from long wavelength scalar modes. However, if we are dealing with propagation effects, the amplitude of Φ is very real as it can be seen in the gravitational lensing of gravitational waves and time-delay effects (for the latter case see, e.g. the analysis in <cit.>).
If we go to the next to leading order in the gradient expansion, we roughly expect that
Φ_ long≈ constant+ O(ϵ^2)Φ_ long ,
where ϵ^2=L_γ^2/L_Φ^2≈ q^2/k^2, where q is the wavenumber of Φ and k the wavenumber of γ_ij. The fact that it starts at second order in ϵ is because of symmetry: the “amplitude" of Φ cannot depend on the direction of q. This is consistent with the superhorizon expansion of Φ, i.e. T_Φ(x≪1)=1+ O(x^2). This argument is also supported by the existence of a local inertial frame. In the so-called Conformal Fermi Coordinates, see e.g. Ref. <cit.> for the local expansion of the metric (although in a very different context), the scalar piece of the spatial component of the metric expanded around the local Fermi frame reads
g_ij(x_F)=a^2(η_F)[δ_ij-1/3R^F_ikjlx_F^kx_F^l] ,
where the subscript “F” refers to evaluation at the Fermi frame. We then have that at leading order in Φ_F
R^F_ikjlx_F^kx_F^l=1/2(x_i,Fx^k_F ∂_k∂_j Φ_F+x_j,Fx^k_F ∂_k∂_i Φ_F-∂_i∂_j Φ_F x_F^k x_k,F-δ_ijx^k_Fx^l_F ∂_k∂_l Φ_F) .
Since ∂_i∂_j Φ_F∼ q^2 Φ_F and x_F≲ L_γ∼ 1/k (i.e. the expansion is valid on the surroundings of the point x_F which are smaller than the tensor wavelength), this is consistent with Eq. (<ref>) since
R^F_ikjlx_F^kx_F^l≈ O(ϵ^2)Φ_F .
However, note that although we understand the dependence on ϵ, the coefficient that would enter in Eq. (<ref>) is not determined. To do that, we need a careful treatment of these subtle coordinate transformations and their relation between gauge transformations. This is out of the scope of this paper and we leave it for future work. Instead, we use the above arguments to propose a phenomenological solution. We interpolate between P_Φ and P_∇^2Φ for short and long wavelength scalar modes by including in the integral Eq. (<ref>) the following function of u:
f(u)≡u^4/d^4+u^4 .
This function goes from f(u≫1)∼ 1 to f(u≪1)∼ (u/d)^4 and cures any divergence. This also removes the need to subtract superhorizon contributions as they are naturally suppressed. The factor d reflects the uncertainty in the right moment for the transition between short and long wavelength scalar modes which should be d∼ O(0.1)- O(1). Thus, we propose the regular form of the scalar-tensor induced GWs as
Ω^st-ind_GW, R/L(k) =1/384∫_0^∞ dv ∫_|v-1|^v+1 du/v^6u^2 Δ^2_Φ(uk) f(u) ℐ_∞^2(u,v)
×[ ((v+1)^2-u^2)^4 Δ^2_γ_0, R/L(vk)+((v-1)^2-u^2)^4 Δ^2_γ_0, L/R(vk)] .
For example, if we take d=1, Δ^2_Φ=A_Φ and Δ^2_γ, R/L=A_γ, R/L we have
Ω^st-ind_GW, R/L, c(k)≈ A_Φ(0.48A_γ, R/L+0.043A_γ, L/R) ,
which is a sensible result if one compares it to the scalar-scalar induced GWs for flat primordial spectrum, which gives Ω^ss-ind_GW≈ 0.82 A_Φ^2 <cit.>. Eq. (<ref>) also tells us that if we have primordial tensor parity violation with a scale invariant spectrum, the scalar-tensor induced GWs are also scale invariant and mostly parity violating.
For a Dirac delta spectrum, we have that u=v≥ 1/2, so that the correction from f(u) is bounded below by f(u)≥ 2^-4/(d^4+2^-4). If we choose d=1, the suppression factor at u=v=1/2 is ∼ 0.06, quickly becoming 1 for u=v>1. However, we do not expect large corrections for the Dirac delta case with the same peak location, which means that the factor d should be O(0.1), giving us f(u) ≳ 0.99, and barely modifying the prediction in the Dirac delta case. This also implies that the suppression effect should only be important when the scalar mode has at least a wavelength ten times larger than the tensor mode.
Note that one may argue that there should be a gauge in which such divergences do not occur and the amplitude of curvature perturbation does not appear directly into the equations of motion for tensor modes. If that is the case, that particular gauge might be more appropriate for calculations. Unfortunately, we have found no standard gauge choice in which that occurs. For example, the third order action in Maldacena's paper in the flat and comoving gauges contain these kind of couplings. So if such gauge exist is by no means a trivial transformation. And, importantly, we would then enter into the discussion of the well-known gauge dependence of the energy density of GWs at second order in perturbation theory. For these reasons, we propose a regularization within the Poisson gauge which, besides some uncertainty, should provide a good estimate for the correct scalar-tensor induced GWs spectrum.
§ SUMMARY AND DISCUSSION
While the focus of GWs hunt has traditionally been on first-order GWs generated during primordial inflation, there is also a growing interest in second-order effects, specifically scalar-induced GWs (SIGWs). These SIGWs arise from the nonlinear interactions between scalar perturbations and GWs, and their detection could provide valuable insights into the early Universe and inflationary models. In this context, we are investigating a nonlinear interaction between first-order scalar perturbations and first-order tensor perturbations. Instead of directly generating gravitational waves from zero, these interactions modulate the existing waves. While similar modulated waves have been studied before, mainly considering non-chiral waves, our analysis focuses on the case of chiral waves. Our work aims to explore the distinctive features and implications of chiral modulated gravitational waves.
We have presented an analysis of both chiral and non-chiral peaked gravitational waves, modulated by a peaked scalar perturbation. Our investigation aimed to determine the extent to which these modulated waves are buried under the dominant SIGWs. Interestingly, while non-chiral waves are expected to surpass SIGWs only in a limited range of wave numbers, we have discovered a distinguishing feature in the case of chiral waves. This feature enables us to differentiate between the two induced polarization modes, providing a potential avenue for detecting and characterizing the modulated waves amidst the dominant SIGWs. We also anticipate the possibility of a universal unpolarised IR behaviour of induced GWs. However, further investigation is needed to confirm and fully understand this effect.
Another significant finding in this study is the identification of an unphysical divergence in the momentum integral of the kernel. This divergence emerges due to the existence of exceptionally long wavelengths in the scalar modes, compared to that of the tensor modes. It is important to note that such divergences do not occur in the SIGWs case, as it involves a source term that includes the gradient of the gravitational potential Φ. However, in the current scenario, the divergence arises from the multiplication of the gravitational potential Φ with the gradient of the tensor mode. To address this divergence issue, a possible solution involves separating the long and short modes of the scalar field and absorbing the long modes into a new background. This approach can help mitigate the divergence problem. However, to fully resolve this issue, further investigation is required, and it will be explored in future work. A factor with a free parameter is introduced in the kernel to address the divergence, but its effectiveness and implications need to be thoroughly investigated in subsequent studies.
§ ACKNOWLEDGEMENTS
We would like to thank J.L. Bernal, A. Caravano, E. Komatsu, A. Ricciardone, F. Schmidt and M. Sasaki for valuable discussions. G.D. is supported by the DFG under the Emmy-Noether program grant no. DO 2574/1-1, project number 496592360. Calculations of the scalar-scalar induced GW spectrum have been done with https://github.com/Lukas-T-W/SIGWfast/releasesSIGWfast <cit.>. P.B. acknowledges funding from Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di eccellenza” project Science of the Universe. N.B. and S.M. acknowledge support from the COSMOS network (www.cosmosnet.it) through the ASI (Italian Space Agency) Grants 2016-24-H.0, 2016-24-H.1-2018 and 2020-9-HH.0. For N.B. and S.M. it is a pleasure to recall that some preliminary notes about the GWs propagation including scalar and tensor perturbations at second-order, on which the starting of this project was based, were already shared with Lev Kofman during a visit at the Physics Department of Padova.
§ EVOLUTION EQUATION OF SCALAR-TENSOR INDUCED GRAVITATIONAL WAVES
We expand the three-space metric in Eq. (<ref>), keeping the tensor terms up to the linear order, as
h_ij =a^2 e^-2Ψ (δ_ij+γ_ij) ,
h^ij =a^-2 e^2Ψ (δ^ij- γ^ij) .
With these metric components, the extrinsic curvature reads <cit.>
K_ij=-1/2N ḣ_ij .
To get an evolution equation of the tensor perturbations sourced by a mixing of first order scalar and tensor ones, we focus on the trace-less part of the ij-th Einstein equation. It can be written as below <cit.>, defining the trace-less part of Eq. (<ref>) as K_ij= K_ij-1/3K h_ij,
∂K^i_k/∂ t =-N^|i_|k+1/3 N^|l_|l δ^i_k+N (KK^i_k +^(3)R^i_k-8π G S^i_k) ,
where ^(3)R^i_k is the Ricci tensor associated with the three-metric h_ik, and S^i_k is the space-space contribution of the matter energy-momentum tensor.
Vertical
bars denote three-space covariant derivatives with connection
coefficients determined from h_ij. Trace and trace-less parts of K_ij are, respectively,
K =-1/2N h^ijḣ_ij=-3/N (H-Ψ̇) ,
K^i_k =h^ijK_jk ,
= -1/2N γ̇^i_k .
We obtain the rest of the terms of the the trace-less ij-th equation Eq. (<ref>) from h_ij,
N^|i_|k =γ^ij N_|jk= Ne^2Ψ/a^2[Φ^,i_,k+Φ^,iΦ_,k-Φ_,mΨ^,mδ^i_k+Φ^,iΨ_,k+Ψ^,iΦ_,k+1/2Φ_,mγ^i,m_k-1/2Φ_,mγ^im_,k.
.-1/2Φ_,mγ^m,i_k+γ^mtΦ_,mΨ_,tδ^i_k-γ^ijΦ_,jk-γ^ijΦ_,jΦ_,k-Ψ_,kΦ_,jγ^ij-Φ_,kΨ_,jγ^ij] ,
N^|l_|l = Ne^2Ψ/a^2[∇^2Φ-Ψ^,lΦ_,l+Φ^,lΦ_,l-Φ_,mγ^lm_,l+γ^mtΦ_,mΨ_,t-Φ_,mlγ^lm-Φ_,mΦ_,lγ^lm] ,
[^(3)]R_jk =∇^2Ψ (δ_jk+γ_jk )-1/2∇^2 γ_jk-1/2γ_jk,mΨ^,m+Ψ_,jΨ_,k+Ψ_,jk+1/2γ^m_j,km+1/2γ^m_k,jm
-γ^mt_,mΨ_,tδ_jk-γ^mtΨ_,tmδ_jk-Ψ^,λΨ_,λδ_jk-Ψ^,λΨ_,λγ_jk-1/2Ψ_,λγ^λ_j,k-1/2Ψ_,λγ^λ_k,j+Ψ_,λΨ_,tγ^λ tδ_jk ,
[^(3)]R^i_k =γ^ij [^(3)]R_jk= e^2Ψ/a^2[∇^2Ψδ^i_k-1/2∇^2 γ^i_k-1/2γ^i_k,mΨ^,m+Ψ^,iΨ_,k+Ψ^,i_,k+1/2γ^mi_,km+1/2γ^m,i_k,m.
. -γ^mt_,mΨ_,tδ^i_k-γ^mtΨ_,tmδ^i_k-Ψ^,λΨ_,λδ^i_k-1/2Ψ_,λγ^iλ_,k-1/2Ψ_,λγ^λ,i_k.
.+γ^λ tΨ_,tΨ_,λδ^i_k-Ψ_,jkγ^ij-Ψ_,kΨ_,jγ^ij] ,
[^(3)]R = [^(3)]R^i_i=e^2Ψ/a^2[4∇^2Ψ-2Ψ^,iΨ_,i+γ^mi_,im -3 γ^m t_,mΨ_,t-3γ^mtΨ_,tm-Ψ_,mγ^im_,i.
.+3γ^m tΨ_,tΨ_,m-Ψ_,jiγ^ij-Ψ_,iΨ_,jγ^ij] ,
[^(3)]R^i_k = [^(3)]R^i_k-1/3[^(3)]R δ^i_k= e^2Ψ/a^2[-1/3∇^2Ψδ^i_k-1/2∇^2γ^i_k-1/2Ψ^,mγ^i_k,m+Ψ^,iΨ_,k+Ψ^,i_,k.
.+1/2γ^mi_,km+1/2γ^m,i_k,m+1/3γ^m tΨ_,tΨ_,mδ^i_k+1/3γ^m tΨ_,mtδ^i_k -1/3Ψ^,mΨ_,mδ^i_k
- 1/2γ^im_,kΨ_,m.
.- 1/2γ^m,i_kΨ_,m-γ^ijΨ_,jk-γ^ijΨ_,kΨ_,j-1/3γ^m t_,mtδ^i_k+1/3γ^mt_,tΨ_,mδ^i_k-1/3∇^2(Φ-Ψ)δ^i_k] .
Hence Eq. (<ref>) becomes
γ̈^i_k-Φ̇e^Φγ̇^i_k+3(H-Ψ̇)γ̇^i_k-e^2(Φ+Ψ)/a^2∇^2 γ^i_k= 2 e^2(Φ+Ψ)/a^2[Φ^,i_,k+Φ^,iΦ_,k-Ψ^,iΨ_,k-Ψ^,i_,k.
.+Φ^,iΨ_,k+Ψ^,iΦ_,k-2/3Φ^,lΨ_,lδ^i_k-1/3Φ^,lΦ_,lδ^i_k+1/3Ψ^,lΨ_,lδ^i_k]+𝒮 ,
where 𝒮 contains scalar-tensor mixed terms like ∼Φ(γ_ij), and contribution from the matter component of the Universe, S^i_k,
𝒮 =2e^2(Φ+Ψ)/a^2[-γ^ij(Φ-Ψ)_,jk+1/3γ^lm(Φ-Ψ)_,lm δ^i_k-1/2γ^im_,k(Φ-Ψ)_,m+1/2γ^i,m_k(Φ-Ψ)_,m.
.-1/2γ^m,i_k(Φ-Ψ)_,m+1/3γ^lm_,l(Φ-Ψ)_,mδ^i_k-γ^ijΨ_,kΦ_,j-γ^ijΦ_,kΨ_,j-γ^ijΦ_,kΦ_,j.
.1/3γ^lmΦ_,l(Φ+2Ψ)_,mδ^i_k+γ^ijΨ_,kΨ_,j-1/3γ^lmΨ_,lΨ_,mδ^i_k-1/2γ^m,i_k,m-1/2γ^im_,km+1/3γ^lm_,lmδ^i_k]
+16π G e^2ΦS^i_k .
These equations lead to Eq. (<ref>), after collecting scalar-scalar and scalar-tensor terms.
§ CALCULATION OF THE KERNEL FUNCTION
The dimension-less power-spectrum for each polarization λ of the correction to GWs, γ_1(k,η), i.e. the second term on the right hand side of Eq. (<ref>), is given by Eq. (<ref>), where we have used the definition of the two-point function of Φ and γ^(σ)_k
⟨Φ_k-k_1(0)Φ_k'-k'_1(0)⟩ = (2π)^3 δ^3(k-k_1+k'-k'_1) 2π^2/|k-k_1|^3Δ^2_Φ(|k-k_1|) ,
⟨γ^(σ)_k_1(0)γ^(σ')_k'_1(0))⟩ = (2π)^3 δ^3(k_1+k'_1) δ_σσ'2π^2/k_1^3Δ^(σ)2_γ_0(k) .
Contraction of the polarisation tensors give
Δ^2_γ_1, R/L(k) = k^3/π∫
d^3k_1Δ^2_Φ(|k-k_1|) /k_1^3
|k-k_1|^3[4 cos^8θ/2 Δ^(σ)2_γ_0, R/L(k_1)+4 sin^8θ/2 Δ^(σ)2_γ_0, L/R(k_1)]
×( ∫_0^η dη̃ G(η, η̃)[k_1^2 T_γ(k_1η̃ ) T_Φ(|k-k_1|η̃ )-T'_γ(k_1η̃ ) T'_Φ(|k-k_1|η̃ )])^2,
which leads to Eq. (<ref>). The Green's function is given by
G(x, x̃) =π/2k x̃ √(x̃/x) (J_1/2(x̃)Y_1/2(x)-J_1/2(x)Y_1/2(x̃)) .
Here the kernel has been written as, taking the upper limit of the time integral to be ∞,
ℐ= ∫_0^∞ k dx̃ G(x, x̃) [v^2 T_γ(vx̃) T_Φ(c_s ux̃)-Ṫ_γ(vx̃) Ṫ_Φ(c_s ux̃)] ,
= π/4v/c_su 1/x {-cosx(1-P^0_2(cosm))Θ(v+c_s u-1)Θ(1-|v-c_s u|)
-2/πsinx[(Q^0_0(coshn)-Q^0_2(coshn))Θ(1-v-c_s u).
.-(Q^0_0(cosm)-Q^0_2(cosm))Θ(v+c_s u-1)Θ(1-|v-c_s u|)]} ,
where 2uvc_scosm=v^2+c_s^2u^2-1 and 2uvc_scoshn=1-v^2-c_s^2u^2, and P^l_m, Q^l_m are the associated Legendre polynomials of the first and second kind.
It gives the oscillation average of the kernel squared (cosm=s=-coshn)
⟨ℐ^2⟩ =9/2^7x^2(v/c_su)^2[π^2(1-s^2)^2 Θ(1-|s|)+(2s+(1-s^2)log|1+s/1-s|)^2] .
§ USEFUL FORMULAE FOR THE CALCULATION OF THE KERNEL FUNCTION IN THE INDUCED GRAVITATIONAL WAVES INTEGRAL
We here write down some formulae used to calculate Eq. (<ref>)
with G(x, x̃) defined in Eq. (<ref>), and using
T_Φ(x) =2^3/2Γ (5/2)(x/√(3))^-3/2J_3/2(x/√(3)) ,
Ṫ_Φ(ux) = -3/xj_2 (ux/√(3))=-3^5/4/x√(x)√(π/2u)J_5/2 (ux/√(3)) ,
T_γ(x) =√(π/2x) J_1/2(x) ,
Ṫ_γ(vx) = -v j_1(vx)=-√(π v/2x) J_3/2 (vx) ,
the kernel turns out to be, for radiation domination (c_s=1/√(3))
ℐ = ∫_0^∞ dx̃(π/2)^2 3^5/4√(v/u)1/√(x x̃)(J_1/2(x̃)Y_1/2(x)-J_1/2(x)Y_1/2(x̃))
×[√(3)v/uJ_1/2(vx̃)J_3/2 (ux̃/√(3))-J_3/2(vx̃)J_5/2 (ux̃/√(3))] .
Using the recurrence relation 2n/z J_n(z)=J_n-1(z)+J_n+1(z), we have, putting n=3/2,
ℐ = ∫_0^∞ dx̃(π/2)^2 v√(v/u/√(3))√(x̃/x)(J_1/2(x̃)Y_1/2(x)-J_1/2(x)Y_1/2(x̃))
×[J_1/2(vx̃)J_1/2 (ux̃/√(3))-J_5/2(vx̃)J_5/2 (ux̃/√(3))],
= (π/2)^2 v√(v/u/√(3))1/√(x){Y_1/2(x)∫_0^∞ dx̃√(x̃)J_1/2(x̃) [J_1/2(vx̃)J_1/2 (ux̃/√(3))-J_5/2(vx̃)J_5/2 (ux̃/√(3))]
-J_1/2(x) ∫_0^∞ dx̃√(x̃)Y_1/2(x̃)
[J_1/2(vx̃)J_1/2 (ux̃/√(3))-J_5/2(vx̃)J_5/2 (ux̃/√(3))]} .
Using the formulae given in <cit.>, we have
∫_0^∞ dτ̃√(x̃) J_1/2(x̃) J_1/2(vx̃)J_1/2 (ux̃/√(3)) = √(√(3)/2π vu) |v-u/√(3)|<1<v+u/√(3)
0 |v-u/√(3)|>1
or v+u/√(3)<1
∫_0^∞ dx̃√(x̃) J_1/2(x̃) J_5/2(vx̃)J_5/2 (ux̃/√(3))= √(√(3)/2π vu) P^0_2(cosm)
for |v-u/√(3)|<1<v+u/√(3)
0
for 1< |v-u/√(3)| or 1>v+u/√(3)
∫_0^∞ dx̃√(x̃) Y_1/2(x̃) J_1/2(vx̃)J_1/2 (ux̃/√(3)) =
-1/π√(2√(3)/π vu) Q^0_0(cosm)
for |v-u/√(3)|<1<v+u/√(3)
1/π√(2√(3)/π vu) Q^0_0(coshn)
for 1>v+u/√(3)
∫_0^∞ dx̃√(x̃) Y_1/2(x̃) J_5/2(vx̃)J_5/2 (ux̃/√(3)) =
-1/π√(2√(3)/π vu) Q^0_2(cosm)
for |v-u/√(3)|<1<v+u/√(3)
1/π√(2√(3)/π vu) Q^0_2(coshn)
for 1>v+u/√(3)
where 2uv/√(3)cosm=v^2+u^2/3-1 and 2uv/√(3)coshn=1-v^2-u^2/3, and
P^0_2(cosm) =3cos^2m-1/2 ,
Q^0_0(cosm) =1/2ln1+cosm/1-cosm ,
Q^0_2(cosm) = 3cos^2m-1/4ln1+cosm/1-cosm-3cosm/2 .
Applying all these, we have Eq. (<ref>).
Remembering that ⟨cos^2x⟩=⟨sin^2x⟩=1/2, we have, for the oscillation average, Eq. (<ref>). The same procedure can be applied to the scalar induced GWs.
§ SCALAR-INDUCED TENSOR PERTURBATIONS
Since we have assumed A_γ_0 < A_Φ, the SIGWs are expected to have larger amplitudes compared to our modulated GWs. To assess the magnitude of this difference and evaluate the prospects of detecting our effect, we compare the amplitudes of SIGWs to those of our modulated GWs.
To get the SIGWs, we need the matter contribution in Eq. (<ref>). Previously it was ignored as we considered linear scalar-linear tensor and it contains no linear tensor if not some anisotropic stress. Now,
S^i_k =h^ijS_jk
= S^i_k-1/3(δ^i_k -γ^ijγ_jk)(δ^n_p -γ^mnγ_np)S^p_n ,
where
S^i_k = (ρ+P)u^i u_j+Pδ^i_j
=(ρ+P)V^2+3(P+δ P_1+δ P_2)-Pγ^mnγ_mn .
Hence,
S^i_k =(ρ+P)(v^iv_k-1/3v^2δ^i_k)
+P(γ^ijγ_jk+1/3γ^mnγ_mnδ^i_k) .
Here ρ,P are the background energy density and pressure respectively. The perturbation in pressure in first and second orders are given by δ P_1, δ P_2 respectively.
For scalar-scalar interaction, we need only the first term. As we are taking transverse trace-less component, there is no first order contribution. The evolution equation becomes
γ̈^i_k+3Hγ̇^i_k+k^2/a^2γ^i_k =4/a^2Φ^,iΦ_,k+16π G(ρ+P)V^iV_k ,
where
V^i=-2/8π G a^2(ρ+P)∂^i(Φ'+ℋΦ) .
In terms of conformal time and Fourier space, it is
γ^”_k+2ℋγ'_k+k^2γ_k =4∫d^3k_1/(2π)^3Φ_k(0)Φ_k-k_1(0) ϵ^k_i(k̂) k_1^i k_1k [T_Φ(k_1η)T_Φ(|k-k_1|η).
. +1/2(ℋT_Φ(k_1η)+T'_Φ(k_1η))(ℋT_Φ(|k-k_1|η)+T'_Φ(|k-k_1|η))] .
The sum of polarization states is
∑_λ=+,×(ϵ^ik_(λ)(k̂)k_1 ik_1 k)^2 =k_1^4 (1-(1+x^2-y^2/2x)^2)^2=k_1^4sin^4θ .
Defining the kernel as
I(k_1, |k-k_1|) = ∫_0^η dη̃ G(η, η̃) [T_Φ(k_1η)T_Φ(|k-k_1|η) .
.+1/2(ℋT_Φ(k_1η)+T'_Φ(k_1η))(ℋT_Φ(|k-k_1|η)+T'_Φ(|k-k_1|η))] ,
the total GWs power-spectrum for both polarization is (τ=kη)
Δ^2_γ_1(k)= 8k^3/π∫ d^3k_1Δ^2_Φ(k_1)Δ^2_Φ(|k-k_1|)/k^3|k-k_1|^3∑_λ=+,×(ϵ^ik_(λ)(k̂)k_1 ik_1 k)^2 ⟨ I^2 ⟩ ,
=8k^3/π∫ d^3k_1Δ^2_Φ(k_1)Δ^2_Φ(|k-k_1|)/k^3|k-k_1|^3k_1^4sin^4θ⟨ I^2 ⟩ ,
=16 k^2 ∫_0^∞ dv ∫_|v-1|^v+1 du v^2/u^2 (1-(1+v^2-u^2/2v)^2)^2Δ^2_Φ(uk)Δ^2_Φ(vk) ( ∫_0^τ dτ̃ G(τ, τ̃).
. ×[ T_Φ(vτ̃)T_Φ(uτ̃) +1/2(ℋT_Φ(vτ̃)+kṪ_Φ(vτ̃))(ℋT_Φ(uτ̃)+kṪ_Φ(uτ̃))])^2 .
The time integral is
I = ∫_0^τ dτ̃ 3π^2/8k√(uv)√(3τ̃/τ) (J_1/2(τ̃)Y_1/2(τ)-J_1/2(τ)Y_1/2(τ̃))
×[J_1/2(vτ̃/√(3))J_1/2 (uτ̃/√(3))+2J_5/2(vτ̃/√(3))J_5/2 (uτ̃/√(3))] ,
and its oscillation average reads (taking the upper limit of the integration as infinity)
⟨ I^2⟩ = 3^4/2^5k^2(1/uvτ)^2{Θ(u+v/√(3)-1)9π^2/4n^4
+(3n^2/2ln1+n/1-n-3n)^2} .
Eq. (<ref>) becomes
Δ^2_γ_1(k)=3^4/2(1/τ)^2 ∫_0^∞ dv ∫_|v-1|^v+1 dy 1/u^4 (1-(1+v^2-u^2/2v)^2)^2
×Δ^2_Φ(uk)Δ^2_Φ(vk){Θ(u+v/√(3)-1)9π^2/4n^4
+(3n^2/2ln1+n/1-n-3n)^2} .
For a Dirac delta input scalar spectrum, we have
⟨Δ^2_γ_1(k)⟩= 16 A^2_Φ (k_*/k)^2 [1-k^2/4k_*^2]^2⟨ I^2⟩_u=v=k_*/k Θ(2k_*-k) .
§ EXPLICIT FORMULAE FOR THE GENERAL KERNEL
Here we present the explicit formulas for the general kernel in <ref>. These are given by:
G_0[ Si [x]]≡ Si[(1 + c_s u - v) x] +
Si[(1 - c_s u + v) x] - Si[(1 + c_s u + v) x]- Si[(1 - c_s u - v) x] ,
G_1[cos x] ≡cos[(1 + c_s u - v) x]/1+c_su-v+cos[(1 - c_s u + v) x]/1-c_su+v -cos[(1 + c_s u + v) x]/1+c_su+v-cos[(1 - c_s u - v) x]/1-c_su-v ,
G_2[sin x] ≡(1+(1+c_su+v)^2/4c_suv)sin[(1 - c_s u - v) x] + (1-(1-c_su+v)^2/4c_suv)sin[(1 + c_s u - v) x]
+(1-(1+c_su-v)^2/4c_suv)
sin[(1 - c_s u + v) x] +(1+(1-c_su-v)^2/4c_suv) sin[(1 + c_s u + v) x] ,
G_3[cos x]≡ -(1 + 3 c_s u +3 v) cos[(1 - c_s u - v) x] + (1 - 3 c_s u +
3 v) cos[(1 + c_s u - v) x]
+ (1 + 3 c_s u -
3 v) cos[(1 - c_s u + v) x] - (1 - 3 c_s u -
3 v) cos[(1 + c_s u + v) x] ,
G_4[sin x]≡sin[(1 + c_s u - v) x] +
sin[(1 - c_s u + v) x] - sin[(1 + c_s u + v) x]-sin[(1 - c_s u - v) x] ,
F_1[sin x] ≡sin[(1 + c_s u - v) x]/1+c_su-v+sin[(1 - c_s u + v) x]/1-c_su+v -sin[(1 + c_s u + v) x]/1+c_su+v-sin[(1 - c_s u - v) x]/1-c_su-v ,
F_2[cos x] ≡(1+(1+c_su+v)^2/4c_suv)cos[(1 - c_s u - v) x] + (1-(1-c_su+v)^2/4c_suv)cos[(1 + c_s u - v) x]
+(1-(1+c_su-v)^2/4c_suv)
cos[(1 - c_s u + v) x] +(1+(1-c_su-v)^2/4c_suv) cos[(1 + c_s u + v) x] ,
F_3[sin x]≡ (1 + 3 c_s u +3 v) sin[(1 - c_s u - v) x] - (1 - 3 c_s u +
3 v) sin[(1 + c_s u - v) x]
- (1 + 3 c_s u -
3 v) sin[(1 - c_s u + v) x] + (1 - 3 c_s u -
3 v) sin[(1 + c_s u + v) x] ,
F_4[cos x]≡cos[(1 + c_s u - v) x] +
cos[(1 - c_s u + v) x] - cos[(1 + c_s u + v) x]-cos[(1 - c_s u - v) x] .
|
http://arxiv.org/abs/2307.03973v1 | 20230708131320 | Autonomy 2.0: The Quest for Economies of Scale | [
"Shuang Wu",
"Bo Yu",
"Shaoshan Liu",
"Yuhao Zhu"
] | cs.RO | [
"cs.RO",
"cs.AI",
"cs.CY"
] |
takeaways[2]
[title=#1, size=fbox,after skip=0.5, colbacktitle=yellow!25,coltitle=black]
#2
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printacmref=false
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Autonomy 2.0: The Quest for Economies of Scale
Shuang Wu, Bo Yu, Shaoshan Liu, Yuhao Zhu
August 12, 2023
==============================================
§ INTRODUCTION
With the advancement of robotics and AI technologies in the past decade, we have now entered the age of autonomous machines. In this new age of information technology, autonomous machines, such as service robots, autonomous drones, delivery robots, and autonomous vehicles, rather than humans, will provide services <cit.>. The rise of autonomous machines promises to completely transform our economy. However, after more than a decade of intense R&D investments, autonomy has yet to deliver its promise <cit.>.
In this article, through examining the technical challenges and economic impact of the digital economy, we argue that scalability is both highly necessary from a technical perspective and significantly advantageous from an economic perspective, thus is the key for the autonomy industry to achieve its full potential. Nonetheless, the current development paradigm, dubbed Autonomy 1.0, scales with the number of engineers, instead of with the amount of data or compute resources, hence preventing the autonomy industry to fully benefit from the economies of scale, especially the exponentially cheapening compute cost and the explosion of available data. We further analyze the key scalability blockers and explain how a new development paradigm, dubbed Autonomy 2.0, can address these problems to greatly boost the autonomy industry.
§ SCALABILITY OF THE DIGITAL ECONOMY
The digital economy refers to the use of information technology to create, market, distribute, and consume goods and services. It has been the key driving force for the world's economic growth in the past two decades.
Consider the internet industry, for instance.
The internet industry has accounted for 21% of the GDP growth in mature economies from 2005 to 2010 <cit.>.
In 2019, the internet industry contributed $2.1 trillion to the U.S. economy, about 10% of the U.S. GDP, and is the fourth largest industry of the U.S. economy (behind only real estate, government, and manufacturing) <cit.>.
Along with its contribution to economy, the internet industry provides nearly 6 million direct jobs, accounting for 4% of U.S. employments.
Two key forces fuel the continuous growth of the digital economy, both of which have to do with scalability:
* The commoditization of computing power, as exemplified by Moore's law <cit.>, is the greatest driving force behind the digital industry.
The most successful digital economy companies have developed core technology stacks that are scale by the available compute resources and data, not by the size of their engineering teams. One remarkable example is WhatsApp: when acquired by Facebook for $19 billion, WhatsApp had only 32 engineers serving over 450 million users.
* The breakthrough of artificial intelligence in the last decade has demonstrated that, in addition to many technical improvements and tuning, scaling neural network models and training datasets has been our most effective strategy for achieving continuous performance gains <cit.>.
Autonomy technologies such as those found in autonomous driving are widely seen as the pillar of the next digital economy era.
However, today's autonomous machines technologies, dubbed Autonomy 1.0, represent everything a scalable industry should not do.
To illustrate the problem facing autonomous driving companies, Figure <ref> analyzes the R&D expenditures and revenue per employee of two leading public digital economy companies, Microsoft representing the software industry and Alphabet representing the internet industry, and two public autonomous driving companies, TuSimple representing the robot truck industry and Aurora representing the robotaxi industry. We selected these autonomous driving companies for the accessibility of their financial data.
Both Alphabet and Microsoft spend less than 20% of their total operating expenditures on R&D. For instance, Google employs less than 30,000 engineers while serving over 4.3 billions of users. Their scalability is mainly constrained by available compute resources and data instead of by the number of engineers.
In comparison, both TuSimple and Aurora spend more than 70% of their operating expenditures on R&D. Often, to reach new users or to deploy services to new locations, autonomous driving companies need to pour additional R&D resources to re-calibrate their existing technology stacks to adapt to new environments.
Hence, their scalability is constrained by R&D investment or, more directly, the number of engineers.
As a result, Alphabet and Microsoft are able to generate $1.5 million and $0.8 million of revenue per employee respectively while maintaining a high growth rate, whereas TuSimple and Aurora generate negligible revenue per employee and struggle with growth. For the autonomy industry to achieve economies of scale, we have to revolutionize the R&D paradigm.
In following sections, we will describe key scalability issues with Autonomy 1.0, and outline promising solutions that are already at the horizon to achieve scalability in Autonomy 2.0.
§ AUTONOMY 1.0: THE END OF THE ROAD OF AN AGING ARCHITECTURE
Current commercial autonomous driving systems mostly inherited the software architecture from competitors in the DARPA Grand Challenges between 2005 and 2007 <cit.>. This software architecture, while represented a great leap of autonomy technology at the time, has showed its age and become difficult to scale after more than a decade of intense industry efforts to improve and adapt.
Figure <ref> illustrates Autonomy 1.0's scalability problems using autonomous driving operation data from California from 2018 to 2022. Over the past five years, although enormous amount of investment has been poured into autonomous driving, we did not observe significant growth of the number of vehicles under operation, which increased only from 400 in 2018 to 1,500 in 2022. The operation mileage per year increased only from 2 million miles to 5 million miles. Most importantly, there are still over 2,000 disengagement incidents per year. Given this trend in Autonomy 1.0, we are still years away from serious commercial operations of autonomous vehicles.
Autonomy 1.0 is modular and consists of functional modules such as sensing, perception, localization, high-definition maps, prediction, planning and control <cit.>, each further consists of several functional sub-modules integrated by explicit and hand-crafted logic. Most decision-making tasks, such as planning, which is responsible for generating optimal and drivable paths, are solved with constraint optimization under a set of hand-tuned rules. When a disengagement incident happens, engineers usually have to go through a long process of debugging to identify which specific module or rule may have been the root cause of the disengagement, then optimize that module or develop logic changes to handle the specific problem. Often, due to intricate dependency and coupling among modules or rules, the new software version leads to other problems that need to be addressed, thus greatly slowing down development process. The Autonomy 1.0 software stack over time became a complicated collection of ad-hoc rules and a set of interdependent modules for handling various long-tail events, which has been increasingly difficult to debug, maintain and evolve for improved performance.
Taking the open-source project Apollo <cit.> as an example, its perception module alone consists of multiple individual leaning-based sub-modules to accomplish object detection in 2D images, LiDAR point cloud segmentation, traffic light detection, lane detection, and others. To integrate information from these perception sub-modules, a post-processing module then fuses 2D and 3D information and outputs an integrated representation of the environment to the downstream prediction module. The planning module makes decisions and plans routes based on the data from the prediction, localization, and map modules. These modules often have strong dependencies among themselves. Making changes to one module not only impacts the overall system performance, possibly violating real-time constraints and resource allocation, but also impacts the algorithmic performance of other downstream modules due distributional shift of data.
The whole system has become complicated and even brittle, demanding enormous amount of engineering resources to maintain, let alone to scale.
We summarize the three Autonomy 1.0's major scalability bottlenecks below.
* Complexity Bottleneck: The design of autonomy 1.0 systems demands extensive engineering efforts to define software interfaces, distribute data among modules, and map various workloads in a heterogeneous computing system. It is challenging, given the complexity, to debug and continuously update the software stack.
The myriad of components also make it challenging to schedule tasks and optimize the latency of the unwieldy stack at run-time.
As a result, typical autonomy 1.0 systems exhibit large latency variations <cit.>, which can harm the reliability of the autonomous driving system.
* Human-Data Bottleneck: Autonomy 1.0 systems depend on fleets of physical vehicles operated by humans to collect data and perform system-level tests. This is a time-consuming and expensive process that is difficult to scale out.
The scalability issue will only get worse as increasingly more modules of autonomy stack adopt data-driven approaches, which requires continuous collection and labeling, because any specific instance of the recorded data reflects only a particular subset of the world states.
* Generalization Bottleneck: Autonomy 1.0 systems consist rule-based processing logic and hand-crafted interfaces, which makes them difficult to generalize to new environments. This is because the complexity and diversity of real-world environments makes it difficult to design the autonomy system to anticipate all possible challenging scenarios, whether for perception or planning. As a result, autonomy 1.0 systems are often over-fitted to frequently operated regions and common situations. To handle new environments and newly encountered rare cases, additional changes to the system are required, which is increasingly difficult and time-consuming.
§ AUTONOMY 2.0: SCALABILITY IS EVERYTHING
Recent research breakthroughs in artificial intelligence, such as Transformer <cit.>, large language models (LLM) <cit.> and offline reinforcement learning <cit.>, have sparked new ideas in architecture design, data and model infrastructure, and engineering practices of autonomous driving, leading to a new development paradigm, which we dub Autonomy 2.0.
The key of Autonomy 2.0 is scalability, which is delivered through two ingredients: 1) a software stack that improves continuously with increasing scale of data and compute resources. 2) a simulation paradigm based on digital twins for algorithmic exploration using large-scale, real-time, realistic data before deployment. Figure <ref> illustrates the differences between Autonomy 1.0 and Autonomy 2.0 system architectures. Table <ref> summarizes how Autonomy 2.0 addresses the three bottlenecks in Autonomy 1.0.
§.§ Learning-Native Software Stack
Any autonomous machine performs two main tasks: perception and action, reflecting the natural dichotomy of the past and the future. The perception task observes the environment and infers its current state based on observations so far. The action task, based on these observations, chooses an appropriate sequence of actions to achieve goals while considering how the environment may evolve in the near future.
The software stack in Autonomy 2.0, thus, naturally consists of a perception module and an action module. Unlike in Autonomy 1.0 where each module is implemented by a number of sub-modules, there is a strong evidence that the two modules, in Autonomy 2.0, will each be implemented as a single large deep learning model, likely based on transformer or its variants due to their ability to generalize, as demonstrated in their recent successes in LLMs.
Benefits.
Before describing how the two-model architecture will look like in Autonomy 2.0, we will first discuss why such an architectural design choice is key to scalability.
The two-model architecture addresses the Complexity Bottleneck by drastically reducing the amount of code that needs to be maintained and reasoned about.
Figure <ref>a) compares the lines of code in the Apollo Perception module <cit.>, which represents the Autonomy 1.0 approach, with an example of the perception module in Autonomy 2.0, BEVFormer <cit.>. The Apollo Perception module's size is ten times larger than BEVFormer, and BEVFormer has achieved state of the art perception results.
The software architecture also handles corner cases through data-driven model learning instead of hand-crafted logic, and thus address the Generalization Bottleneck in Autonomy 1.0.
In Figure <ref>b), we analyze over 400 issues associated with the Apollo planning modules, 47% of the issues are related to Apollo failing to handle a specific usage case, and 30% of the issues are related to software engineering problems such as interfaces with other modules. In Autonomy 1.0, many hand crafted rules are implemented to handle specific use cases. As the rules accumulate, software quality naturally becomes an issue.
Architectural Design.
The perception and action modules have different goals and traditionally require distinctive algorithmic approaches.
The perception module is trained using supervised learning and self-supervised learning to infer one unique ground truth of world states.
In contrast, the action module needs to search and choose from many acceptable action sequences, while anticipating the behaviors of other agents.
Therefore, the action module makes use of methods from reinforcement learning, imitation learning, and model predictive control.
Interestingly, while the fundamental distinctions of the two modules have not changed in Autonomy 2.0, there is a growing convergence of the implementation of the two modules: recent successes of large language models (LLM) <cit.> to comprehend a large amount of information to perform multiple sub-tasks suggest that both modules can be implemented using a similar architecture based on Transformer <cit.>.
Transformer is a great algorithmic substrate for both the perception and action modules because of its ability to generalize.
For perception, a transformer can effectively fuse perceptual data from multiple sensors and multiple moments into a unified representation, avoiding information loss from sparsification and module serialization. For action, the sequential nature of transformer makes it a perfect fit for processing and generating temporal data, especially for sampling multiple possible future paths.
Perception.
In Autonomy 1.0, the perception module consists of multiple DNNs, each trained separately to support individual tasks such as 2D/3D object detection, segmentation, and tracking.
In contrast, the perception module in Autonomy 2.0 uses a single transformer backbone to provide a unified representation of the ego-vehicle's environment (e.g., 2D Bird's Eye View (BEV) <cit.> or 3D occupancy <cit.>), which is then attached to a number of decoder “heads”, each of which is tuned for an individual task.
This single-transformer approach toward the perception module has been gaining popularity across the AV industry.
For instance, this is the approach described by Tesla engineers in their “AI Day 2022” event <cit.>, and has been deployed by another leading intelligent electric vehicle company XPENG <cit.>.
Action. The action module anticipates a combinatorially large number of possible “world trajectories”, hypothesizes multiple action sequences, and evaluates them to send the optimal one to actuators.
In Autonomy 1.0, the action module is implemented as a set of sub-modules for prediction, planning, and control.
The action module in Autonomy 2.0 is end-to-end learned using transformer-inspired architectures for sequential decision making <cit.>.
The action transformer incorporates two models: a policy model and a world model.
First, the pre-trained, transformer-based policy model leverages the large amount of historical data for agent behavior prediction and ego vehicle decision making and trajectory planning <cit.>.
Second, the world model is essentially a behaviorally realistic simulator (validated against real-world data) of the world.
The two models are connected with a closed-loop in the transformer so that the policies can be fine-tuned online <cit.>.
§.§ Digital-Twin Based Development and Deployment
Autonomy 1.0 relies almost exclusively on human efforts for tasks such as manual data labeling and physical testing, posing a scalability bottleneck.
Autonomy 2.0 addresses the “Human-Data Bottleneck” using an emerging simulation technology called digital twins, where a virtual representation acts as the counterpart of the physical world.
As highlighted by the recent National Artificial Intelligence R&D Strategic Plan 2023 published by the White House <cit.>, digital twins have fueled many real-world applications (e.g., urban planning/management of smart cities and additive manufacturing), and is a main strategy to sustain AI technologies.
Under the digital-twins paradigm, one instruments the physical system to collect real-world, real-time data, which is then interactively shared with the digital counterpart.
In the digital world, one could further synthesize scenarios (e.g., traffics) with a statistically significant fidelity with a similar behavioral distributions as that in human driving behaviors.
Developing and testing autonomous driving software using synthesized virtual scenarios accelerates the evaluation process by 10^3 to 10^5 times <cit.> and reduces the testing costs by two orders of magnitude <cit.> compared to the physical-only approach in Autonomy 1.0.
Figure <ref>c) demonstrates the R&D cost efficiency in Autonomy 1.0, which costs $180/hr through physical testing, vs. in Autonomy 2.0, which costs $2/hr through virtual testing, an 100-fold improvement <cit.>. Figure <ref>d) demonstrates the R&D efficiency in Autonomy 1.0, which takes around 3 kilo miles per physical vehicle per year through physical testing<cit.>, vs. in Autonomy 2.0, which takes over 3 million miles per virtual vehicle per year through simulation, a 1000-fold improvement <cit.>. Combining these two factors would bring over 10^5 times improvement under the same engineering investment in Autonomy 2.0, and scalability is thus only constrained by the available compute resources instead of number of engineers, effectively eliminating the human-data bottleneck.
§ SUMMARY
The autonomy economy, or the use of autonomous machines to provide goods and services, will fuel the world's economic growth in the coming decades.
Huge investments are pouring into the autonomy economy. Such a huge investment will only be justified if autonomous machines can reach, and provide utility for, every person on planet.
Similar to today's digital economy, scalability will necessarily be the winning formula in this process.
The current practice of developing and deploying autonomous machines carries the historical baggage of complexity bottleneck, human-data bottleneck, and generalization bottleneck, and is thus unscalable.
We must start from a clean slate and rethink the architecture design of autonomous machines.
We posit that Autonomoy 2.0 will embrace a learning-native software stack, which addresses the complexity bottleneck through software simplicity and addresses the generalization bottleneck through end-to-end learning.
The digital twins technologies will have to be integrated throughout the development, evaluation, and deploymemt cycle in Autonomy 2.0 to address the human-data bottleneck.
ieeetr
|
http://arxiv.org/abs/2307.04366v1 | 20230710064648 | A New Wind Farm Active Power Control Strategy to Boost Tracking Margins in High-demand Scenarios | [
"Simone Tamaro",
"Carlo L. Bottasso"
] | physics.flu-dyn | [
"physics.flu-dyn",
"cs.CE",
"cs.SY",
"eess.SY"
] |
Explanation Needs in App Reviews: Taxonomy and Automated Detection
Max Unterbusch
University of Cologne
[email protected]
Mersedeh Sadeghi
University of Cologne
[email protected]
Jannik Fischbach
Netlight Consulting GmbH | fortiss GmbH
[email protected]
Martin Obaidi
Leibniz University Hannover, Software Engineering Group
[email protected]
Andreas Vogelsang
University of Cologne
[email protected]
August 12, 2023
============================================================================================================================================================================================================================================================================================================================================================================================================================
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empty
This paper presents a new active power control algorithm designed to maximize the power reserve of the individual turbines in a farm, in order to improve the tracking accuracy of a power reference signal. The control architecture is based on an open-loop optimal set-point scheduler combined with a feedback corrector, which actively regulate power by both wake steering and induction control. The methodology is compared with a state-of-the-art PI-based controller by means of high-fidelity LES simulations. The new wind farm controller reduces the occurrence of local saturation events, thereby improving the overall tracking accuracy, and limits fatigue loading in conditions of relatively high-power demand.
§ INTRODUCTION
The growth of wind energy penetration in the electricity mix requires new control algorithms to keep the electrical grid in balance <cit.>. When operating in active power control (APC) mode, a wind farm intentionally extracts less than the available power from the wind, in order to meet the demands of the transmission system operator (TSO). The application of APC to a wind farm is not trivial and introduces new challenges. In fact, the maximum available power dependents on ambient conditions, which vary dynamically in uncertain ways <cit.>. Additionally, wind may suddenly drop, possibly leaving not enough power reserves to track a given reference signal <cit.>. In a wind farm, the situation is further complicated by the presence of low-momentum turbulent wakes, which are responsible for power losses and fatigue loading of waked turbines <cit.>. Various solutions have been proposed to mitigate wake effects, such as induction and yaw control <cit.>. The latter consists of “steering” the wake away from downstream rotors, and its effectiveness for power boosting has been demonstrated numerically <cit.>, experimentally in the wind tunnel <cit.>, as well as in field trials <cit.>.
Different APC approaches have been presented in the literature. An open-loop APC strategy is discussed in <cit.>. The authors showed that the lack of feedback poses a limitation on the power tracking accuracy of the method, especially in conditions of strong waking. Furthermore, an equal dispatch of power sharing among the turbines proved to be suboptimal, due to the different local power reserves induced by the heterogeneity of the flow.
Recently, various authors have used model predictive control (MPC) for APC <cit.>. The main drawback of such methods lies with the need of a dynamic farm flow model, which can be computationally expensive.
Simpler control structures based on classical PI (proportional integral) loops have also been extensively investigated <cit.>. While lacking the sophistication of MPC, such methods do not need a wind farm flow model and can provide fast response times with simple implementations. The APC PI controller of ref. <cit.> operates on the tracking error and adjusts the power demands to follow a reference, sharing power in an arbitrary, static manner among the turbines. The method includes gain scheduling based on the fraction of saturated wind turbines, defined as the ones whose available power is smaller than the demanded one. This method was improved in ref. <cit.> by dynamically adjusting the set-points of the wind turbines, with the goal of equalizing their loading. The authors tested this methodology with an actuator disk model using large eddy simulations (LES). Later, this approach was also demonstrated with the more sophisticated actuator line method (ALM) in LES <cit.>. So far these PI-based methods have been applied only to induction control, and they are not necessarily optimal. Moreover, saturation conditions are problematic, due to the possible local lack of power reserves (margins), which are not explicitly accounted for nor monitored in the existing implementations.
In this paper, a new wind farm control architecture is presented to improve the power tracking accuracy in conditions of strong persistent wakes, when the wind farm power demand is close to the maximum available power. An improved tracking performance is obtained by explicitly maximizing the power margin, in order to hedge against wind lulls. This novel methodology combines wake steering with induction control. Wake steering is used because of its ability to increase power margins by mitigating wake effects <cit.>. Wake steering is implemented through an open-loop model-based set-point optimal scheduler, closely following the standard implementation that has recently become popular in power-boosting wind farm control <cit.>. Induction control is implemented through a fast closed-loop corrector to improve tracking accuracy. The new methodology is demonstrated in a partial wake impingement scenario of a cluster of turbines, using a TUM-modified version of NREL's ALM-LES Simulator fOr Wind Farm Applications (SOWFA) <cit.>.
The paper is structured as follows. First, the novel APC methodology is presented. Second, the simulation model is described and finally, results are discussed for steady-state and unsteady conditions.
§ METHODOLOGY
The core of the proposed wind farm control architecture is an open-loop model-based set-point optimal scheduler. This control element determines the yaw misalignment of each turbine and its contribution to the demanded value (i.e. power share), given the power demand required by the TSO and the ambient conditions. The latter can be obtained in real time from SCADA data or with wind sensing methods <cit.>. A feedback loop serves the main purpose of correcting tracking errors, which will inevitably arise from the open loop control element during operation. A sketch of the overall control architecture is shown in fig. <ref>. The closed and the open loops are executed at two distinct time rates, since their outputs involve physical phenomena characterized by different time scales. Specifically, the open loop updates the yaw-set points and the power shares at a slower rate, due to the time required by the wake to propagate downstream. On the other hand, the closed loop changes the turbine inductions at a faster pace, to reduce tracking errors.
§.§ Open-loop set-point optimal scheduler
The open-loop component of the algorithm provides the optimal set-points in terms of yaw misalignment and power share. These are computed by a gradient-based optimization that maximizes the smallest power reserve within the wind turbines of the farm, for a given overall power demand.
The power of the ith turbine is noted P_i = P_i (A_i,u_i), where A_i indicates the local ambient conditions (here assumed to include wind speed, wind direction and turbulence intensity), and u_i are the control inputs (namely, induction and yaw misalignment). Power is computed using a wind farm flow model, which here is based on the FLOw Redirection and Induction in Steady-state (FLORIS v2) tool <cit.>.
The maximum power that can be captured by turbine i by adjusting its control set-point u_i (while keeping the set-points of the other turbines fixed) is computed as
P_a,i = arg max_u_i P_i (A,u_i) = 1/2ρπR^2 C_p U^3 cos^P_p(γ),
where ρ is the air density, R is the wind turbine radius, U is the undisturbed free-stream velocity, and P_p is the cosine exponent relating the yaw misalignment angle γ to power. The algorithm looks for the combination of set-points that produce the maximum possible minimum power ratio P_i/P_a,i across all turbines in the farm, while satisfying the power demand of the TSO. This can be expressed as
min_u max_i ∈ [1,N]P_i/P_a,i
such that ∑_i=1^N P_i=P_ref.
In fact, the smaller the power ratio P_i/P_a,i, the larger the margin m_i = 1-P_i/P_a,i that is available to compensate against drops in the wind. Equation (<ref>) represents a constrained optimization problem, which is solved with the gradient-based Sequential Quadratic Programming (SQP) method <cit.>. The optimization does not need to be performed in real time during operation. Rather, it is executed offline for a set of ambient conditions and relative wind farm capacities. Results are collected in a look-up table, which is then interpolated at run-time, similarly to what is routinely done for power-boosting wind farm control <cit.>.
In the example shown later in this work, the open loop is executed every 30 seconds.
§.§ Closed-loop corrector
The closed-loop corrector is directly taken from the work of ref. <cit.>, and it is executed every 0.01 seconds. The corrector consists of a simple PI feedback loop that operates on the power tracking error, which arises from the open-loop component of the control structure. The tuned PI gains used in this work are K_P,APC=0.2 and K_I,APC=0.05^-1.
§.§ Identification of saturation conditions
On each turbine, the occurrence of saturations is determined by a condition that combines tracking error and pitch angle. In particular, a saturation is detected when the blade pitch is at its optimal value and the tracking error exceeds a given negative threshold, set to the value of 100 in this work. The magnitude of this threshold determines the aggressiveness of the wind farm controller. This method was chosen because it can be implemented based on standard information that is readily available on board wind turbines, and does not rely on uncertain and difficult-to-estimate parameters such as thrust coefficient or axial induction.
§ NUMERICAL MODEL
§.§ Steady-state model
The engineering farm flow model FLORIS v2 <cit.> is used here both to synthesize the open-loop part of the controller and to perform steady-state analyses, prior to testing in the dynamic higher-fidelity LES-ALM environment. The standard FLORIS implementation is extended with the option to derate the turbines by modifying the C_p and C_t tables, following a basic curtailment approach. Moreover, a linear dependency of the power loss exponent P_p with C_t is also included in the model <cit.>, so that
P_p=A C_t+B,
where A=-1.56 and B=3.16, based on experimental and numerical observations. This dependency between the power loss exponent and the thrust coefficient is particularly relevant when combining derating and yaw misalignment, since the wind turbines operate at a wide range of C_t values due to their dynamic curtailment.
§.§ Unsteady simulations
LES-ALM simulations are used for testing the performance of the new APC formulation, because they are able to deal with the complex dynamics typical of wind turbine wakes and their interactions <cit.>.
The filtered ALM of refs. <cit.> is used to model the blades, by projecting forces computed along the lifting lines onto the LES mesh grid. Simulations are run with a turbulent wind obtained from a precursor generated in stable atmospheric conditions. The Cartesian mesh consists of approximately 13.5 million cells, and uses six refinement levels. The smallest cells measure 1, and are located in correspondence of the rotors. The computational domain, grids and turbine layout are shown in fig. <ref>.
§ RESULTS AND ANALYSIS
The scenario analyzed in this paper consists of a cluster of three IEA 3.4 wind turbines <cit.>, installed at a distance of 4 diameters and misaligned by half a diameter relatively to the incoming wind vector. The scenario is adapted from <cit.>, and it is chosen to mimic the typical operating conditions of an onshore wind plant with close spacings and partial wake overlaps. The inflow is characterised by a turbulence intensity of 6% at hub height, a shear of 0.2, and a mean wind speed of 9.5, equal to the rated speed of the turbines.
§.§ Steady-state conditions
First, the open-loop optimal scheduler is demonstrated in steady-state conditions. For each turbine, fig. <ref> reports the yaw set-points and power share percentage that maximize the smallest power margin.
The figure shows that the most upstream turbines are misaligned relatively to the wind, with the goal of increasing the power reserves of the downstream ones. Moreover, power share is not distributed equally, because of different local inflow conditions and wake effects.
These margin-optimal set-points (noted induction + yaw in the following) are compared to the ones of two alternative strategies in fig. <ref>. In the first of these strategies (noted induction), only induction is used to match the demand (i.e. the turbines are always aligned with the incoming wind vector). In the second (noted first yaw then induction), the turbines are first misaligned to maximize power capture, and then induction control is used to match the demand. In both cases, the power share is computed in order to maximize the smallest power margin in the wind farm.
The figure shows that —as expected— the margin drops to zero in correspondence of the maximum power of the plant, and increases as the power demand is lowered and the wind turbines are derated. Compared to the induction case, the methods featuring wake steering are able to significantly increase the power margin for a wide range of wind farm power demands. Furthermore, the first yaw then induction strategy generates similar margins to the induction + yaw case at relatively high TSO demands. However, its performance drops slightly as the power demand is lowered, because of the power losses caused by its larger persistent yaw misalignments. These losses are particularly enhanced by the low thrust coefficient at which the turbines operate, due to curtailment <cit.>. Because of its better ability to generate large margins, only the induction + yaw strategy is considered in the remainder of this work.
§.§ Unsteady simulations
Next, the methodology is tested with unsteady CFD simulations. Results are compared with the controller developed in ref. <cit.>, which is assumed here as the state-of-the-art benchmark.
A dynamic reference power signal typical of automatic generation control (AGC) is used as input signal. AGC is the secondary response regime of grid frequency control, and it consists in the modification of the power output of a plant depending on the dynamically changing requests by the transmission system operator <cit.>. A similar signal has been considered by other authors <cit.>.
Fig. <ref> presents the average velocity fields in the wind farm obtained with the benchmark control and with the proposed induction+yaw approach. The effect of yaw misalignment can be clearly observed, as the wakes of the upstream turbines appear to have been deflected in Fig. <ref>.
Fig. <ref> shows a comparison of the power tracking error obtained with the benchmark method and the newly proposed one.
The figure shows that the benchmark method presents frequent negative deviations from the reference signal. These deviations are due to the power saturation of the wind turbines operating in waked inflow conditions. On the other hand, the controller featuring wake steering is capable of reducing the frequency of occurrence of these phenomena, thereby improving the overall tracking accuracy. For the results of fig. <ref>, the new wind farm controller reduces the root-mean-square of the tracking error by 42.6% relatively to the benchmark. In the latter, the significant error occurring at t≈760 is due to a simultaneous saturation of all the wind turbines in the cluster.
In order to better understand how the local power margin is increased by the new method, the pitch angles commanded by the wind turbine controllers are plot in fig. <ref>.
For a standard curtailment derating strategy, larger power reserves are obtained for larger absolute differences between the commanded pitch angle and the optimal value. Figures <ref> and <ref> show that waked turbines display the highest margin increase compared to the benchmark case, due to the lowered impact of the impinging wakes. On the other hand, the most upstream wind turbine (see fig. <ref>) generally displays a lower margin with the new control strategy, because of its yaw misalignment. Nevertheless, for the benchmark controller, the frequent saturation of the downstream turbines number 2 and 3 forces the upstream turbine number 1 to compensate, and in these conditions its margin drops relatively to the new proposed formulation.
Finally, the effect of the new methodology on loads is briefly considered. Fig. <ref> shows the damage equivalent loads (DEL), computed by rainflow counting (<cit.>), for the tower base fore-aft bending moment of each turbine.
Results indicate that the new control strategy reduces fatigue compared to the benchmark one. These results can be explained by the fact that the benchmark controller is unable to maintain load balancing within the farm in high-power-demand conditions, due to the frequent saturation events. Conversely, the new controller reduces the extent of the saturation phenomena, thereby suppressing the abrupt controller actions that are responsible for high-amplitude fatigue cycles.
§ CONCLUSIONS
A new wind farm control methodology for power tracking was presented. The methodology combines wake steering and induction control with the aim of maximizing the lowest power margin within a wind farm. The implementation is based on a slow-rate open-loop optimal set-point scheduler, combined with a fast feedback loop corrector. Compared to a state-of-the-art benchmark, the new methodology is capable of reducing the root-mean-square of the tracking error in conditions of power demand close to the maximum capacity of the plant. In such conditions, the fatigue of the individual wind turbines is also mitigated, because of less frequent saturation phenomena.
-2.5cm
§ ACKNOWLEDGMENT
The authors acknowledge the support of the German Federal Ministry for Economic Affairs and Climate Action (BMWK) through the PowerTracker project. The authors express their appreciation to the Leibniz Supercomputing Centre (LRZ) for providing access and computing time on the SuperMUC Petascale System under Projekt-ID pr84be “Large-eddy Simulation for Wind Farm Control”.
IEEEtran
|
http://arxiv.org/abs/2307.04127v1 | 20230709085036 | Self-healing unitarity is an Optical illusion: Comment on "Self-healing of unitarity in effective field theories and the onset of new physics" | [
"Archit Vidyarthi"
] | hep-ph | [
"hep-ph",
"hep-th"
] |
Department of Physics, Indian Institute of Science Education and Research Bhopal, Madhya Pradesh - 462066, India
Among the vast variety of proposals put forward by the community to resolve tree-level unitarity violations in Higgs inflation models, there exists the concept of self-healing. This mechanism helps cancel out tree-level violations for elastic scattering processes by summing over successive vacuum polarization loop corrections. In this comment, we shall see how self-healing is a manifestation of the optical theorem for a theory tailored to behave in a certain way.
Self-healing unitarity is an Optical illusion: Comment on `Self-healing of unitarity in effective field theories and the onset of new physics'
Archit Vidyarthi [email:[email protected]]
August 12, 2023
==============================================================================================================================================
§ INTRODUCTION
Unitarity is one of several properties at the heart of a quantum theory, and essentially implies that the probability of an event cannot exceed unity. Along with other properties such as positivity, causality, etc., it helps provide us with useful bounds on a theory (for example: perturbative bounds, Froissart bounds, etc.) in the form of constraints on a parameter, or on the domain within which the theory is valid, without needing to introduce new degrees of freedom (DsOF).
Tree-level unitarity violations, estimated using perturbative unitarity bounds, are immensely helpful in pointing out missing pieces in a theory. For a non-renormalizable theory, these may imply that the loop corrections might become relevant as we approach the apparent violation scale in describing the complete process <cit.>. For others, they may indicate that the theory is incomplete. Beyond Standard Model (BSM) physics helps fill in gaps stemming from the incompatibility of the Standard Model and gravity, and provides us with possible candidates for the missing DsOF, often motivated by the existence of dark matter and dark energy that make up the majority of the energy content of the universe.
Given how Higgs driven inflation has been one of the prime candidates for a theory describing the birth of the universe, the fact that it faces unitarity violations far below the Planck scale is something the scientific community has been trying to explain away for a long time (see our recent work <cit.> and cited works for more info). After several decades of search, though, we have as of yet not been able to resolve the issue completely. Among the several approaches suggested towards resolving the issue is `self-healing' of unitarity proposed in <cit.> and later applied in the context of Higgs inflation in <cit.>, which are at the heart of what we discuss in this work.
This paper is organized as follows: in Sec.<ref>, we introduce the reader to the optical theorem and partial wave unitarity bounds as presented in <cit.>; in Sec.<ref>, we briefly review the idea of self-healing as it was put forward in <cit.>; followed by our explanation how self-healing is a special case of the optical theorem in Sec.<ref>; and lastly, some discussions in Sec.<ref>.
§ BRIEF RECAP
Imposing that the action is unitary, we obtain the famous optical theorem, which equates the imaginary part of the scattering amplitude to the total scattering cross section.
ℳ(i→ f)-ℳ^*(f→ i)=i∑_X∫ dΠ_X (2π)^4δ^4(p_i-p_X)ℳ(i→ X)ℳ^*(f→ X).
In its generalized form (<ref>), this theorem states that order-by-order in perturbation theory, imaginary parts of higher loop amplitudes are determined by lower loop amplitudes. For instance, the imaginary part of one-loop amplitude could be determined by the tree-level amplitude. A special case arises from this using the assumption that the initial and final states are the same state |A>:
Imℳ(A→ A)=2E_CM|p_i|∑_Xσ(A→ X).
Optical theorem puts a constraint on how large a scattering amplitude can be. From the approximate form,
Imℳ≤|ℳ|^2|ℳ|<1.
Now, using the partial wave expansion of the scattering amplitude to impose constraints on coefficients of the Legendre polynomials. To recap, we first expand the scattering amplitude as:
ℳ(θ)=16π∑_j a_j (2j+1) P_j(cosθ),
where P_j(cosθ) are Legendre polynomials, with P(1)=1, and
∫_-∞^∞ P_j(cosθ) P_k(cosθ) dcosθ=2/2j+1δ_jk.
For a case where the initial and final states are the same, we can write the cross section in the center of mass frame as:
σ_CM_tot= 16π/E_CM^2∑_j |a_j|^2 (2j+1).
Employing the optical theorem at θ=0, we have,
Imℳ(A B → A B at θ=0) =2 E_CM|p⃗_i| ∑_X σ_tot(A B → X)
≥ 2 E_CM|p⃗_i| σ_tot(A B → A B),
where a simple inequality has been introduced owing to the fact that |AB>∈|X>. Then,
∑_j=0^∞(2 j+1) Im(a_j) ≥2|p⃗_i|/E_CM∑_j=0^∞(2 j+1)|a_j|^2 .
This, coupled with the inequality |a_j|≥Im(a_j), means that the magnitude of a_j is now constrained as |a_j|≤1, 0≤Im(a_j)≤ 1, and |Re(a_j)| ≤ 1/2, as seen in Fig. (<ref>).
§ PROPOSITION: SELF-HEALING UNITARITY
Preceding <cit.>, authors of <cit.> worked with a set of complex scalar fields, nonminimally coupled with gravity, and tried to estimate the scattering amplitude for the process ss→ s's', where they set s≠ s' to make sure that only the s-channel graviton exchange diagram contributed to the process, and they could avoid collinear divergences in the t and u channels. They, then, try to estimate the scale at which the standard model of particle physics and the minimal supersymmetric standard model, both coupled with gravity, would similarly violate unitarity at tree-level. They claim that in the limit where the number of particles is large, the leading order loop corrections are successive vacuum polarization diagrams and that these violations could be fixed by considering such higher-loop corrections.
Following this, authors of <cit.> consider a similar Lagrangian as <cit.> involving a nonminimal coupling between gravity and multiple scalar fields and provide a useful confirmation for the results presented in <cit.>. They first use partial wave analysis to do so, and then verify the result using a summation of the infinite series loops diagrams. Please note that <cit.> focused on j=2 partial wave amplitudes only.
Authors of <cit.> expanded on the work of the preceding authors and verified the results for a theory involving the Higgs doublet. Instead of sticking to just one process, however, the authors considered certain combinations of these scalars for initial and final states, making sure the combinations adhered to the rules set forth in <cit.> mentioned earlier. Later, they summed over the contributions from all of these processes to show explicitly that the self-healing phenomenon could be applied to j=0 level as well.
§ EQUALITIES
The most important step in order to proceed with Eq.(<ref>) is to fix the initial state |AB>. |X> would, then, contain all possible states that |AB> could transform to, with |AB> itself being one of them. This is what causes the inequality in Eq.(<ref>). What happens if we constrain the theory in such a way that the only possible state is |AB>? Then, instead of an inequality we'd get the equality |a_j|=Im(a_j) for all j. This is exactly what's observed in <cit.>, though they only show it for j= 2. So while the iterative sums are novel and useful in explicitly demonstrating the idea of self-healing, it is, for all intents and purposes, simply an artefact of the optical theorem. This could be visualized easily in Fig.(<ref>).
Additionally, if we fix the initial state, find out all the elements of the corresponding scattering matrix, and sum over their contributions, we revert to the initial form of the optical theorem Eq.(<ref>) and, again, instead of a partial wave bound (inequality), we get an equality as proposed originally. This can be seen in the result of <cit.> for all j, though it was shown explicitly only for j=0. Again, another manifestation of the optical theorem. This latter result covers theories that could be transformed to different forms using field transformations where the `ideal' structure (as required in <cit.>) of these theories could get ruined as more interaction terms show up, meaning more varied final states.
Also note that it was stated in <cit.> that the contribution from vacuum polarization corrections far exceeded that from other sources only when the number of particles was large. For a limited number of DsOF, contribution from other loop diagrams, such as vertex corrections or embedded loops, might be of the same order as the vacuum polarization corrections. Nevertheless, the optical theorem should still be able to restore unitarity in those theories. One example of this is <cit.> where, as previously mentioned, the authors have considered four DsOF in the form of the Higgs doublet.
§ DISCUSSION
Well-behaved gravitational theories are expected to face unitarity violations close to the Planck scale, where the loop diagrams start to contribute. Any violations below this scale imply either that the loops from DsOF (other than graviton) already present in the theory may be contributing, or that some new DsOF need to be introduced. It was seen to be the former in theories mentioned in this work, where summing over loop contributions was able to restore unitarity through the self-healing mechanism, which turned out to be a special case of the optical theorem.
The results of the optical theorem Eq.(<ref>) and Eq.(<ref>), and even the partial wave analysis Eq.(<ref>), are independent of whether the collisions are elastic or inelastic. Therefore, this analysis should be applicable to those cases as well, i.e. even the inelastic versions of the processes considered in <cit.> should be able to `self-heal' adequately. This could be explicitly verified as an independent work.
§ ACKNOWLEDGEMENT
This work is partially funded by DST (Govt. of India), Grant No. SERB/PHY/2021057.
unsrtnat
|
http://arxiv.org/abs/2307.04328v1 | 20230710034732 | Where to Drop Sensors from Aerial Robots to Monitor a Surface-Level Phenomenon? | [
"Chak Lam Shek",
"Guangyao Shi",
"Ahmad Bilal Asghar",
"Pratap Tokekar"
] | cs.RO | [
"cs.RO",
"cs.DM"
] |
Where to Drop Sensors from Aerial Robots
to Monitor a Surface-Level Phenomenon?
This work is supported in part by National Science Foundation Grant No. 1943368.
^* indicates equal contribution and authors are listed alphabetically
Chak Lam Shek^*, Guangyao Shi^*, Ahmad Bilal Asghar, and Pratap Tokekar
University of Maryland, College Park, MD 20742 USA [cshek1, gyshi, abasghar, tokekar]@umd.edu
August 12, 2023
========================================================================================================================================================================================================================================
empty
empty
We consider the problem of routing a team of energy-constrained Unmanned Aerial Vehicles (UAVs) to drop unmovable sensors for monitoring a task area in the presence of stochastic wind disturbances. In prior work on mobile sensor routing problems, sensors and their carrier are one integrated platform, and sensors are assumed to be able to take measurements at exactly desired locations. By contrast, airdropping the sensors onto the ground can introduce stochasticity in the landing locations of the sensors. We focus on addressing this stochasticity in sensor locations from the path planning perspective. Specifically, we formulate the problem (Multi-UAV Sensor Drop) as a variant of the Submodular Team Orienteering Problem with one additional constraint on the number of sensors on each UAV. The objective is to maximize the Mutual Information between the phenomenon at Points of Interest (PoIs) and the measurements that sensors will take at stochastic locations. We show that such an objective is computationally expensive to evaluate. To tackle this challenge, we propose a surrogate objective with a closed-form expression based on the expected mean and expected covariance of the Gaussian Process. We propose a heuristic algorithm to solve the optimization problem with the surrogate objective. The formulation and the algorithms are validated through extensive simulations.
§ INTRODUCTION
Multi-robot systems have been widely used in scientific information gathering including exploring the ocean <cit.>, tracking algal blooms <cit.>, and monitoring soil <cit.>. The planning problem on this topic is usually named Informative Path Planning (IPP), in which the research focus is on how to design planning algorithms to coordinate multiple robots to collect as much useful information as possible given the limited onboard resources (e.g., sensing and battery).
In some cases, the robotic platform and the sensors for scientific monitoring are integrated systems and are treated as mobile sensors as a whole <cit.>.
In other cases, the robotic platforms are treated as carriers of sensors <cit.>, and they are separable. The research efforts for such cases are mainly devoted to finding collaborative route strategies for these mobile platforms to serve the sensors to finish the sampling tasks. Our research is also along this line and we are interested in how to airdrop sensors to an area of interest with a team of Unmanned Aerial Vehicles (UAVs).
Specifically, we consider the problem of airdropping multiple sensors to the ground with a team of budget-constrained UAVs to reduce the uncertainty of Points of Interest (PoIs) as shown in Fig. <ref>. If the UAVs can precisely drop the sensors to the desired locations, such a problem is closely related to the classic Team Orienteering Problem (TOP) <cit.>. However, due to wind disturbances, when we release one sensor from the UAV, its landing location, i.e., the sampling location, is stochastic. This is the main difference from the existing research on mobile robotic sensors, in which authors usually assume that robots can take samples at precisely the desired location. Such a difference requires to rethink of the underlying optimization for planning.
To this end, we propose a new variant of the TOP for airdropping sensors with UAVs, in which the stochasticity of the sensor landing position is explicitly considered. However, the resulting optimization objective is computationally expensive to evaluate. To address this challenge, we resort to a Gaussian approximation approach <cit.> to obtain one surrogate objective with one closed-form expression. With this surrogate objective, we show that the problem can be solved in polynomial time and near optimally.
In summary, the main contribution of this paper is:
* We propose a variant of the Submodular Team Orienteering Problem to model the sensor dropping problem with aerial robots.
* We propose one computationally efficient surrogate objective function for the proposed problem and propose a heuristic algorithm to solve it.
* We demonstrate the effectiveness of our formulation and algorithm through simulations.
The rest of the paper is organized as follows. We first
give a brief overview of the related work in Section <ref>. Then, we explain the problem setup and formulation in Section <ref>. We introduce the technical approach in Section <ref> and validate the formulation and the proposed framework in Section <ref>.
§ RELATED WORK
In this section, we present the work most closely related to ours. We first discuss the related work on airdropping sensors, followed by stationary sensor placement and mobile sensor planning, and finally on estimating stationary fields with Gaussian Processes.
§.§ Airdroping sensors
Dropping resources from an aerial vehicle has long been of interest, particularly for military and search-and-rescue operations. For example, in military resupply missions, aircrafts are required to accurately deliver supplies to the target areas, taking into account geological factors and weather conditions. Extensive research has been conducted on low-level optimization of the release trajectory to achieve high precision in airdrop operations <cit.>. In this work, we focus on the complementary high-level planning of where to drop the sensors from multiple UAVs to monitor a surface-level phenomenon. We abstract the low-level trajectory control by assuming that for any given airdrop trajectory planner, the associated uncertainty of the landing position of the sensor is known. Specifically, we focus on route-level planning for multiple UAVs to deploy multiple sensors to the area of interest for environmental monitoring applications.
Our work is closely related to that of Gerlach et al. <cit.>. They formulate the problem of dropping multiple payloads to multiple targets as a Traveling Salesperson Problem (TSP). However, there are two key differences between their work and ours. First, our objective is to reduce the uncertainty at Points of Interest (PoIs) by dropping sensors and we use an information-theoretic metric. In contrast, the objective in <cit.> is to minimize the risk encountered by the soldiers. Second, our problem involves multiple energy-constrained UAVs, which cannot be modeled as TSP or its variants.
§.§ Sensor Placement and Mobile Sensor Planning
The sensor placement problem aims to maximize the information gain or sensing quality by strategically selecting sensor deployment locations. The typical approach is to model the phenomenon as a Gaussian Process <cit.> and use information theoretic measures for placing the sensors. The foundational work was done by Krause el al. <cit.> who showed that the partial monotonicity and submodularity allows a greedy placement to achieve a constant-factor approximation algorithm. This work was later extended to mobile sensor planning (also termed as informative path planning). Binney et al. <cit.> introduced the additional constraint of identifying a feasible path that connects these selected sensing locations. One approach to finding such paths is to convert the problem into an orienteering instance with submodular rewards. In <cit.> this problem is solved by constructing an additive approximation for the coverage objective to find a UAV path for image acquisition. A recursive greedy algorithm <cit.> is used in <cit.> to solve the submodular orienteering problem for informative path planning. This approach provides guarantees for the submodular objective but
runs in quasi-polynomial time, limiting its use for large problem instances.
In the context of a multi-robot setting, the orienteering problem can be solved iteratively, where the single robot performance guarantee can be extended to the multi-robot scenario <cit.>. Our work closely aligns with this body of work on informative path planning with a key difference. Because we are airdropping sensors, the exact sensing location depends on the wind field and is not known, unlike existing work. We show how to deal with this additional source of uncertainty.
§.§ GP with Uncertain Inputs
We use Gaussian Processes <cit.> to model the spatial function that is to be estimated by the sensors. Since we do not know the exact locations the sensors will fall at before planning UAV paths, the input to GP regression is uncertain. It is shown that the predictive distribution for Gaussian processes with uncertain inputs may not be Gaussian in general in <cit.>. Various approaches have been used to deal with input uncertainty in GPs. In the Bayesian approach, the distribution with uncertain input locations can be obtained by integrating over the uncertainty of the locations <cit.>. However, these integrals are analytically intractable in general. Taylor expansion about the uncertain locations is used in <cit.> to present an approximate method that requires the derivative of the mean of f. The Gaussian Approximation method <cit.> assumes that the posterior distribution is Gaussian and finds its expected mean and expected covariance by integrating over the uncertainty of the locations. For certain kernel functions, these co-variances can be computed analytically. We employ the Gaussian approximation method in this paper to handle the random sensor locations.
§ PROBLEM STATEMENT
Consider a weighted graph G = (V, E), where the vertex set V represents locations that can be visited by a team of m UAVs. The weight w(u,v) of an edge (u, v) ∈ E represents the time taken or energy spent by the UAVs to travel from vertex u to vertex v. Let (x_v, y_v, z_v) represent the coordinates of vertex v. Each vertex corresponds to a location where one of the UAVs can drop a sensor onto the ground below to observe the spatial field.
The sensor's landing position on the surface, denoted by q_v, can vary depending on the wind conditions at the drop location v and the height of the drop location z_v. We assume that q_v follows a normal distribution, specifically q_v ∼𝒩(q̅_v, Σ_v), and that s̅_v and Σ_v are known for each v∈ V. Each UAV i∈[m] has a given number of sensors k_i and limited amount of time (or energy) T_i to visit some locations in V and to drop the sensors from those locations. The path of UAV i must start and end at its designated depot location r_i∈ V.
The purpose of dropping sensors is to observe the value of a spatial function f at specific points of interest (POI) U on the ground. Each sensor obtains a measurement of the underlying field with additive Gaussian noise. Since we may have fewer sensors than POI, and due to the stochastic nature of sensor drop, we will need to estimate the value of f at POI. Consequently, there will be inherent uncertainty associated with these estimates.
Gaussian Processes associate a random variable with each POI in U and the joint distribution over U can be used to quantify the information gained by the sensors dropped by the UAVs.
Given paths P = {P_1,…, P_m} for the UAVs, let S(P) = {S_1,…,S_k} represent the corresponding sensor drop locations, and let Q(P) be the random variable representing the sensor locations, i.e., for every drop location v∈ S, the sensor location q_v ∈ Q. Also, let the length of the path ℓ(P_i) denote the total time taken by the UAV i to visit all the locations in P_i. Let η be the time required to drop a sensor. Therefore, the total time of a path P_i is given as C(P_i)=ℓ(P_i) + S(P_i)η.
Let ℱ_U represent the random variable associated with POI U and let ℱ_Q represent the random variable associated with sensor readings at locations in Q. Then Pr(ℱ_U|ℱ_Q(P)=f_Q) is the prediction at U given sensor readings at locations in Q(P). To simplify notation, we will use S and Q going forward, without explicitly indicating their dependence on UAV paths P. We focus on the offline planning problem <cit.> where the plan must be decided before dropping any sensor.
The mutual information – as a function of the UAVs' paths – between the random variables ℱ_U and ℱ_Q is defined as,
MI(P) = H(ℱ_U) - H(ℱ_U|ℱ_Q),
where H(𝒳) represents the entropy of random variable 𝒳. We now formally define the multi-UAV sensor drop problem.
[Multi UAV Sensor Drop]
Given the points of interest U, sensor drop locations in G=(V, E) along with the mean q̅_v and covariance Σ_v of sensor's location associated with each v ∈ V, k_i sensors and budget T_i for each UAV i∈[m], find path P_i rooted at the depot r_i along with drop locations S_i for each UAV i∈[m] to maximize the mutual information, i.e.,
max_P_1,…,P_m MI(P) = H(ℱ_U) - H(ℱ_U|ℱ_Q)
s.t. C(P_i) ≤ T, ∀ i∈ [m]
|S_i| ≤ k_i, ∀ i∈ [m].
Note that given drop locations SS, the sensor locations in Q are random. If the locations in Q are deterministic, i.e., the sensors fall at the exact locations desired, and if points of interest U are the same as the vertices in V, we get the traditional informative path planning problem <cit.>.
Since the locations in Q are themselves random variables, evaluating the probability distribution Pr(ℱ_U|ℱ_Q) and its entropy is challenging. In the next section, we discuss how we address this challenge and present the planning algorithm.
§ TECHNICAL APPROACH
In this section, we discuss how to evaluate the objective function given in Problem <ref>. We then propose the planning algorithm to solve the problem.
§.§ Gaussian Process with Stochastic Drop Locations
Trying to agree on random variables and notation/abuse of notation
Q random variable for sensor locations
q realization of QQ, a vector of sensor locations
S random variable representing drop locations
s realisation of SS, again a vector
U with abuse of notation, joint random variable for f(U)f(U)
Then
Pr(U|S=s) =∫ Pr(U|Q=q)Pr(q_i=q_i|s=s_i)dq_i
or
Pr(U|S=s) =∫ Pr(U|Q=q)Π_i=1^a( 𝒩(q_i,Σ_i) dq_i)
In order to evaluate the objective function (<ref>), we need to calculate the entropy of the random variable (ℱ_U|ℱ_Q). If the sensor locations in Q were deterministic, this random variable would be a multivariate Gaussian, and its covariance matrix could be used to determine the entropy. However our data is of the form {q_i, f(q_i)+ϵ_i}_i=1^∑_j |S_j| and q_i∼𝒩((q̅_i, Σ_i)). Then, since the locations of sensors are independent of each other, the probability distribution Pr(ℱ_U|ℱ_Q) is given by integrating the distribution given fixed locations over random sensor locations, i.e.,
Pr(ℱ_U|ℱ_Q) =
∫∫ Pr(ℱ_U|ℱ_Q,{q_1, , q_a})∏_i=1^a(Pr(q_i) )dq_i dq_a.
This distribution is not Gaussian and there is generally no closed
form expression for this integral <cit.>. Existing literature on Gaussian Processes with input uncertainty <cit.> resorts to approximations in order to solve this integral. A Monte Carlo approach by drawing samples of q from uncertain location distributions is considered in <cit.>. Taylor expansion about q̅ is used in <cit.> to present an approximate method that requires the derivative of the mean of f. The Gaussian approximation method <cit.> assumes that the posterior distribution is Gaussian and finds its expected mean and expected covariance by integrating over the uncertainty of the locations q.
For the squared exponential covariance, the expected covariance for normally distributed sensor locations can be analytically computed exactly using the following expression <cit.>.
Σ_QQ(i,j) =
σ^2 exp( -1/2 (q̅_i - q̅_j)^⊤ (W+Σ_i +Σ_j)^-1 (q̅_i - q̅_j))/| I+W^-1(Σ_i+Σ_j)(1-δ_ij) |^1/2
Here q̅_i and Σ_i are the mean and covariance of the normally distributed sensor location q_i in Q, and W is a diagonal matrix where each diagonal element corresponds to a characteristic length scale for the respective input variable.
We use the Gaussian approximation method in this paper because it does not require sampling and is computationally tractable with a simple analytical expression for the covariance matrix. Moreover, since we are planning paths for UAVs offline, before getting any sensor readings, we can use this method to find the mutual information by just using the expected covariance as discussed below.
Since the Gaussian approximation method assumes that the distribution of ℱ_U|ℱ_Q is a Gaussian distribution, and because ℱ_U and ℱ_Q are jointly Gaussian, the mutual information is given by
MI = H(ℱ_U)- H(ℱ_U|ℱ_Q)
= H(ℱ_U) + H(ℱ_Q) - H(ℱ_U,ℱ_Q)
= 1/2log( (Σ_UU) (Σ_QQ)/(Σ̅)),
where
Σ̅ = [Σ_UU Σ_UQ
Σ_QU Σ_QQ].
We can use the expression (<ref>) to evaluate Σ_UQ(i,j) by replacing x_i with the known location of i^th point of interest in U and Σ_i by the null matrix.
The Objective function (<ref>) and the surrogate objective defined in Equation (<ref>) are submodular and monotonically non-decreasing set functions in S.
The objective function is a submodular function.
I(f_PoIs4pt,X) = H(f_PoIs4pt) + H(X) - H(f_PoIs4pt∪ X)
I(f_PoIs4pt,X') = H(f_PoIs4pt) + H(X') - H(f_PoIs4pt∪ X')
The increment of MI denotes EE, such that
E_x = H(X ∪ z) - H(f_PoIs4pt∪ X ∪ z)
- H(X) + H(f_PoIs4pt∪ X)
E_x' = H(X' ∪ z) - H(f_PoIs4pt∪ X' ∪ z)
- H(X') + H(f_PoIs4pt∪ X')
E_x - E_x' = [ H(X ∪ z)) - H(X) - H(X' ∪ z) + H(X')_(1) ] +35pt
[ H(f_PoIs4pt∪ X ∪ z)) - H(f_PoIs4pt∪ X)
- H(f_PoIs4pt∪ X' ∪ z) + H( f_PoIs4pt∪ X')_(2) ] ≥ 030pt
There are a few nice properties of mutual information.
Monotonicity Clearly, the MI objective is also a monotonic function because the conditioning always reduces entropy:
H(f_PoIs4pt | X) ≤ H(f_PoIs4pt)
In other words, the additional sensor can provide extra information which always helps.
§.§ Planner
The submodularity and monotonicity of the surrogate objective function allow us to formulate Problem <ref> as a submodular TOP. However, there is one additional constraint in Problem <ref> that is not present in standard submodular TOP, that of the number of sensors k_i that each robot is able to deploy. We address this problem using the following observation.
In a complete graph with N≥ k_i vertices for all i, there always exists an optimal solution where the robot i's path consists of no more than k_i vertices, excluding the starting vertex.
The proof follows by contradiction. Suppose there is an instance where no optimal solution has at most k_i vertices along robot i's path. The robot is allowed to deploy at most k_i sensors. Therefore, there must be one or more vertices along the robot path that no sensor is dropped. Since the graph is a complete metric graph, we can “shortcut” such vertices without increasing the cost of the path. Therefore, we can recover a solution that consists of exactly k_i vertices. This is a contradiction proving the original claim.
With this insight, we present our algorithm (Algorithm <ref>) to solve the Problem <ref>.
We first take the metric completion of the input graph. Recall that for a weighted graph G(V, E), each edge (u,v) ∈ E is associated with a cost w(u,v). In the preprocessing step, we generate a complete graph G^'=(V, E^') using G, where the edge cost w^'(u,v) is defined as the length of the shortest path between u and v in G. Then, we sequentially call a subroutine, Generalized Cost-Benefit (GCB), to compute a path for each robot. Compared to the original GCB algorithm <cit.>, in Algorithm <ref>, we add one extra control condition in the while loop to account for the constraint, Eq. (<ref>), on the number of available sensors using Lemma <ref>.
The constraints imposed on the paths of UAVs, which limit them to at most k_i vertices and a maximum length of T_i for UAV i, can be regarded as a partition matroid constraint. It has been shown in <cit.> that an α-approximate greedy step for submodular maximization over a matroid yields an approximation ratio of 1/α+1. Hence, given an α-approximation algorithm to solve the submodular orienteering problem for a single UAV, Algorithm <ref> results in a 1/α + 1 approximation ratio for maximizing Objective (<ref>) for multiple UAVs. When the paths of all the UAVs are constrained to be of at most T length and k vertices, we get a uniform matroid resulting in 1-1/e^α approximation ratio.
A quasi-polynomial time recursive greedy algorithm to solve the single vehicle orienteering problem with submodular rewards is given in <cit.>, resulting in α=Olog(). In this paper we use Generalized Cost Benefit (GCB) algorithm to solve the single UAV problem as it has better runtime than the recursive greedy algorithm <cit.>.
§ EVALUATION
In this section, we evaluate the performance of our algorithm through a series of numerical experiments.
We first explain the setup for the simulation. Then, we will show one qualitative example to illustrate the difference between the proposed approach and the baseline. Next, we will quantitatively evaluate the performance of the proposed approaches w.r.t. the uncertainty reduction of PoIs. Moreover, we will show the running time of the proposed algorithm w.r.t. the number of robots.
§.§ Experimental Setup
The flying object model used in this study is based on the work described in <cit.>. This model captures the motion of the sensors, considering the gravity, the sensors' surface area, and the speed of the wind. The sensor mass is set to 10kg. The surface coefficient is 1 and the vertical height is 500m.
We begin by defining the map, ground truth, and wind field, as shown in Fig. <ref>. The map provides labels for all the potential dropping points and PoIs. The ground truth is generated by combining multiple Gaussian functions. Data points sampled from the ground truth are used to learn the kernel function, where we employ the RBF function. The wind field indicates the speed at specific locations on the map. By combining the sensor motion model with the wind field, we can estimate the landing position of the sensors.
Using a given kernel, the Algorithm <ref> is applied to search for a set of sensor dropping locations which is an approximate solution to the main problem. The final sensor locations are determined by sampling from the flying object model with uncertainty. Once the sensor locations are obtained, we can measure the environmental values and compute the posterior of PoIs based on these measurements.
§.§ An qualitative example
In the following, we present a comparison between a baseline approach and our proposed method using the defined settings. The experiment focuses on a scenario with two UAVs, where each UAV is equipped with four sensors. The UAVs are allocated a distance budget of 870 units to drop all the sensors along their respective paths.
§.§.§ Baseline
In the baseline case (Fig <ref>), the UAVs tend to drop a higher number of sensors in areas with a higher concentration of PoIs. The objective is to ensure that each sensor can cover one or more PoIs. However, due to the uncertainty introduced by the wind, the sensors tend to cluster in smaller regions. As a result, the four sensors located around coordinates (0,100) are only capable of accurately estimating two PoIs' value, while the remaining PoIs are not sufficiently covered. This can be observed in Fig <ref>, where the two PoIs in the lower right corner exhibit a significantly higher error of estimation.
§.§.§ Our Approach
Our approach, on the other hand, considers the impact of wind uncertainty and prefers to drop sensors in a wider area. As shown in Fig. <ref>, the wind blows the sensors to a broader coverage area, allowing them to reach and cover more PoIs. This broader coverage results in a significant reduction in the error of PoI estimation compared to the baseline case. Additionally, it is worth noting that the areas where the sensors are dropped but do not have high concentration of PoIs exhibit high error rates. This demonstrates the effectiveness of our approach in adapting to the wind uncertainty and achieving better coverage of the target area.
§.§ Comparisons with Baselines
In this section, we compare the MSE of three different approaches across three different scenarios. The MSE is computed as the sum of the square of the difference between the posterior of the PoIs and the ground truth values of the PoIs. In the first two scenarios, we assume that the wind speed is uniform and the variance of landing location is the same for all dropping nodes. In the first scenario, the final location of a sensor follows a Gaussian distribution with a variance of 900. Two UAVs are deployed, with each carrying 4 sensors. In the second scenario, the final location of a sensor follows a Gaussian distribution with a variance of 820. Two UAVs are deployed, with each carrying 3 sensors. In both of these scenarios, our approach demonstrates approximately a 10% improvement in MSE compared to the baseline approach. The random selection approach, on the other hand, results in an MSE of 1.
The third scenario introduces non-uniform uncertainty w.r.t. the drop point location, where the variance is a function of the non-uniform wind speed. Once again, our approach consistently outperforms the baseline approach, achieving a 12% improvement in MSE. These results highlight the effectiveness of our approach in mitigating the impact of uncertainty in different scenarios and achieving more accurate sensor placements.
§.§ Running Time
Lastly, we demonstrate the scalability of our approach. In comparison to the baseline approach, our approach may have a slightly longer running time in each scenario. However, both approaches grow polynomially in run time with the number of sensors per UAV.
To further evaluate the computational performance, we also simulated a brute-force approach. The brute-force approach generates all possible combinations of sensor dropping points within the budget constraint and selects the set with the highest objective value. The runtime of the brute-force approach grew exponentially, taking hours to days to complete due to the factorial computation of all possible combinations. This stark contrast highlights the effectiveness and efficiency of our approach in finding nearly optimal solutions for sensor placement in a timely manner.
§ CONCLUSION
This paper studies the problem of routing a team of UAVs to drop sensors to reduce the uncertainty of PoIs. The problem is formulated as a variant of TOP. To reduce the computational cost in the evaluation of the objective, we propose one surrogate objective with closed-form expression based on Gaussian approximation. A heuristic algorithm (SGA) is proposed to solve the relaxed problem with the surrogate objective. The formulation and the algorithm are validated in numerical simulation.
IEEEtran
|
http://arxiv.org/abs/2307.04502v1 | 20230710114651 | Modular Completely Dirichlet forms as Squares of Derivations | [
"Melchior Wirth"
] | math.OA | [
"math.OA",
"math-ph",
"math.FA",
"math.MP",
"quant-ph"
] |
We prove that certain closable derivations on the GNS Hilbert space associated with a non-tracial weight on a von Neumann algebra give rise to GNS-symmetric semigroups of contractive completely positive maps on the von Neumann algebra.
Distributed Decisions on Optimal Load Balancing
in Loss Networks
Qiong Liu1, Chenhao Wang2, Ce Zheng1
1Télécom Paris, Institut Polytechnique de Paris, France
2Beijing Normal University, China
Email: [email protected], [email protected], [email protected]
==========================================================================================================================================================================================================================
§ INTRODUCTION
The interplay between derivations and symmetric semigroups of unital (or contractive) completely positive maps has proven fruitful for applications in quantum information theory <cit.>, operator algebras <cit.> and beyond. Using the framework of completely Dirichlet forms, this connection is particularly well-understood in the case of tracially symmetric semigroups after the seminal work of Cipriani and Sauvageot <cit.>.
In many situations however one encounters non-tracial reference states or weights: In quantum statistical mechanics, the reference state is typically a Gibbs state, which is not a trace at finite temperature; in quantum probability in the study of Lévy processes on compact quantum groups, the natural reference state is the Haar state, which is only a trace for the class of compact quantum groups of Kac type; and in the structure theory of von Neumann algebras, one is faced with non-tracial states when the von Neumann algebra has a non-trivial type III summand.
In the non-tracial setting, the connection between derivations and symmetric semigroups of completely positive maps is much less understood. Recently, it was shown by the author that every GNS-symmetric semigroup of unital completely positive maps gives rise to a canonical derivation via its associated Dirichlet form <cit.>. This result was (partially) extended to KMS-symmetric semigroups by Vernooij and the author <cit.>.
There has also been work in the opposite direction – starting with a derivation to construct a completely Dirichlet form <cit.>. However, these results all rely on additional structural assumptions on the derivation, usually some form of (approximate) innerness. This means that natural examples like derivations arising from cocycles on non-unimodular groups or Voiculescu's derivation in non-tracial free probability could not be treated in this framework.
In this article, we prove in a general context that closable derivations give rise to GNS-symmetric semigroups of completely bounded maps. More precisely, our main result is the following.
Let Å be a Tomita algebra, $̋ a normal Tomita bimodule overÅandδÅ→$̋ a closable symmetric derivation. Let ℰ be the closure of the quadratic form _0 given by (_0)=(δ) and _0(a)=δ(a)_^̋2. Then the strongly continuous semigroup associated with is the GNS implementation of a GNS-symmetric semigroup of contractive completely positive maps on the left von Neumann algebra generated by Å.
Here a normal Tomita bimodule is a bimodule over a Tomita algebra that additionally carries a complex one-parameter group (_z) and an involution satisfying some compatibility conditions, and a symmetric derivation δÅ→$̋ is a map that intertwines the complex one-parameter groups and involutions onÅand$̋ and satisfies the product rule
δ(ab)=aδ(b)+δ(a)b.
These objects were introduced in <cit.> and appear to be the natural non-tracial analogs of the Hilbert bimodule and derivation occurring in the context of completely Dirichlet forms on tracial von Neumann algebras.
Combined with the results from <cit.>, we thus obtain a comprehensive picture of GNS-symmetric quantum Markov semigroups analogous to the result of Cipriani and Sauvageot for tracially symmetric semigroups. Among other potential applications, we hope that this result opens the gate for applications to non-tracial free probability and deformation/rigidity theory of type III von Neumann algebras similar to recent work in this direction in the tracial case.
One main difficulty when trying to prove that closable derivations generate completely Dirichlet forms (or semigroups of completely positive maps) is that the property defining derivations, the product rule, is an algebraic property, while Dirichlet forms are defined in terms of order properties, and the domain of a derivation is not necessarily closed under order operations. As such, the problem of properly dealing with domains is crucial. Note that it is unavoidable to allow for unbounded derivations as everywhere defined derivations yield norm continuous semigroups of completely positive maps, which is too restrictive for many applications.
In the tracial case, this difficulty can be overcome since order operations such as taking the positive part can be expressed in terms of functional calculus and as such can be approximated by polynomials. In the non-tracial case, the order operations can still be expressed in terms of functional analysis in the setup of Haagerup L^p spaces, but the product rule is formulated in terms of Hilbert algebra multiplication, which is different from the product of two operators in Haagerup L^2 (which is only in L^2 if it is zero). Therefore it is not clear how to connect the two.
Instead of trying to follow the proof in the tracial setting, our proof strategy instead relies on Haagerup reduction method, which allows to embed a von Neumann algebra as an expected subalgebra of a bigger von Neumann algebra that can be approximated by finite von Neumann algebras. As it turns out, this reduction method is well-suited to reduce the problem at hand to the known case of tracial von Neumann algebras. One key challenge are again domain issues: For the Haagerup construction one has to extend the derivation to a domain on a crossed product that is sufficiently big, but such that the extension still satisfies the product rule. The essential new technical ingredient to overcome this kind of domain problems lies in the introduction of a new locally convex topology on the domain of a derivation that allows to extend derivations to derivations on a completion.
As a final note, considering the results from <cit.>, it is a natural question whether the results from the present article can be extended to cover KMS-symmetric semigroups. For one, our methods crucially use commutation with the modular group, which fails for KMS-symmetric maps if they are not GNS-symmetric. But more severely, it seems like there are additional algebraic obstructions, already in finite dimensions: It is shown in <cit.> that if is a completely Dirichlet form on L^2(M_n(),ϕ), then there exist self-adjoint matrices v_j∈ M_n() such that
(ρ^1/4xρ^1/4)=∑_j tr(ρ^1/4[v_j,x]ρ^1/4^2),
where ρ is the density matrix inducing the state ϕ on M_n(). However, without further assumptions on the operators v_j, the quadratic form on the right side of the previous equation is not necessarily a completely Dirichlet form.
§.§ Outline of the article
In Section <ref> we recall some basics regarding modular theory, completely Dirichlet forms on standard forms of von Neumann algebras and Tomita bimodules and derivations. In Section <ref> we introduce a topology on the domain of a derivation, the δ-topology, and show that derivations can be extended to derivations on the completion in the δ-topology. In Section <ref> we give a closability criterion for derivations in our setting. In Section <ref> we discuss how derivations can be extended to crossed products and discuss how completely Dirichlet forms behave with respect to change of the reference weight. Then we state and prove the main result of this article, Theorem <ref>, showing that the quadratic form associated with a closable derivation is a modular completely Dirichlet form. Finally, in Section <ref> we discuss several classes of examples, including inner derivations, derivations arising in non-tracial free probability and derivation induced by cocycles on (possibly non-unimodular) locally compact groups.
§.§ Acknowledgments
The author was funded by the Austrian Science Fund (FWF) under the Esprit Programme [ESP 156]. For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
§ BASICS
In this section we briefly recap some material concerning modular theory and in particular Hilbert and Tomita algebras, completely Dirichlet forms, Tomita bimodules and derivations that is used in the later sections.
§.§ Modular theory
As our approach is formulated in the language of Hilbert and Tomita algebras, we summarize the relevant definitions here. Our treatment mostly follows <cit.>.
An algebra Å with involution ^♯ (resp. ^♭) and inner product ⟨ · ,· ⟩ is called left (resp. right) Hilbert algebra if
* for every a∈Å the map π_l(a)Å→Å, b↦ ab (resp. b↦ ba) is bounded,
* ⟨ ab,c⟩=⟨ b,a^♯ c⟩ (resp. ⟨ ab,c⟩=⟨ b,ca^♭⟩) for all a,b,c∈Å,
* the involution ^♯ (resp. ^♭) is closable,
* the linear span of all products ab with a,b∈Å is dense in Å.
Let M be a von Neumann algebra and ϕ a normal semi-finite faithful weight on M. We write _ϕ for the definition ideal {x∈ M|ϕ(x^∗ x)<∞} and (π_ϕ,L^2(M,ϕ),Λ_ϕ) for the associated semi-cyclic representation.
The prototypical example of a left Hilbert algebra is Å=Λ_ϕ(_ϕ∩_ϕ^∗) with the product Λ_ϕ(x)Λ_ϕ(y)=Λ_ϕ(xy), the involution Λ_ϕ(x)^♯=Λ_ϕ(x^∗) and the inner product inherited from L_2(M,ϕ), that is, ⟨Λ_ϕ(x),Λ_ϕ(y)⟩=ϕ(x^∗ y). In this case, π_l(Å)^''=π_ϕ(M). We write Å_ϕ for this left Hilbert algebra.
Conversely, every left Hilbert algebra Å gives rise to a von Neumann algebra π_l(Å)^'' acting on the completion of Å and a weight
ϕπ_l(Å)^''_+→ [0,∞], ϕ(x)=ξ^2 if x^1/2=π_l(ξ),
∞ otherwise.
If Å is a full left Hilbert algebra <cit.>, then ϕ is a normal semi-finite faithful weight on π_l(Å)^'', and Å is canonically isomorphic to Å_ϕ.
Let be the completion of the left Hilbert algebra Å. Since the involution ^♯ on Å is closable, its closure S on exists and has a polar decomposition S=JΔ^1/2. The operator Δ is a non-singular positive self-adjoint operator, called the modular operator, and J is an anti-unitary involution, called the modular conjugation. If Å is the left Hilbert algebra associated with a weight ϕ, we write Δ_ϕ and J_ϕ for the associated modular operator and modular conjugation. We write Λ_ϕ^'_ϕ^∗→ L_2(M,ϕ) for the map x↦ J_ϕΛ_ϕ(x^∗).
If Å is full, the modular conjugation J gives rise to the positive self-dual cone P={π_l(a)Ja| a∈Å} and π_l(Å)^'' is in standard form <cit.>.
The modular operator Δ gives rise to a point weak^∗ continuous group of automorphisms x↦Δ^itxΔ^-it on π_l(Å)^''. If ϕ is a normal semi-finite faithful weight on M, the group σ^ϕ given by σ^ϕ_t(x)=π_ϕ^-1(Δ_ϕ^itπ_ϕ(x)Δ_ϕ^-it) is called the modular group associated with ϕ.
If (α_t)_t∈ is a point weak^∗ continuous group of ∗-automorphisms on M, then an element x∈ M is called entire analytic if the map t↦α_t(x) has an extension z↦α_z(x) to the complex plane such that z↦ω(α_z(x)) is analytic for every ω∈ M_∗. The entire analytic elements form a weak^∗ dense ∗-subalgebra of M.
A Tomita algebra is a left Hilbert algebra Å endowed with a complex one-parameter group (U_z)_z∈ of algebra automorphism such that
* z↦⟨ a,U_z b⟩ is analytic for all a,b∈Å,
* (U_z a)^♯=U_z̅(a^♯) for all a∈Å, z∈,
* ⟨ U_z a,b⟩=⟨ a,U_-z̅b⟩ for all a,b∈Å, z∈,
* ⟨ a^♯,b^♯⟩=⟨ U_-ib,a⟩ for all a,b∈Å.
Note that every Tomita algebra becomes a right Hilbert algebra when endowed with the involution
Å→Å,a↦ a^♭=U_-i(a^♯).
For a full left Hilbert algebra Å let
Å_0={ξ∈⋂_n∈D(Δ^n) | Δ^nξ∈Å for all n∈}.
For every ξ∈Å_0 the map t↦Δ^itξ has an entire analytic extension z↦ U_zξ with U_zξ∈Å_0 for all z∈. This makes Å_0 into a Tomita algebra such that π_l(Å_0)^''=π_l(Å)^''.
In particular,
(Å_ϕ)_0={Λ_ϕ(x)| x∈_ϕ∩_ϕ^∗, x entire analytic for σ^ϕ}.
§.§ Completely Dirichlet forms
Completely Dirichlet forms in the non-tracial setting were introduced by Goldstein and Lindsay <cit.> in the language of GNS Hilbert spaces of states (or weights) and by Cipriani <cit.> in the language of standard forms with a fixed cyclic vector. Our approach is somewhat different from both of these formulations in that we use left Hilbert algebras, but in view of the previous subsection it is equivalent to the formulation by Goldstein–Lindsay (and to that of Cipriani in case the left Hilbert algebra has a unit).
Let Å be a full left Hilbert algebra with completion . Let C be the closure of {Δ^1/4a| a∈Å, 0≤π_l(a)≤ 1} and let P_C be the metric projection onto C. We say that a closed densely defined quadratic form on is a Dirichlet form with respect to Å if ∘ J= and (P_C(a))≤(a) for all a∈ with Ja=a.
The Dirichlet form is called completely Dirichlet form if for every n∈ the quadratic form
^(n)⊗ M_n()→ [0,∞], ^(n)([ξ_ij])=∑_i,j=1^n (ξ_ij)
is a Dirichlet form with respect to Å⊙ M_n(). Here M_n() carries the normalized Hilbert–Schmidt inner product and the multiplication and involution on Å⊙ M_n() are given by [a_ij][b_ij]=[∑_k a_ikb_kj], [a_ij]^♯=[a_ji^♯].
A (completely) Dirichlet form with respect to Å is called modular (or GNS-symmetric) if ∘ U_t= for all t∈.
Completely Dirichlet forms are of particular interest for their connection to semigroups of contractive completely positive maps on von Neumann algebras. Let us briefly sketch this correspondence. Proofs can be found in <cit.> for the wider class of KMS-symmetric semigroups. The result for GNS-symmetric semigroups follows from the fact that GNS symmetry is equivalent to KMS symmetry and commutation with the modular group (see <cit.> for example).
Let M be a von Neumann algebra. A quantum dynamical semigroup is a semigroup of normal contractive completely positive operators on M that is continuous in the point weak^∗ topology. If ϕ is a normal semi-finite faithful weight on M, a quantum dynamical semigroup (P_t) is called GNS-symmetric with respect to ϕ if ϕ∘ P_t≤ϕ for all t≥ 0 and
ϕ(P_t(x)^∗ y)=ϕ(x^∗ P_t(y))
for all x,y∈_ϕ and t≥ 0.
Every GNS-symmetric quantum dynamical semigroup gives rise to a strongly continuous semigroup (T_t) on L^2(M,ϕ), its GNS implementation, acting by T_tΛ_ϕ(x)=Λ_ϕ(P_t(x)) for x∈_ϕ, and the associated quadratic form is a modular completely Dirichlet form with respect to Å_ϕ. Vice versa, the strongly continuous semigroup associated with a modular completely Dirichlet form is the GNS implementation of a GNS-symmetric quantum dynamical semigroup.
We call a completely Dirichlet form a quantum Dirichlet form if the associated quantum dynamical semigroups consists of unital maps. A criterion in terms of the form itself is given in <cit.>.
§.§ Tomita bimodules and derivations
Tomita bimodules were introduced in <cit.> as codomains of the derivations associated with modular completely Dirichlet forms.
Let Å be a Tomita algebra. A Tomita bimodule over Å is an inner product space $̋ endowed with non-degenerate commuting left and right actions ofÅ, an anti-isometric involution→̋$̋ and a complex one-parameter group (_z) of isometries such that
* aξ b≤π_l(a)π_r(b)ξ for a,b∈Å, ξ∈$̋,
*⟨aξb,η⟩=⟨ξ, a^♯ηb^♭⟩fora,b∈Å,ξ,η∈$̋,
* _z(aξ b)=(U_z a)(_z ξ)(U_z b) for a,b∈Å, ξ∈$̋,z∈,
*(aξb)=(Jb)(ξ)(Ja)fora,b∈Å,ξ∈$̋,
* _z =_z̅ for z∈.
Let ̋̅ be the completion of $̋. The first two bullet points imply thatπ_l(a)↦(ξ↦aξ)extends to a non-degenerate∗-homorphism fromπ_l(Å)toB(̋̅). If this map can be extended to a normal∗-homomorphism fromπ_l(Å)^''toB(̋̅), then we say that$̋ is a normal Tomita bimodule. Requiring normality for the right action instead leads to the same notion of normal Tomita bimodule.
If Å is a Tomita algebra and $̋ a bimodule overÅ, we call a linear mapδÅ→$̋ a derivation if it satisfies the product rule
δ(ab)=aδ(b)+δ(a)b
for a,b∈Å. If $̋ is a Tomita bimodule overÅ, we say that a derivationδÅ→$̋ is symmetric if δ∘ J=∘δ and δ∘ U_z=_z∘δ for all z∈.
If Å is a full left Hilbert algebra and a modular quantum Dirichlet form with respect to Å, it is shown in <cit.> that
Å_={a∈Å_0| U_z a∈() for all z∈}
is a Tomita subalgebra of Å_0 and a core for . Moreover, by <cit.> there exists a Tomita bimodule $̋ overÅand a symmetric derivationδÅ_→$̋ such that
(a,b)=⟨δ(a),δ(b)⟩_
for a,b∈Å_.
Under the minimality condition =̋lin{δ(a)b| a,b∈Å_}, the pair (,̋δ) is uniquely determined by up to isometric isomorphism preserving the Tomita bimodule structure and intertwining the derivations <cit.>. By a slight abuse of notation, any such pair (,̋δ) is called the first-order differential structure associated with . If $̋ is a normal Tomita bimodule, the quantum Dirichlet formis called Γ-regular. A characterization in terms of the carré du champ is given in <cit.>.
§ Δ-TOPOLOGY AND COMPLETENESS
In this section we introduce a locally convex topology on the domain of a closable symmetric derivation, called theδ-topology. This topology is strong enough to ensure that the derivation extends to a derivation on the completion, which is a key technical ingredient in the proof of the main theorem later.
For the definition of theδ-topology recall that the Mackey topologyτ(M,M_∗)on a von Neumann algebraMis the finest linear topology𝒯onMsuch that the topological dual of(M,𝒯)isM_∗. Equivalently, it is the finest locally convex topology onMthat coincides with the strong^∗topology on norm bounded sets <cit.>. It has the advantage over the other usual locally convex topologies onMof being complete, which is convenient for several of the following arguments.
LetÅbe a Tomita algebra with completion, let$̋ be a normal Tomita bimodule over Å and δÅ→$̋ a symmetric derivation. We define theδ-topology𝒯_δonÅas the coarsest locally convex topology that makes the maps
Å→⊕̋̅, a↦(Δ^n a,δ(Δ^n a))
continuous with respect to the norm topology on⊕̋̅for alln∈and the maps
Å→ B(), a↦π_l(Δ^n a)
continuous for the Mackey topology onB()for alln∈. Clearly, theδ-topology is stronger than the topology induced by the graph norm(·_^2+δ(·)_^̋2)^1/2.
If is a Γ-regular modular quantum Dirichlet form and (,̋δ) the associated first-order differential structure, then Å_ is complete in the δ-topology.
Let (a_j) be a Cauchy net in Å_ with respect to the δ-topology. In particular, (Δ^n a_j,δ(Δ^n a_j))_j is Cauchy in ⊕̋̅ for all n∈. Since Δ^n and δ are closable on Å_, it follows that there exists a∈ such that (Δ^n a_j,δ(Δ^n a_j))→ (Δ^n a,δ̅(Δ^n a)) for all n∈. In particular, a∈⋂_n∈(Δ^n) and Δ^n a∈(δ̅)=() for all n∈.
Moreover, as the Mackey topology is complete, for n∈ there exists x_n∈ B() such that π_l(Δ^n a_j)→ x_n with respect to τ(B(),B()_∗). For b∈Å_ we have
x_n b=lim_j π_l(Δ^n a_j)b=lim_j π_r(b)Δ^n a_j=π_r(b)Δ^n a.
Since Å_ is dense in , it follows that Δ^n a∈Å_^'' and π_l(Δ^n a_j)→π_l(Δ^n a) for all n∈.
Altogether we conclude that a∈Å_ and a_j→ a in the δ-topology.
If Å is a Tomita algebra, $̋ a Tomita bimodule overÅandδÅ→$̋ a closable symmetric derivation, the inclusion of Å into its completion extends to an injective map from the completion of Å in the δ-topology to .
We have to show that if (a_j) is a Cauchy net in Å with respect to the δ-topology and a_j→ 0 in , then a_j→ 0 in the δ-topology. Since Δ^n, n∈, and δ are closable, we have (Δ^n a_j,δ(Δ^n a_j))→ 0 for all n∈. Furthermore, using the completeness of the Mackey topology and a similar argument as in the previous lemma, one sees that π_l(Δ^n a_j)→ 0 in τ(B(),B()_∗) for all n∈. Hence a_j→ 0 in the δ-topology.
LetÅ^δdenote the set of all elementsa∈for which there exists a net(a_j)inÅsuch thata_j→ainand(a_j)is Cauchy in theδ-topology. By the previous lemma,Å^δis a completion ofÅin theδ-topology, and we call it simply theδ-completion ofÅ. It is not hard to see thatÅ^δis a Tomita subalgebra of(Å^'')_0and contained in(δ̅).
Recall that ifis a normal Tomita bimodule overÅ, we can continuously extend the left and right action ofÅand the mapsand_t,t∈, to the Hilbert completion̋̅. This is usually not possible for_z,z∈∖. We define
^̋a={ξ∈̋̅| t↦_tξ has an entire extension}.
If it exists, this entire extension is unique and will be denoted byz↦_zξ. Clearly,⊂̋^̋aand_z⊂_zfor allz∈.
If we endow ^̋a with the coarsest locally convex topology that makes
^̋a→̋̅, ξ↦_inξ
continuous for all n∈, then ^̋a is complete.
If A is the unique non-singular positive self-adjoint operator in ̋̅ such that _t=A^it for t∈, then ^̋a=⋂_n∈(A^n) and ^̋a is the projective limit of the Banach spaces ((A^n)∩(A^-n),·_̋̅+A^n · _̋̅+A^-n · _̋̅) in the topology described in the lemma. In particular, ^̋a is complete.
Since$̋ is a normal Tomita bimodule over Å, the Hilbert completion ̋̅ has a canonical structure of a π_l(Å)^''-π_l(Å)^'' correspondence determined by
π_l(a)·ξ· Jπ_r(b)^∗ J=aξ b
for a,b∈Å and ξ∈$̋.
Ifa∈Å^δ,(a_j)is a net inÅsuch thata_j→ain theδ-topology andξ∈̋̅, then
_t(π_l(a)·ξ)=lim_j _t(π_l(a_j)ξ)=lim_j π_l(U_t a_j)_tξ=π_l(U_t a)·_tξ.
Thus, ifξ∈^̋a, thenz↦π_l(U_z a)·_zξis an entire continuation oft↦_t(π_l(a)·ξ), which impliesπ_l(a)ξ∈^̋a. Likewise, ifb∈Å^δ, thenξ·Jπ_r(b)^∗J∈^̋a. It is then routine to check that the bimodule structure given byaξb=π_l(a)·ξ·Jπ_r(b)^∗J, then group(_z)_z∈and the restriction ofmake^̋ainto a Tomita bimodule overÅ^δ.
If Å is a Tomita algebra, $̋ is a Tomita bimodule overÅandδÅ→$̋ is a closable symmetric derivation with closure δ̅, then δ̅(Å^δ)⊂^̋a and δ̅Å^δ→^̋a is a symmetric derivation.
If a∈Å^δ and (a_j) is a net in Å such that a_j→ a in the δ-topology, then
_tδ̅(a)=lim_j _t δ(a_j)=δ(U_t a_j)=δ̅(U_t a).
It follows that t↦_tδ̅(a) has the entire continuation z↦δ̅(U_z a), which implies δ̅(a)∈̋̅^a. Again, routine computations show that the restriction of δ̅ to Å^δ is a symmetric derivation from Å^δ to ^̋a.
§ CLOSABILITY OF DERIVATIONS
In this section we give a simple criterion for the closability of derivations inspired by a well-known result (see <cit.> and <cit.> for the non-tracial case) on the closability of the derivation used in free probability.
IfÅis a Tomita algebra and$̋ is a Tomita bimodule over Å, we say that ξ∈$̋ is a bounded vector if there existsC>0such thataξb≤Cabfor alla,b∈Å. In this case, the mapsa↦aξandb↦ξbextend to bounded linear operators from the completionofÅto$̋, which we denote by R(ξ) and L(ξ), respectively.
Let Å be a Tomita algebra, $̋ a normal Tomita bimodule overÅandδÅ→$̋ a derivation. If δ(Å) is contained in the space of bounded vectors, then (δ^∗) is a subbimodule of $̋ and
δ^∗(aξ b)=a^∗δ^∗(ξ)b-L(δ(a^∗))^∗(ξ b)-R(δ(b^∗))^∗ (aξ)
fora,b∈Åandξ∈(δ^∗).
Let a,b,c∈Å and ξ∈(δ^∗). By the product rule,
⟨ aξ b,δ(c)⟩ =⟨ξ,a^∗δ(c)b^∗⟩
=⟨ξ,δ(a^∗ c b^∗)-δ(a^∗)cb^∗-a^∗ cδ(b^∗)⟩
=⟨ aδ^∗(ξ)b-L(δ(a^∗))^∗(ξ b)-R(δ(b^∗))^∗ (aξ),c⟩.
Thus aξ b∈(δ^∗) and the claimed identity for δ^∗(aξ b) holds.
Let Å be a Tomita algebra, $̋ a normal Tomita bimodule over$̋ and δÅ→$̋ a derivation. Ifδ(Å)is contained in the space of bounded vectors and(δ^∗)is a cyclic subset, thenδis closable.
By the previous lemma, (δ^∗) is a subbimodule of $̋. Hence, if(δ^∗)is cyclic, then it is dense in$̋. Therefore, δ is closable.
§ COMPLETELY DIRICHLET FORMS ASSOCIATED WITH CLOSABLE DERIVATIONS
In this section we prove the main theorem of this article, Theorem <ref>, showing that the closure of the quadratic form associated with a closable symmetric derivation is a modular completely Dirichlet form.
As mentioned in the introduction, we rely on Haagerup's reduction method. To set up the stage for its use, we first discuss crossed products of Tomita algebras and Tomita bimodules. To extend closable symmetric derivations to a sufficiently large domains on the crossed product, we use theδ-completion technique developed in Section <ref>. Further, to reduce the problem to the tracial case, we need a “change of reference weight” argument and an analysis of approximation properties of completely Dirichlet forms. This will be dealt with in the following lemmas. Finally, in Proposition <ref> we discuss the relation between the derivation we started with and the first-order differential structure of the associated completely Dirichlet form.
LetÅbe a Tomita algebra,$̋ a normal Tomita bimodule over Å and δÅ→$̋ a closable symmetric derivation. Throughout this section we endowÅwith theδ-topology and$̋ with the projective topology induced by the maps →̋̋̅, ξ↦_inξ for n∈, and we assume that Å and $̋ are complete in these topologies. As discussed in Section <ref>, this can always be achieved by passing to the completions.
LetGbe a countable subgroup of, viewed as discrete group. The vector spaceC_c(G;Å)≅ C_c(G)⊙Åcan be made into a Tomita algebra by the operations
(a∗ b)(g) =∑_h∈ GU_-ha(g-h)b(h),
a^♯(g) =U_-g(a(-g)^♯),
(U_z a)(g) =U_z a(g).
Moreover, the vector spaceC_c(G;)̋becomes a normal Tomita bimodule overC_c(G;Å)with the operations
(aξ)(g) =∑_h∈ GU_-h a(g-h)ξ(h)
(ξ b)(g) =∑_h∈ G_-hξ(g-h)b(h)
(ξ)(g) =_-gξ(-g)
(_z ξ)(g) =_z ξ(g).
Furthermore,1_C_c(G)⊙δ C_c(G;Å)→ C_c(G;)̋is a closable symmetric derivation, whose closure we denote by1⊗δ̅.
We writeÅ̃for the(1⊙δ)-completion ofC_c(G;Å),̋̃forC_c(G;)̋^aandδ̃for the restriction of1⊗δ̅toÅ̃. By Lemma <ref> the mapδ̃is a (closable) symmetric derivation fromÅ̃tő̃.
If x∈ L(G)⊗ 1_̋̅ and a∈Å̃, then x a, a x∈Å̃ and δ̃(x a)=xδ̃(a), δ̃(a x)=δ̃(a)x.
Let x=y⊗ 1 with y∈ L(G), let (y_i) be a bounded net in [G] such that y_i→ y in the strong^∗ topology and let x_i=y_i⊗ 1. Clearly, x_i→ x in the Mackey topology.
If a∈ C_c(G;Å), then x_i a∈ C_c(G;Å) and Δ^n(x_i a)=x_i Δ^n a, (1⊙δ)(x_i a)=x_i(1⊙δ)(a), π_l(Δ^n (x_i a))=x_iπ_l(Δ^n a). It follows that x_i a→ xa in ℓ^2(G;), the net (x_i a) is Cauchy in the δ̃-topology and (1⊙δ)(x_i a)→ x(1⊙δ)(a). Thus xa∈Å̃ and δ̃(xa)=xδ̃(a).
A similar argument shows that if (a_j) is a Cauchy net in C_c(G;Å) with respect to 𝒯_δ̃ and a_j→ a in ℓ^2(G;)̋, then (x a_j) is Cauchy with respect to 𝒯_δ̃ and xa_j→ xa in ℓ^2(G;)̋. Hence if a∈Å̃, then xa∈Å̃ and δ̃(xa)=xδ̃(a). The statement for ax can be proven analogously.
For the next lemma recall thatÅ_ϕ=Λ_ϕ(_ϕ∩_ϕ^∗)is the full left Hilbert algebra induced by the weightϕ, the coneC_ϕis the closure of{Δ_ϕ^1/4a| a∈Å_ϕ,0≤π_l(a)≤ 1}, andM_ϕdenotes the centralizer ofϕ.
Let M be a von Neumann algebra, ϕ a normal semi-finite faithful weight on M and x∈ M_ϕ be positive and invertible. Let ψ=ϕ(x^1/2· x^1/2).
If is a modular (completely) Dirichlet form on L^2(M) with respect to Å_ϕ, x()⊂() and (xa,b)=(a,xb) for all a,b∈(), then is also a modular (completely) Dirichlet from with respect to Å_ψ.
Since x is invertible, the weight ψ is faithful and J_ψ=J_ϕ, and since x commutes with (Δ_ϕ^it), we have Δ_ψ^it=x^itΔ_ϕ^it(·)x^-it=Δ_ϕ^it(x^it· x^-it).
Let A be the positive self-adjoint operator associated with and T_t=e^-tA. The commutation relation x()⊂() and (xa,b)=(a,xb) for a,b∈() implies that x commutes strongly with A^1/2. Hence T_t(xa)=x T_t(a) for all a∈ H and t≥ 0. Since T_t commutes with J_ϕ, we also have T_t(ax)=T_t(a)x for a∈ H, t≥ 0. In particular, (T_t) and (Δ_ψ^is) commute.
Moreover, C_ψ=x^1/4C_ϕ x^1/4. Indeed, a direct computation shows that _ψ=_ϕ and Λ_ψ(y)=Λ_ϕ(y)x^1/2 for y∈_ϕ. Hence, if y∈_ϕ with 0≤ y≤ 1, then
Δ_ψ^1/4Λ_ψ(y)=x^1/4Δ_ϕ^1/4Δ_ϕ(y)x^1/4∈ x^1/4C_ϕ x^1/4.
The converse inclusion follows by swapping the roles of ϕ and ψ.
Therefore, if a∈ C_ψ, then
T_t(a)=x^1/4T_t(x^-1/4a x^-1/4)x^1/4∈ C_ψ.
Thus is a Dirichlet form with respect to Å_ψ by <cit.>. The result for completely Dirichlet forms follows easily by applying the same argument to the forms ^(n) on L^2(M⊗ M_n()).
Let M be a von Neumann algebra and ϕ a normal semi-finite faithful weight on M. Let (M_n) be an increasing sequence of von Neumann subalgebras with weak^∗ dense union and assume that M_n is the range of a ϕ-preserving conditional expectation E_n on M. Let H_n denote the closure of Λ_ϕ(_ϕ∩ M_n) and let P_n denote the orthogonal projection from H to H_n.
If is a closed densely defined quadratic form on H such that for every n∈ the quadratic form |_H_n is a Dirichlet form with respect to Λ_ϕ(_ϕ∩_ϕ^∗∩ M_n) and ∘ P_n≤, then is a Dirichlet form with respect to Å_ϕ.
Let (T_t) be the strongly continuous semigroup associated with . Since ∘ P_n≤, we have T_t(H_n)⊂ H_n by Ouhabaz' theorem <cit.>. Thus T_t commutes with P_n, and it is easy to see that (T_t P_n) is the semigroup associated with |_H_n, viewed as semigroup on H. In particular, T_t P_n J_ϕ=J_ϕ T_t P_n. In the limit we obtain T_t J_ϕ=J_ϕ T_t.
It remains to show that T_t(C_ϕ)⊂ C_ϕ for all t≥ 0. A direct computation shows that E_n and P_n are related by P_nΛ_ϕ(x)=Λ_ϕ(E_n(x)). Moreover, E_n is GNS-symmetric with respect to ϕ, which implies that P_n commutes with (Δ_ϕ^it). Thus P_n(C_ϕ) is the closure of {Δ_ϕ^1/4Λ_ϕ(x)| x∈_ϕ∩_ϕ^∗∩ M_n, 0≤ x≤ 1}. In particular, P_n(C_ϕ)⊂ C_ϕ.
Since _n is a Dirichlet form with respect to Λ_ϕ(_ϕ∩_ϕ^∗∩ M_n) and (T_t P_n) is the associated semigroup, we have T_t P_n(C_ϕ)⊂ P_n (C_ϕ). Moreover, since ⋃_n M_n is weak^∗ dense in M, we have P_n→ 1 strongly by Kaplansky's density theorem. Therefore, T_t(C_ϕ)⊂ C_ϕ.
To prove that the quadratic form associated with a closable symmetric derivation is a completely Dirichlet form, we will reduce the problem to the tracially symmetric case by means of Haagerup's reduction method. We only recall the necessary definitions here and refer to <cit.> for proofs in the case of states and to <cit.> for the extension to weights.
LetM=π_l(Å)^'', letϕbe the weight induced by the full left Hilbert algebraÅ^''onM, letG=⋃_n∈2^-n, letM̃=M⋊_σ^ϕG=π_l(Å̃)^''and letϕ̃be the dual weight ofϕonM̃. Let(a_n)be a sequence of self-adjoint elements ofL(G)⊗ 1⊂𝒵(M̃_ϕ̃),ϕ_n=ϕ e^-a_n,M_n=M̃_ϕ_nandτ_n=ϕ_n|_M_n. HereN_ψdenotes the centralizer of the weightψonNand𝒵(M)is the center of the von Neumann algebraN.
By <cit.> the sequence(a_n)can be chosen such that
* M_n is semi-finite with normal semi-finite faithful trace τ_n,
* for each n∈ there exists a conditional expectation E_n from M onto M_n such that ϕ̃∘ E_n=ϕ̃ and σ^ϕ̃_t∘ E_n=E_n∘σ^ϕ̃_t for all t∈,
* E_n(x)→ x strongly^∗ for every x∈ M.
In the following we fix a sequence(a_n)with these properties. The concrete construction is irrelevant for our purposes.
Let Å be a Tomita algebra with completion , let $̋ be a normal Tomita bimodule overÅandδÅ→$̋ a closable symmetric derivation. The closure of the quadratic form
→ [0,∞], a↦δ(a)_^̋2 if a∈Å,
∞ otherwise
is a modular completely Dirichlet form with respect to Å^''. If moreover Å is unital, then is a modular quantum Dirichlet form.
We continue to use the notation from the previous discussion. The derivation δ̃Å̃→̋̃ is a restriction of 1⊗δ̅. Let denote the closure of the quadratic form
ℓ^2(G;)→ [0,∞], a↦δ̃(a)_̋̃^2 if a∈Å̃,
∞ otherwise.
It is clear that (a)=(1⊗δ̅)(a)_ℓ^2(G;̋̅)^2 for a∈(). Furthermore, ()=(1⊗δ̅) and the strongly continuous semigroups (T_t) and (T̃_t) associated with and , respectively, are related by T̃_t=𝕀_ℓ^2(G)⊗ T_t.
The map ι→ℓ^2(G;), a↦_0⊗ a is an isometric embedding such that ι(C_Å^'')= C_Å̃^''∩ι(). Thus, if is a (completely) Dirichlet form with respect to Å̃^'', then is a (completely) Dirichlet form with respect to Å^''.
Since M is in standard form on and M̃ is in standard form on ℓ^2(G;), these spaces can be canonically identified with L^2(M,ϕ) and L^2(M̃,ϕ̃), respectively, and we will tacitly do so in the following. Under these identifications, Å^''=Å_ϕ, Å̃^''=Å_ϕ̃ and Δ_ϕ̃^it=𝕀_ℓ^2(G)⊗Δ_ϕ^it.
Let
𝒜_n={x∈_ϕ̃∩_ϕ̃^∗∩ M_n|Λ_ϕ̃(x e^-a_n/2)∈Å̃}.
Since e^a_n/2∈M̃_ϕ̃, if x∈𝒜_n, then
Λ_ϕ̃(x)=Λ_ϕ̃(x e^-a_n/2)e^a_n/2∈Å̃
by Lemma <ref>. Reversing the roles of e^-a_n/2 and e^a_n/2 we get
𝒜_n={x∈_ϕ̃∩_ϕ̃^∗∩ M_n|Λ_ϕ̃(x)∈Å̃}.
Since Å̃ is a Tomita algebra, it follows easily that 𝒜_n is a ∗-algebra. Define an 𝒜_n-𝒜_n-bimodule structure on ℓ^2(G;̋̅) by
xξ y=Λ_ϕ̃(x)·ξ·Λ_ϕ̃^'(y).
Using that ̋̃ is a Tomita bimodule over Å̃, it is not hard to see that this left and right action are contractive (anti-) ∗-homomorphisms. Moreover, extends to an anti-unitary involution on ℓ^2(G;̋̅) intertwining the left and right action. We still denote this extension by .
Let
∂_n𝒜_n→ L^2(M̃,ϕ̃), ∂_n(x)=δ̃(Λ_ϕ̃(xe^-a_n/2)).
Since e^-a_n/2∈M̃_ϕ̃ and x∈M̃_ϕ_n, we have
∂_n(x^∗) =δ̃(Λ_ϕ̃(x^∗ e^-a_n/2))
=δ̃(Λ_ϕ̃^'( e^-a_n/2x^∗))
=δ̃(J̃Λ_ϕ̃(xe^-a_n/2))
=δ̃(Λ_ϕ̃(xe^-a_n/2))
=∂_n(x).
Moreover, it follows from Lemma <ref> combined with e^-a_n/2∈M̃_ϕ̃ and x,y∈M̃_ϕ_n that
∂_n(xy) =δ̃(Λ_ϕ̃(xye^-a_n/2))
=Λ_ϕ̃(x)·δ̃(Λ_ϕ̃(ye^-a_n/2))+δ̃(Λ_ϕ̃(x))·Λ_ϕ̃(y e^-a_n/2)
=Λ_ϕ̃(x)·δ̃(Λ_ϕ̃(ye^-a_n/2))+δ̃(Λ_ϕ̃(xe^-a_n/2))·Λ_ϕ̃(e^a_n/2 y e^-a_n/2)
=Λ_ϕ̃(x)·δ̃(Λ_ϕ̃(ye^-a_n/2))+δ̃(Λ_ϕ̃(xe^-a_n/2))·Λ_ϕ̃^'(σ^ϕ_n_-i/2(y))
=Λ_ϕ̃(x)·δ̃(Λ_ϕ̃(ye^-a_n/2))+δ̃(Λ_ϕ̃(xe^-a_n/2))·Λ_ϕ̃^'(y)
=x∂_n(y)+∂_n(x)y.
The operator ∂_n is closable when viewed as operator in L^2(M_n,τ_n) since δ̃ is closable and the map Λ_τ_n(x)↦Λ_ϕ̃(xe^-a_n/2) extends to an isometry ι_n from L^2(M_n,τ_n) to L^2(M̃,ϕ̃).
Since τ_n is a trace, <cit.> implies that the closure Q_n of the quadratic form
L^2(M_n,τ_n)→ [0,∞], a↦∂_n(x)^2 if a=Λ_τ_n(x), x∈𝒜_n,
∞ otherwise
is a completely Dirichlet form.
Let H_n=Λ_ϕ̃(_ϕ̃∩_ϕ̃^∗∩ M_n) and let _n be the closure of the quadratic form
H_n→ [0,∞], a↦δ̃(a)^2 if a∈Å̃,
∞ otherwise.
In other words, _n=Q_n∘ι_n^-1.
Note that ι_n maps {Λ_τ_n(x)| x∈_ϕ̃∩_ϕ̃^∗∩ M_n, 0≤ x≤ 1} onto {Λ_ϕ̃(x e^-a_n/2)| x∈_ϕ̃∩_ϕ̃^∗∩ M_n, 0≤ x≤ 1}. Since ϕ_n is a trace on M_n, the latter set coincides with {Λ_ϕ_n(x)| x∈Å_ϕ_n∩ H_n, 0≤ x≤ 1}. It follows that _n is a completely Dirichlet form with respect to Å_ϕ_n∩ H_n.
Moreover,
_n(Δ_ϕ_n^it a) =_n(e^-i a_n/2 t(Δ_ϕ̃^it a)e^ia_n/2 t)
=e^-ia_n/2 t(_tδ̃(a))e^ia_n/2 t^2
=δ̃(a)^2
=_n(a)
for a∈Å̃. This can easily be extended to the closure so that _n is a modular completely Dirichlet form.
By Lemma <ref> we have e^-a_n/2(_n)⊂(_n) and _n(e^a_n/2a,b)=_n(a,e^-a_n/2b) for a,b∈(_n). Furthermore, e^-a_n/2∈M̃_ϕ̃. Hence _n is also a modular completely Dirichlet form with respect to Å_ϕ̃∩ H_n=Λ_ϕ̃(_ϕ̃∩_ϕ̃^∗∩ M_n) by Lemma <ref>.
Let P_n denote the orthogonal projection from ℓ^2(G;) onto H_n. By definition, |_H_n=_n. To apply Lemma <ref>, we have to check that ∘ P_n≤.
Let (T_t) be the strongly continuous semigroup associated with . As discussed above, (𝕀_ℓ^2(G)⊗ T_t) is the strongly continuous semigroup associated with . The modular group of ϕ_n is given by Δ_ϕ_n^it=e^-ita_n(𝕀_ℓ^2(G)⊗Δ_ϕ^it)(·)e^ita_n. Since (T_t) commutes with (Δ_ϕ^it) and e^a_n∈ L(G)⊗ 1_, the semigroup (𝕀_ℓ^2(G)⊗ T_t) commutes with (Δ_ϕ_n^it).
Since M_n is the centralizer of ϕ_n, the subspace H_n is the fixed-point set of (Δ_ϕ̃_n^it). In particular, (𝕀_ℓ^2(G)⊗ T_t)(H_n)⊂ H_n. From Ouhabaz's theorem <cit.> we deduce ∘ P_n≤.
Now Lemma <ref> shows that is a modular completely Dirichlet form with respect to Å_ϕ̃.
If Å is unital with unit 1_Å, then the left and right action of Å on $̋ are unital since they are non-degenerate by definition. Thus
δ(1_Å)=1_Å·δ(1_Å)+δ(1_Å)· 1_Å-δ(1_Å)=0
and hence(1_Å)=0. ThusT_t(1_Å)=0, which implies thatis a quantum Dirichlet form.
In the situation of the previous theorem, we callthe completely Dirichlet form associated with δ.
If Å is not unital, the completely Dirichlet form associated with a derivation is not necessarily a quantum Dirichlet form, even in the commutative case. For example, this is the case for the standard Dirichlet energy (f)=∫_Ω∇ f^2 with domain H^1_0(Ω)∩ L^∞(Ω) if Ω is a bounded Lipschitz domain.
If Å is a unital Tomita algebra, $̋ a normal Tomita bimodule overÅandδÅ→$̋ a closable symmetric derivation with associated completely Dirichlet form , then the first-order differential calculus associated with is a corestriction of (^̋a,δ̅|_Å_). In particular,
δ̅(ab)=aδ̅(b)+δ̅(a)b
for a,b∈Å_.
Since Å is unital, is a modular quantum Dirichlet form by <ref>. Let (_̋,δ_) be a first-order differential calculus associated with . By definition, Å⊂Å_⊂Å^'' and the graph norm of δ̅ coincides with the graph norm of δ_ on Å_. Thus π_l(Å)^'' is strong^∗ dense in π_l(Å_)^'' and Å is a core for δ_. It follows that the linear hull of {δ_(a)b| a,b∈Å} is dense in _̋.
Let
Ulin{δ_(a)b| a,b∈Å}→,̋ U(δ_(a)b)=δ(a)b.
By <cit.> the map U is well-defined and extends to an isometric Å_-bimodule map from ̋̅_ to ̋̅ such that U(δ_(a))=δ(a) for a∈Å.
If a∈Å_, let (a_n) be a sequence in Å such that a_n-a_δ̅→ 0. As discussed above, this implies δ_(a_n)→δ_(a). Hence U(δ_(a))=δ̅(a). If a,b∈Å_, then
δ̅(ab) =U(δ_(ab))
=U(aδ_(b)+δ_(a)b)
=a U(δ_(b))+U(δ_(a))b
=aδ̅(b)+δ̅(a)b.
Moreover, δ J=δ can be extended by continuity to δ̅J=δ̅, and
→̋̅, z↦δ̅(U_z a)
is an entire continuation of t↦_t δ̅(a) for a∈Å_ by <cit.>.
Thus δ̅(Å_)⊂^̋a and δ̅ is a symmetric derivation on Å_. The statement now follows from the uniqueness of the first-order differential calculus associated with a modular completely Dirichlet form <cit.>.
The previous result holds more generally with the same proof if Å is not necessarily unital, but the completely Dirichlet form associated with δ is still a quantum Dirichlet form.
In the light of Lemma <ref>, one has Å^δ⊂Å_ in the situation of the previous proposition. It is an interesting question if one always has equality or if different derivations with δ-complete domains can have the same associated completely Dirichlet form.
§ EXAMPLES
In this section we present several classes of derivations that give rise to modular completely Dirichlet forms according to Theorem <ref>. The first three classes of examples concern inner derivations, before we treat derivations arising in non-tracial free probability in Example <ref> and derivations induced by cocycles on locally compact groups in Example <ref>.
Let Å be a Tomita algebra, $̋ a normal Tomita bimodule overÅandξ∈$̋ be a bounded vector. Assume that there exists ω∈ such that _t ξ=e^iω tξ for all t∈.
The map
δÅ→⊕̋,̋ a↦ i(ξ a-aξ,(ξ)a-a(ξ))
is a bounded derivation, and it is symmetric when ⊕̋$̋ is endowed with the involution(η,ζ)↦ (ζ,η)and the complex one-parameter group(e^-iω z_z,e^iω z_z).
It follows that the closure of the quadratic form
Å→ [0,∞), a↦ξ a-aξ_^̋2+(ξ)a-a(ξ)_^̋2
is a (bounded) modular completely Dirichlet form with respect toÅ^''. In the case=̋Å, this was first proven by Cipriani <cit.>. See also <cit.> for arbitrary Tomita bimodules$̋ over Å.
The next example is a (partial) extension of the previous example allowing for vectors implementing the inner derivation that are not necessarily bounded.
Let Å be a Tomita algebra with Hilbert completion . For ξ∈(Δ^1/2) the operator
π_l^0(ξ)Å→, a↦ξ a
is closable since π_l^0(ξ^♯)⊂π_l^0(ξ)^∗. Likewise, if ξ∈(Δ^-1/2), then
π_r^0(ξ)Å→, a↦ aξ
is closable with π_r^0(ξ^♭)⊂π_r^0(ξ)^∗. Hence if ξ∈(Δ^-1/2)∩(Δ^1/2), then π_l^0(ξ)-π_r^0(ξ) is closable with π_l^0(ξ^♯)-π_r^0(ξ^♭)⊂ (π_l^0(ξ)-π_r^0(ξ))^∗.
Now assume that there exists ω∈ such that Δ^itξ=e^iω tξ for all t∈. This implies in particular ξ∈(Δ^-1/2)∩(Δ^1/2).
Similar to the last example, one can turn Å⊕Å into a Tomita bimodule over Å if one equips it with the usual bimodule structure, the involution (η,ζ)↦ (Jζ,Jη) and the complex one-parameter group (e^-iω zU_z,e^iω zU_z).Then the map
δÅ→Å⊕Å, a↦ i(ξ a-aξ,(Jξ)a-a(Jξ))
is a closable symmetric derivation.
Thus the closure of the quadratic form
Å→ [0,∞), a↦ξ a-aξ_^2+(Jξ)a-a(Jξ)_^2
is a modular completely Dirichlet form with respect to Å^''. This result has first been obtained by Cipriani and Zegarlinski <cit.>.
The previous examples require eigenvectors of the modular group to construct a symmetric derivation, which may be hard to find. In the following examples we show that in certain situations one can start with an arbitrary element if one “averages” the action of the modular group to ensure modularity.
Let M be a von Neumann algebra with separable predual. A normal semi-finite weight faithful weight ϕ on M is called integrable<cit.> if
_ϕ={x∈ M: ∫_σ^ϕ_t(x^∗ x) dt exists in the σ-strong topology}
is weak^∗ dense in M.
If ϕ is integrable, the set
Å={Λ_ϕ(x)| x∈ M analytic for σ^ϕ, σ^ϕ_z(x)∈_ϕ∩_ϕ^∗∩_ϕ∩_ϕ^∗ for all z∈}
is a Tomita subalgebra of (Å_ϕ)_0 with Hilbert completion L^2(M) and π_l(Å)^''=M, as can be seen from <cit.> together with a standard mollifying argument.
Let (V_t) be the translation group on L^2(), that is, V_t f(s)=f(s+t), and let L^2()^a be the set of all entire analytic elements for (V_t). Endow L^2()^a⊙Å with the left and right action of Å given by a(f⊗ b)c=f⊗ abc, the complex one-parameter group (V_z⊙ U_z)_z∈ and the involution f⊗ a↦f̅⊗ Ja. It can be checked that this makes L^2()^a⊙Å into a normal Tomita bimodule, which we denote by $̋.
Leta∈(Δ_ϕ^1/2)∩(Δ_ϕ^-1/2)withJa=aand define
δÅ→ L^2(;L^2(M)), δ(b)(s)=(U_-sa)b-b(U_-sa).
We have
δ(U_t b)(s) =(U_-sa)(U_t b)-(U_t b)(U_-sa)
=U_t((U_-(s+t)a)b-b(U_-(s+t)a))
=U_t δ(b)(s+t).
Thusδ∘ U_t=(U_t⊗ V_t)∘δ. In particular,δmaps into^̋a.
It is not hard to check thatδÅ→^̋ais a symmetric derivation. To show closability, first note that for every fixeds∈the mapb↦δ(b)(s)is closable as seen in the previous example. Ifb_n→ 0andδ(b_n)→ξ, then there exists a subsequence such thatδ(b_n_k)(s)→ξ(s)for a.e.s∈. Closability of the mapb↦δ(b)(s)impliesξ(s)=0for a.e.s∈, which proves the closability ofδ.
Thus the closure of the quadratic form
Å→ [0,∞), b↦∫_(U_-sa)b-b(U_-sa)^2 ds
is a modular completely Dirichlet form.
If we drop the assumptionJa=a, a similar argument shows that the closure of the quadratic form
Å→ [0,∞), b↦∫_((U_-sa)b-b(U_-sa)^2+(U_-sJa)b+b(U_-sJa)^2) ds
is a modular completely Dirichlet form with respect toÅ^''.
A similar construction is possible if one starts with a weight with periodic modular group instead of an integrable weight and integrates over a period of the modular group.
The following class of examples of derivations was introduced by Nelson <cit.> in the context of non-tracial free probability.
Let M be a von Neumann algebra, ϕ a normal faithful state on M and B⊂ M a ∗-subalgebra. Let ∂ B→ M⊗ M be a linear map such that
∂(xy)=(x⊗ 1)·∂(y)+∂(x)· (1⊗ y)
for x,y∈ B. Note that Nelson works with M⊗ M^ instead, but under the identification x⊗ y↦ x⊗ y^, the M-bimodules M⊗ M and M⊗ M^ (with the bimodule structure used in <cit.>) are isomorphic.
Let ω∈ and write M_∞ for the set of entire analytic elements for σ^ϕ. Nelson <cit.> calls the map ∂ an e^ω-modular derivation if B⊂ M_∞, B is invariant under σ^ϕ_z for all z∈, ∂(B)⊂ M_∞⊙ M_∞ and
∂(σ^ϕ_z(x))=e^iω z(σ^ϕ_z⊗σ^ϕ_z)(∂(x))
for all x∈ B and z∈.
One example given by Nelson is the free difference quotient from free probability (see <cit.> in the non-tracial case). Given a ∗-subalgebra B of M and an element a∈ M that is algebraically free from B (and a^∗ is algebraically free from a if a≠ a^∗), let
∂_a B[a]→ B[a]⊙ B[a], ∂_a(a)=1⊗ 1, δ|_B=0
(and δ_a(a^∗)=0 if a^∗≠ a). If a is an eigenvector of Δ_ϕ to the eigenvalue e^ω, then ∂_a is an e^ω-modular derivation.
Let us see how an e^ω derivation gives rise to a symmetric derivation in our sense. For x,y∈ M let (x⊗ y)^†=y^∗⊗ x^∗. The conjugate derivation of ∂ is the map
∂̂ B→ M_∞⊙ M_∞, ∂̂(x)=∂(x^∗)^†.
Let Å=Λ_ϕ(B). Since B is consists of the analytic elements for σ^ϕ and is invariant under σ^ϕ_z for z∈, the set Å is a Tomita subalgebra of (Å_ϕ)_0=Λ_ϕ(M_∞).
Let =̋(Λ_ϕ(M_∞)⊙Λ_ϕ(M_∞))^⊕ 2 with left and right action of Å given by
a(ξ_1⊗η_1,ξ_2⊗η_2)b=(aξ_1 ⊗η_1 b,aξ_2 ⊗η_2 b),
involution given by (ξ_1⊗η_1,ξ_2⊗η_2)↦ (Jη_2⊗ Jξ_2,Jη_1⊗ Jξ_1), and complex one-parameter group (_z)=(e^iω zΔ_ϕ⊗ϕ^iz,e^-iω zΔ_ϕ⊗ϕ^iz). One can check that this makes $̋ into a normal Tomita bimodule overÅ.
Let
δÅ→,̋ δ(Λ_ϕ(x))=(Λ_ϕ⊗ϕ(∂(x)),Λ_ϕ⊗ϕ(∂^†(x))).
The product rule for∂and∂̂translate to the product rule forδ, thee^ωmodularity of∂ensuresδ∘Δ_ϕ^iz=_z ∘δand the definition of∂̂andare tailored to guaranteeδ∘ J_ϕ=∘δ.
All of these properties follow by routine calculations, let us just show the product rule (for the first component of)δas illustration. Letδ_1(Λ_ϕ(x))=Λ_ϕ⊗ϕ(∂(x)). By the product rule for∂we have
δ_1(Λ_ϕ(xy)) =Λ_ϕ⊗ϕ((x⊗ 1)∂(y)+∂(x)(1⊗ y))
=Λ_ϕ⊗ϕ((x⊗ 1)Λ_ϕ⊗ϕ(∂(y))+∂(x))(1⊗σ^ϕ_-i/2(y))
=(π_l(Λ_ϕ(x))⊗ 1)δ_1(Λ_ϕ(y))+(1⊗π_r(Λ_ϕ(y)))δ_1(Λ_ϕ(x)).
Thus, ifδis closable, the closure of the associated quadratic form is a completely Dirichlet form with respect toÅ_ϕon the GNS Hilbert spaceL^2(M,ϕ).
To compare that to the result of Nelson, he showed <cit.> that one gets a completely Dirichlet form on the GNS Hilbert spaceL^2(M_ϕ,ϕ)of the centralizerM_ϕofϕ, which is of course a tracial von Neumann algebra.
Our methods allow to extend this result to the “fully” non-tracial setting in that we obtain a modular completely Dirichlet form on the GNS Hilbert space ofMon not just of the centralizer. Note however that Nelson's definition of the mapδbetweenL^2spaces seems slightly different, owing to the use ofM⊗M^instead ofM⊗M.
The last example concerns group von Neumann algebras. The case of discrete groups was treated in <cit.>, but to cover general locally compact groups, possibly non-unimodular, one needs the theory for non-tracial reference weights as developed here.
Let G be a locally compact group with left Haar measure μ and modular function Δ_G. As discussed in <cit.>, the space C_c(G) of compactly supported continuous function on G with the L^2 inner product, the convolution product, the involution f^♯(g)=Δ_G(g)^-1f(g^-1) and the complex one-parameter group U_z f(g)=Δ_G(g)^izf(g) forms a Tomita algebra. We write λ and ρ for the associated left and right action of C_c(G) on L^2(G) and Å_G for the associated full left Hilbert algebra.
Let π be a strongly continuous orthogonal representation of G on the real Hilbert space H. A continuous map b G→ H is called 1-cocycle if b(gh)=b(g)+π(g)b(h) for all g,h∈ G. We extend π to a unitary representation of G on the complexification H^ of H and write ξ↦ξ̅ for the anti-unitary involution induced by H⊂ H^.
On C_c(G;H^) define a left and right action of C_c(G) by
(f∗ξ)(g) =∫_G f(h)π(h)ξ(h^-1g) dμ(h)
(ξ∗ f)(g) =∫_G f(h^-1g)ξ(h) dμ(h),
an anti-unitary involution by (ξ)(g)=-Δ_G(g)^-1/2π(g)ξ(g^-1) and a complex one-parameter group by _z ξ(g)=Δ_G(g)^izξ(g). One can check that C_c(G;H^) with this operations is a Tomita bimodule over C_c(G).
Let
δ C_c(G)→ C_c(G;H^), δ(f)(g)=f(g)b(g).
Using the cocycle property of b, one gets
δ(f_1∗ f_2)(g) =∫_G f_1(h)f_2(h^-1g) dμ(h) b(g)
=∫_G f_1(h)f_2(h^-1g)(π(h)b(h^-1g)+b(h)) dμ(h)
=(f_1∗δ(f_2))(g)+(δ(f_1)∗ f_2)(g).
It is readily verified that δ also satisfies δ∘ J=∘δ and δ∘ U_z=_z∘δ for all z∈. Hence δ is a symmetric derivation. As a multiplication operator, it is clearly closable.
Therefore,
L^2(G,μ)→ [0,∞], (f)=∫_G f(g)^2b(g)^2 dμ(g)
is a modular completely Dirichlet form with respect to Å_G. The associated quantum dynamical semigroup on L(G) is given by
P_t(∫_G x̂(g)λ(g) dμ(g))=∫_G e^-tb(g)^2x̂(g)λ(g) dμ(g).
In this case, complete positivity of P_t also follows directly from Schönberg's theorem as g↦b(g)^2 is a conditionally negative definite function on G.
[article]citetitle#1[article]title#1 |
http://arxiv.org/abs/2307.08613v1 | 20230714150421 | Brain in the Dark: Design Principles for Neuro-mimetic Learning and Inference | [
"Mehran H. Bazargani",
"Szymon Urbas",
"Karl Friston"
] | cs.NE | [
"cs.NE",
"cs.LG"
] |
[
Brain in the Dark: Design Principles for Neuro-mimetic Learning and Inference
equal*
Mehran H. Bazarganiequal,comp
Szymon Urbasequal,stats
Karl Fristonactinf
compSchool of Computer Science, University College Dublin (UCD), Dublin, Ireland
statsDepartment of Mathematics and Statistics, University College Dublin (UCD), Dublin, Ireland
actinfWellcome Centre for Human Neuroimaging, Institute of Neurology, University College London (UCL), London, UK
Mehran H. [email protected]
Machine Learning, ICML
0.3in
]
Even though the brain operates in pure darkness—within the skull—it can infer the most likely causes of its sensory input. An approach to modelling this inference is to assume that the brain has a generative model of the world, which it can invert to infer the hidden causes behind its sensory stimuli, that is, perception. This assumption raises key questions: how to formulate the problem of designing brain-inspired generative models, how to invert them for the tasks of inference and learning, what is the appropriate loss function to be optimised, and, most importantly, what are the different choices of mean field approximations (MFA) and their implications for variational inference (VI).
§ INTRODUCTION
It is remarkable that even though the brain resides in pure darkness in our skull, it is still capable of understanding and analysing the world out there, plan for an unseen future and even make decisions that could affect and change the world. For decades, there has been a popular view of the brain as a predictive machine that is constantly inferring the hidden causes behind its sensory inputs.
This view dates back to Helmholtz <cit.>, who proposed the idea of “perception as unconscious inference”– a view that has emerged as the “Bayesian brain” hypothesis <cit.>. This approach formulates perception as an inferential process based on a generative model of how the brain believes its sensations are generated, where the brain is thought of a statistical organ that updates probabilistic beliefs about the external states of the world, given the observed sensory data. This formulation appeals to Bayes’ rule, which allows for optimal belief updates, given the sensory stimuli <cit.>. More technically, given a sensory observation, o, the goal of perception is to infer the most likely hidden cause, s, which led to this observation, which can be formulated through the Bayes’ theorem.
In order to define an appropriate generative model—and its inversion, one needs to consider several aspects of the problem at hand. Crucially, one needs to address some foundational questions: are we dealing with continuous or discrete hidden states? And are we looking at continuous or discrete time? Is the task of inference limited to the parameters of the generative model (i.e. learning), or just the hidden states (i.e. inference), or both? What is the most suitable objective function—whose optimisation entails learning and inference—and how to extremise it? Does one commit to a functional form for the posteriors? Is there a role for mean-field approximations (MFA)? Should one use sampling schemes or analytic variational inference (VI)? and so on. In this paper, we offer a detailed investigation of these questions and provide a road map towards an accurate and efficient formulation of neuro-mimetic probabilistic generative modelling.
§ VARIOUS PROBLEM FORMULATIONS AND THEIR IMPLICATIONS
There are different problem spaces when designing generative models and the method for their inversion. In this section, we will discuss the implications and general considerations to keep in mind before implementing these models.
§.§ Inference and learning: estimating the hidden states or estimating the parameters
It is important to clarify if the task of model inversion is in the service of inference, i.e. inferring the most likely distribution over the hidden state (assuming fixed/learned model parameters), given some noisy observations, and/or learning the parameters of the generative model as well. Interestingly, the ML community normally focuses on estimating the unknown parameters, where the issue of state estimation is suppressed — it does not matter if the states are unknown random variables (i.e., with random fluctuations), or whether they are fixed variables conditioned on the parameters (i.e. a deterministic State Space Model (SSM)).
We use z to denote the collection of all quantities to be inferred (estimated); e.g. z is the set of hidden states and model parameters for inference and learning. The posterior distribution of z based on all observed data, 𝒟, is obtained through the Bayes' theorem, which states: p(z|𝒟)=p(𝒟|z)p(z)/p(𝒟). Apart from special cases, the posterior is not readily available; the normalising constant p(𝒟) can involve a difficult and often high-dimensional sum or integral and has no closed form. This quantity is often referred to as model evidence. In the VI framework we wish to identify a surrogate distribution q which resembles the true posterior. This approximating posterior is found by using the variational free energy (VFE) <cit.> defined as
F(q;𝒟) = D_KL[q(z)||p(z)]-𝔼_q(z)[ln p(𝒟|z)]
= -𝔼_q(z)[ln p(𝒟,z)/q(z)],
where D_KL is the Kullback-Leibler divergence.
The VFE quantity is the negative of the evidence lower bound (ELBO). Variational inference is based on choosing a distribution q from some prespecified class of distributions. Indeed, if, and only if, we had access to the true posterior p(z|𝒟), the VFE would become exactly zero; if we have two distributions, the one closer to the p(z|𝒟) achieves a lower VFE value — the minimum of VFE can be a proxy for the intractable model evidence, enabling Bayesian model selection. This converts the impossible marginalisation problem into an optimisation problem. As VFE is a functional of q (i.e. it takes in a function and returns a scalar), calculus of variation is used for minimisation <cit.>. By inverting the generative model through VFE minimisation one can accomplish: (i) unknown parameter estimation, where there is no interest in hidden-states estimation, and only parameter estimation is of interest (i.e. learning); and (ii) unknown state and parameter estimation, where model inversion solves a dual estimation problem in partially observed or stochastic systems, where both the hidden states and parameters are estimated (i.e. inference and learning).
§.§ State-space model formulations
For inference, a key question is whether we are working with discrete states or continuous states. We consider a sequence of states s_1:T that we wish to infer, based on potentially noisy observations o^τ:=o_1:T. A hidden Markov model (HMM) is characterised by the following properties: (i) p(s_τ | s_1, s_2,...,s_τ -1)=p(s_τ | s_τ -1) and (ii) p(o_τ | s_1, s_2,...,s_τ)=p(o_τ | s_τ). The former specifies the Markov transitions of the hidden states and the latter the partial observation process. In Appendix C, Fig. 1 <cit.> provides an example of a HMM for inference; this example will be discussed in detail in later sections.
Here, one assumes some particular dynamic generative model composed of: an initial (prior) distribution, i.e. s_1∼μ_θ( · ); a transition mechanism, s_τ|s_τ-1∼ f_θ( · |s_τ-1), τ>1; and an observation (emission) mechanism o_τ|s_τ∼ g_θ( · |s_τ), τ≥1; θ encompasses all model parameters, and we use τ to simply denote any of the time points where variables are generated (τ=1,2,...). By construction, this generative model is a HMM.
Inference of HMM may concern different posterior distributions: (i) p(s_1:τ|o_1:τ, θ) (smoothing); (ii) p(s_τ|o_1:τ, θ) (filtering); or (iii) p(s_τ+1|o_1:τ,θ) (prediction). Here, we suppose the model parameters could be unknown and thus will need to be included in the inference: to inform parameter learning we will require the whole smoothing distribution (at least in principle). An additional complication arises when we carry out this inference online, i.e. using streaming data: e.g. when the brain continuously assimilates data from the sensorium.
To deploy variational inference, we must decide on a particular form of our MFA, q^ψ(s_1:τ, θ)≈ p(s_1:τ, θ|o_1:τ), where the joint distribution q is defined by sufficient statistics or hyperparameters ψ; for example, in discrete-state models these would be probability vectors of a categorical distribution, and equivalently the means and variances of a Laplace approximation in continuous-state models <cit.>. With the notation z=(s_1:τ,θ), the first factorisation is to separate the latent variables and θ, q^ψ(s_1:τ, θ) = q^ψ(s_1:τ)q^ψ(θ). In this general setup, we focus on identifying a θ^* which maximises the ELBO; this can be read as inference through the expectation-maximisation algorithm, or maximum a posteriori estimation if a prior for θ were to be used. In terms of the MFA, we suppose q^ψ(θ) is a Dirac function; we could, however, use a Gaussian family instead <cit.>.
Particular care must be taken when deciding on the MFA form used for the latent variables, q^ψ(s_1:τ). The ELBO at time τ, as a function of model and MFA parameters, is
L_τ(θ,ψ):=𝔼_q^ψ(s_1:τ|o^τ)[ln(p(s_1:τ, o_1:τ|θ)/q^ψ(s_1:τ|o^τ))];
this is our objective function based on all the information available at that time under a particular form for the MFA. The goal of the scheme is to obtain max_θ,ψ L_τ(θ,ψ) at each time point τ as the data appear. If ψ and θ are specified to be real-valued, this will require the gradients ∇_θ L_τ(θ,ψ) and ∇_ψ L_τ(θ,ψ).
The ensuing Bayesian belief updating for filtering or smoothing will have different functional forms depending on the type of state space in question: continuous states or discrete states.
§.§.§ Discrete State Space Models
Suppose the hidden state, s_τ, can take K possible values. In this case, the initial prior probability regarding the hidden state is encoded in a K-dimensional vector D expressed as a categorical distribution at time τ = 1 as P(s_1). Since both the states and outcomes are categorical variables, the likelihood has also a categorical distribution, parameterised by the K× K matrix A: P(o_τ) = 𝖢𝖺𝗍(A) where A_ij = P(o_τ = i|s_τ = j). Furthermore, the transition probability for the states is parameterised by the K × K matrix B: P(s_τ + 1 | s_τ) = 𝖢𝖺𝗍(B_τ). Finally, one can compute the prior over the sequence of hidden states, denoted as s̃, using the prior over the initial state, expressed by vector D, and state transition beliefs denoted by the matrix B: P(s̃)=P(s_1)∏_τ=1^ P(s_τ+1| s_τ). The prior, likelihood, and state transitions probabilities together, constitute the HMM generative model for leaning and inference. The learning is possible since by defining priors over the parameters of the model, we can now update these belief (See Fig. 2 in Appendix C). Here, we present a finite state-space but the methods covered in this work could be applied to infinite state-space problems with an appropriate transition distributions motivated by an assumed data-generating model (e.g. random walk on all integers, s_τ∈ℤ); the VFE construction would involve the same steps.
§.§.§ Continuous State Space Models
As opposed to the discrete SSM, in a continuous state space model, both the states and observations can take real continuous values. Using the following pair of stochastic equations, at a given time τ, we can determine how hidden states, s_τ, generate observation, o_τ, and how states evolve over time, as parametrised by ν_τ:
ṡ_τ = f(s_τ, ν_τ) + ω_τ^1 o_τ = g(s_τ) + ω_τ^2,
where ṡ_τ is the first-order time derivative of the hidden state s_τ, representing the rate of change of the hidden state. Furthermore, ω_τ^1 and ω_τ^2 represent the random fluctuations corresponding to the states and observations, respectively; the two random processes are assumed to be independent (e.g. <cit.>). In the most basic case, these could be Wiener processes, with independent increments, but other smoother processes such as the Matérn process could be used here <cit.> The first equation describes the evolution of hidden states over time through a deterministic function f(s_τ, ν_τ) and stochastic fluctuations ω_τ^1. Here, we suppose the evolution of the hidden states can be modelled as differential equations, i.e. the change from τ=1 to τ=2 comprises infinitesimally small increments in time. The second equation expresses how the observations are believed to be generated from the hidden state. Interestingly, if we assume the fluctuations to be normally distributed, these two equations form a generative model that underwrites Kalman-Bucy filter <cit.> in engineering literature.
It is interesting to note that, even though we are observing this continuous state space model at discrete times, the underlying dynamics of the system are continuous in time (e.g. the evolution of the hidden states, VFE minimisation, etc.). By collapsing the hidden states and their motion into one state variable s_τ={ṡ_τ, s_τ}, the approximate posterior, q, can now be written as q(s_τ) where s_τ is now an augmented variable. Then the standard VFE can be derived and minimised in the usual way, during the time intervals between observations. More specifically, after observing o_τ, we can minimise the integration of point estimates of VFE along a continuous time interval, T, until the next observation o_τ + T. This quantity is called Free Action and is defined as 𝒜[q(s)]=∫_τ^τ+TVFE[q(s_t)] dt, and it is an upper bound on the accumulated surprise, -ln(P(o))=-∫_τ^τ+Tln(P(o_t)) dt, over the same time period. Thus, by minimising 𝒜 in-between observations, the generative model is constantly minimising VFE of a path of length T, and thus continuously striving to improve the estimation of the posterior over the hidden states and/or parameters.
Interestingly, random fluctuations in the data-generating mechanism, ω, are generally assumed to have uncorrelated increments over time (i.e. Wiener assumption), however, in most complex systems (e.g. biological systems)—where the random fluctuations themselves are generated by some underlying dynamical —they possess a certain degree of smoothness. Indeed, by relaxing the Wiener assumption and imposing smoothness on the model functions f and g, we have the opportunity to not only consider the rate of change of the hidden state and the observation, but also their corresponding higher order temporal derivatives (i.e. acceleration, jerk, etc.); see, for example, <cit.>. The resultant pair of {s, ṡ, s̈,...} and {o, ȯ, ö,...} are called the generalised coordinates of motion <cit.>, which provides an opportunity for further capturing the dynamics that govern the evolution the hidden states and observations. An estimated trajectory over time can be calculated using a Taylor series expansion around the present time, which results in a function that can extrapolate to the near future as well as the recent past. Now, the mapping from continuous to discrete time is possible using this expansion, where one can map from the generalized coordinates of motion to the discretised time.
§.§ Perception modelling: online variational inference in practice
Let us now examine some choices for the MFA and practical considerations that may arise. The simplest, and perhaps naive, choice is the fully decoupled factorisation q^ψ(s_1:τ) = ∏_t=1^τ q^ψ_t(s_t). However, a more natural option — motivated by the Markov process —- is q^ψ(s_1:τ) = q^ψ_1(s_1)∏_t=2^τ q^ψ_t(s_t|s_t-1). Unfortunately, these two MFAs do not readily lend themselves to online inference: the ELBO in these cases involve an integral over the true filtering distributions p_θ(s_t-1|o_t-1) which itself has no closed form <cit.>. Instead, an approximation to the ELBO can be used: L_τ(θ,ψ) = ∑_t=1^τ𝔼_q_t^ψ(s_t-1,s_t)[lnf_θ(s_t|s_t-1)g_θ(o_t|s_t)/q_t^ψ(s_t-1,s_t)]; this allows for gradient calculations at a constant computational cost at each time point. Where L_τ is an expectation over the full approximate smoothing distribution of s_1:τ, L_τ is instead a sum of pairwise expectations. However, it can be shown that L_τ≤ L_τ — VFE will not be truly minimised. One way around that is to employ a reversed version of the MFA, q^ψ(s_1:τ) = q^ψ_τ(s_τ)∏_t=1^τ q^ψ_t(s_t-1|s_t); the formulation is for the mathematical convenience, and does not change the original HMM. As outlined in Proposition 1 of <cit.>, the ELBO under this MFA has a recursive form: L_τ(θ,ψ) = 𝔼_q_τ^ψ(s_τ)[V_τ^θ,ψ(s_τ)], where V_τ^θ,ψ(s_τ) = 𝔼_q^ψ(s_1:τ-1|s_τ)[ln p(s_1:τ,o^τ|θ)/q^ψ(s_1:τ)]; it is free of the problematic integral which appears for other MFA options, and V_τ can be expressed in terms of V_τ-1 — VFE calculations at time τ reuse the quantities from the previous time point. The MFA parameters are indexed with a subscript; calculations may involve the current iterations MFA, ψ_τ, as well as the previous iteration ψ_τ-1.
We now revisit the discrete state-space example. In this setting, each component of the MFA, q_t(s_t), t=1,…, τ, at time τ, is a categorical distribution with a probability vector π_τ^t = (π_τ^t(1),...,π_τ^t(K))[The expectations with respect to distributions q_t will take the form of summations; e.g. 𝔼_q^ψ(s_t| o^τ)[h(s_t)]=∑_k=1^K π_τ^t(k) h(k), for some integrable function h.], where P(s_t=k|q_t,o_τ)=π_τ^t(k), the probability of the hidden state at time t≤τ having value k, conditional on the information available at time τ; Appendix <ref> gives the exact parametrisation for this vector through ψ. The transition matrix 𝐁 and emission matrix 𝐀 follow similar notation where [𝐁]_ij = β^i(j) and [𝐀]_ij = α^i(j); Appendix <ref> details the parameterisation of the model through θ.
Learning and inference based on streaming data, through optimising the exact ELBO (or equivalently VFE) involves gradient-based updates in between the receipt of packets of data; e.g. suppose, 5 seconds after receiving o_τ, o_τ+1 appears, and that this permits 80 updates on ψ followed by 50 updates on θ. Using the reverse version of MFA, the gradients in the updates on ψ will have the form ∇_θ L_τ(θ,ψ) = ∑_l=1^K π_τ^τ(l)U_τ^θ,ψ(l), which is calculated by recursion
U_t^θ,ψ(l) = ∑_k=1^K π_t^t-1(k)[U_t-1^θ,ψ(l) + u_t^θ(k,l)],
U_1^θ,ψ(l) = ∇_θlnμ(l)g_θ(o_1|l)= ∇_θlnα^l(o_1),
u_t^θ(k,l) = ∇_θln f_θ(l|k)g_θ(o_t|l)= ∇_θlnβ^k(l)α^l(o_t),
where t=2,…,τ <cit.>.
To focus on the online inference, we only update q_τ and q_τ-1 when a new observation comes in at time τ.[For example, at time τ=4, we infer the current hidden state, s_4 and use this information to improve our hidden-state inference for the previous time point, s_3. Then at time 5, we infer s_5 and improve inference on s_4, without changing the posterior approximation of s_3; and so on.] This is akin to only updating our short-term memory along with what we currently perceive, leaving the long-term memory fixed. The gradient of the ELBO with respect to the state-space MFA parameters can also be computed recursively, ∇_ψ_τ L_τ(θ,ψ) = ∇_ψ_τ∑_l=1^K π_τ^τ(l)V_τ^θ,ψ_1:τ(l), where
V_t^θ,ψ_1:t(l) = ∑_k=1^K π_t^t-1(k)[V_t-1^θ,ψ_1:(t-1)(k) + v_t^θ,ψ_1:t(k,l)],
V_1^θ,ψ_1(l) = lnμ(l)g_θ(o_1|l)
= lnμ(l)α^l(o_1),
v_t^θ,ψ_1:t(k,l)=lnf_θ(l|k)g_θ(o_t|l)/m_t^ψ_1:t(l|k)=lnβ^k(l)α^l(o_t)/m_t^ψ_1:t(l|k),
where t=2,…,τ, and the conditional m_t^ψ_1:t quantity is detailed in Appendix <ref>.
§ CONCLUSION
A generative model can help us model our beliefs about the data generating process in the world, given uncertain observations. It is by inverting the generative model that we can estimate: 1) the hidden states that cause these observations, and 2) the parameters of the generative model to explain how observations are caused.
This paper offers a comprehensive guide on designing and inverting generative models for both inference and learning, as well as loss function selection and most importantly, different choices of mean-field approximation (MFA) for variational inference. We have illustrated the discrete SSM; however, the foundational concepts are transferable to the continuous SSM.
§ FUTURE WORK
We are planning to develop a brain-inspired (neuro-mimetic) framework for inverting generative or world models, for the task of perception. The framework of choice is called Predictive Coding (PC), which provides a powerful mathematical framework for describing how the cortex extracts information from noisy stimuli <cit.>. PC assumes that the brain entails a generative model of the world, under which it constantly makes predictions about the hidden causes behind sensory inputs.
PC is a special case of variational inference where it is assumed that the mean-field factors and posterior probabilities follow Gaussian and Dirac distributions, respectively. Because PC can be formulated as variational inference — and VFE provides a bound on model evidence—one can use Bayesian model comparison to evaluate different MFA factorisations.
We are also aiming to finesse the variational inference process by capturing higher-order temporal derivatives of hidden states and observations in generalised coordinates of motion. Using this generalised dynamics, variational inference can, in principle, provide a more accurate and efficient estimation of the true posterior over the hidden states, especially in on-line learning under analytic (i.e. smooth) random fluctuations.
§ ACKNOWLEDGEMENTS
Mehran H. Bazargani is supported by Enterprise Ireland and the Department of Business, Enterprise and Innovation through the Disruptive Technologies Innovation Fund (DT 2018 0185A) and the Science Foundation Ireland through the Insight Centre for Data Analytics (12/RC/2289_P2).
Szymon Urbas is supported by funding from Science Foundation Ireland and the Department of Agriculture, Food and Marine on behalf of the Government of Ireland under Grant Number [16/RC/3835] - VistaMilk.
Karl Friston is supported by funding for the Wellcome Centre for Human Neuroimaging (Ref: 205103/Z/16/Z), a Canada-UK Artificial Intelligence Initiative (Ref: ES/T01279X/1) and the European Union’s Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3).
icml2023
§ MFA CHANGING WITH TIME
In the setting of streaming data the mean-field approximation is being augmented each time new data arrives. We consider an MFA of the form
q^ψ(s_1:τ) = q_τ^ψ(s_τ)∏_t=1^τ-1 q_t^ψ(s_t|s_t+1)
which allows an update from q^ψ(s_1:τ) to q^ψ(s_1:τ+1) via
q^ψ(s_1:τ+1) = q^ψ(s_1:τ) m^ψ_τ+1(s_τ+1|s_τ) m_τ+1^ψ(s_τ+1|s_τ) = q_τ+1^ψ(s_τ|s_τ+1)q_τ+1^ψ(s_τ+1)/q_τ^ψ(s_τ).
In the online inference of the discrete state-space model considered in the main article, we update the MFA hyperparameters, that is, there is no single ψ used throughout and instead we have a sequence ψ_1:τ which itself is augmented at each time point. The gradients of the ELBO (and equivalently VFE) will involve the conditional quantities
m_t+1^ψ_1:t+1(s_t+1|s_t) =q_t+1^ψ_1:t+1(s_t|s_t+1)q_t+1^ψ_1:t+1(s_t+1)/q_t^ψ_1:t(s_t) (t=1,...,τ-1)
=π_t+1^t(s_t)π_t+1^t+1(s_t+1)/π_t^t(s_t),
where the second equality follows for the discrete state space model.
§ DISCRETE STATE SPACE MODEL PARAMETRISATION
In order to allow gradient-based updates on the parameters governing the mean-field approximation q^ψ(s_1:τ), we use the following parameterisation: π_τ^t(k)=exp(ρ_τ^t(k))/∑_lexp(ρ_τ^t(l)), t=1,...,τ, where ρ_τ^t(1)=0 and ρ_τ^t(k)∈ℝ for all τ, t and k≠1. The constraint on the first element of the row vector ensures identifiability. With this notation, the MFA parameters at time τ are ψ_τ=(ρ_τ^1,...,ρ_τ^τ)^⊤.
The transition matrix 𝐁 and emission matrix 𝐀 follow similar a parametrisation where [𝐁]_ij = β^i(j) = exp(β̃^i(j))/∑_kexp(β̃^i(k)) and [𝐀]_ij = α^i(j) = exp(α̃^i(j))/∑_kexp(α̃^i(k)), with β̃^i(1)=0 and α̃^i(1)=0, and α̃^i(k),β̃^i(k)∈ℝ for k=2,...,K, for all i. We suppose a fixed initial prior on the process s_t, that is, s_1∼μ which itself is free of θ. The model parameters are θ = (α̃^1,...,α̃^K,β̃^1,...,β̃^K)^⊤.
§ HMM FOR INFERENCE/LEARNING
The HMM in Fig.1, represents the evolution of a sequence of hidden states, s_τ, over time. At each time step, τ, a hidden state emits an observation, o_τ, and the state at any one time depends only on the state at the previous time where this dependency is encoded in the matrix B. The initial prior probability regarding the hidden state is encoded in the vector D, and finally, the matrix A encodes the likelihood distribution of generating outcomes under each state <cit.>. Here, it is assumed that the parameters of the generative model is learned and we are only interested in inferring the hidden states.
The HMM in Fig.2, represents the evolution of a sequence of hidden states, s_τ, over time, with priors over the parameters of the model, A, B, and D <cit.>. Here, we are interested in inferring the hidden states and learning the parameters of the generative model.
|
http://arxiv.org/abs/2307.06260v1 | 20230712160156 | UGCANet: A Unified Global Context-Aware Transformer-based Network with Feature Alignment for Endoscopic Image Analysis | [
"Pham Vu Hung",
"Nguyen Duy Manh",
"Nguyen Thi Oanh",
"Nguyen Thi Thuy",
"Dinh Viet Sang"
] | cs.CV | [
"cs.CV"
] |
An Architecture for Control Plane Slicing in Beyond 5G Networks
Rashmi Yadav
Department of Electrical Engineering
Indian Institute of Technology Kanpur,
India
[email protected]
Rashmi Kamran
Department of Electrical Engineering,
Indian Institute of Technology Bombay,
India
[email protected]
Pranav Jha
Department of Electrical Engineering,
Indian Institute of Technology Bombay,
India
[email protected]
Abhay Karandikar
Department of Electrical Engineering,
Indian Institute of Technology Bombay, India
[email protected]
Director, Indian Institute Technology Kanpur, India
[email protected]
August 12, 2023
==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
Gastrointestinal endoscopy is a medical procedure that utilizes a flexible tube equipped with a camera and other instruments to examine the digestive tract. This minimally invasive technique allows for diagnosing and managing various gastrointestinal conditions, including inflammatory bowel disease, gastrointestinal bleeding, and colon cancer.
The early detection and identification of lesions in the upper gastrointestinal tract and the identification of malignant polyps that may pose a risk of cancer development are critical components of gastrointestinal endoscopy's diagnostic and therapeutic applications. Therefore, enhancing the detection rates of gastrointestinal disorders can significantly improve a patient's prognosis by increasing the likelihood of timely medical intervention, which may prolong the patient's lifespan and improve overall health outcomes.
This paper presents a novel Transformer-based deep neural network designed to perform multiple tasks simultaneously, thereby enabling accurate identification of both upper gastrointestinal tract lesions and colon polyps.
Our approach proposes a unique global context-aware module and leverages the powerful MiT backbone, along with a feature alignment block, to enhance the network's representation capability. This novel design leads to a significant improvement in performance across various endoscopic diagnosis tasks. Extensive experiments demonstrate the superior performance of our method compared to other state-of-the-art approaches.
§ INTRODUCTION
Digestive tract diseases, particularly colorectal cancer, pose a significant threat to human health <cit.>. Accurate diagnosis and classification of these diseases are critical for effective treatment and improvement of patient outcomes <cit.>. Despite the increasing incidence of these diseases, manual analysis by expert physicians is still the gold standard for diagnosis and classification. However, this method is time-consuming and subject to human error <cit.>.
Recent advancements in computer-aided diagnosis (CAD) have demonstrated considerable potential in enhancing the precision of disease classification <cit.>. A notable technique in this regard is polyp segmentation, which aims to accurately identify and isolate lesions in images of the digestive tract <cit.>. The extracted information can subsequently aid in diagnosing and classifying digestive tract diseases, including potential precursors to cancer. It is imperative to examine the stomach and its relevant structures thoroughly. These cutting-edge advancements are poised to revolutionize disease detection and diagnosis in the medical field.
Accurate anatomical site classification is a crucial aspect of medical image analysis <cit.>, enabling the identification of specific structures within the human body. This information is essential for precise diagnosis and effective treatment planning, providing invaluable insights into areas of anomalies or lesions. Recent advancements in deep learning techniques have contributed significantly to the field of anatomical site classification <cit.>. Nonetheless, challenges persist due to the variability of imaging modalities and the inherent complexity of the human anatomy. Overcoming these hurdles is vital for advancing medical image analysis and improving patient outcomes.
Accurate classification of lesions and Helicobacter pylori (HP) has garnered increasing attention in the medical field. Lesion classification is crucial in improving diagnoses and treatments for various medical conditions. Image segmentation is sometimes necessary to understand the lesion or polyp location clearly. Polyp segmentation is particularly valuable in the early detection of digestive tract diseases. Manual analysis of medical images is prone to human error and time-consuming, highlighting the need for automated methods. Our study focuses on classification and segmentation, with findings indicating their equal significance and mutual complementarity.
In this paper, we propose a new model that can well solve two tasks at the same time. The experimental results demonstrate superior performance of our model in both single-task and multi-task learning scenarios.
Our main contributions are:
* We identified that the existing MiT backbone needed to adequately leverage the channel attention mechanism, resulting in the loss of context information in deeper layers. To overcome this, we incorporated the CGNL module with channel attention in groups to establish local relationships between channel groups.
* However, relationships between groups were still lacking, prompting us to add the SE module to bridge the gap between the groups. The combination of these two modules enhanced the channel characteristics.
* By combining these two contributions, we present UGCANet, which can extensively leverage the channel attention mechanism and propagate through the FaPN decoder <cit.>, resulting in segmented output. Additionally, the Fully-connected layer processes some of the feature information to address the classification task.
The remaining sections of the paper are structured as follows. We briefly review related work in Section <ref>. Then, we present our proposed method in Section <ref>. In Section <ref>, we showcase the conducted experiments and discuss results. Finally, we conclude the paper and discuss potential future directions in Section <ref>.
§ RELATED WORK
§.§ Deep learning and medical image segmentation
Deep learning. Deep learning has been widely used in various fields, such as computer vision, natural language processing, and speech recognition. In recent years, deep learning has made significant progress in these fields and has been applied to various tasks, including image classification, object detection, and segmentation. In the field of image classification, Convolutional Neural Networks (CNNs) have proven to be highly effective in achieving impressive results on large-scale image classification datasets, such as ImageNet <cit.>, CIFAR-100 <cit.>. This highlights the significance of these models as a benchmark for deep learning and image classification research. The advancements in CNNs have paved the way for continued progress in both fields and serve as a foundational foundation for future research <cit.>. Densenet <cit.> uses dense blocks, where each layer receives inputs from all previous layers in the block. This helps reduce the vanishing gradient problem and makes it easier for the network to propagate information through the network. In ResNet <cit.>, the network learns residual connections instead of trying to directly learn the desired mapping from inputs to outputs. Subsequently, in the ensuing years, there have been continued advancements in image classification and deep learning. A new method for scaling CNNs that is more efficient and effective than traditional methods. The method, called EfficientNet, balances the trade-off between accuracy and computational cost by scaling the network's depth, width, and resolution in a consistent manner. The authors show that EfficientNet outperforms previous state-of-the-art models on several benchmark datasets for image classification <cit.>. The progression has not stopped at that point. A breakthrough as the first transformers architecture appeared when the attention mechanism was first introduced by <cit.>. The introduction of the attention mechanism had a significant impact on the thinking of computer vision researchers, leading to the utilization of the self-attention mechanism in computer vision and the emergence of transformer-based proposals for machine vision.
The Vision Transformer (ViT) demonstrates remarkable performance compared to cutting-edge CNNs and requires significantly fewer computational resources for training <cit.>. Unlike traditional CNNs, In ViT <cit.>, the input image is first divided into a sequence of non-overlapping patches, which are then treated as tokens. Which
allows the network to focus on different regions of the input image when
making predictions. While ViT is effective for image classification, adapting it for dense predictions at the pixel level, like object detection and segmentation, presents a significant challenge. The Pyramid Vision Transformer (PVT) <cit.> is introduced as a solution to that issue through the creation of a shrinking pyramid and the implementation of spatial-reduction attention (SRA).
Focal <cit.> used a new mechanism called Focal Self-attention that incorporates both fine-grained local and coarse-grained global interactions, CaiT emphasized the significance of utilizing Layer-scale to scale up the depth dimension <cit.>, LeViT <cit.> astutely applied a combination of CNNs and Transformers to create a novel hybrid neural network, resulting in both precise and rapid inference. While ViT and PVT can be limited by distant dependent information DAT <cit.> introduced deformable self-attention module select key-value pairs depending on the data.
Medical image segmentaton. Unet <cit.> has gained popularity in medical image segmentation. The model is designed to handle tasks with multiple scales and uses a combination of downsampling and upsampling layers to maintain high resolution in the segmented output. The key feature of Unet is its symmetrical architecture, which allows for precise localization and preservation of fine details in the segmented image. However, it has a few limitations. One of the main limitations is that it may struggle with small or fine structures, such as tiny vessels or tumors, in medical images.
Additionally, the symmetrical architecture of Unet can lead to a lack of context information and decreased performance in complex or large-scale segmentation tasks. In order to address these limitations, Unet++ <cit.> and DoubleUnet <cit.> were proposed. Unet++ introduces a multi-scale learning mechanism, which helps capture contextual information from different scales. This leads to improved performance for small and fine structures in medical images. DoubleUnet, on the other hand, uses two parallel Unets, each with different levels of abstraction and context, to make predictions. This dual-stream approach helps the model capture local and global context information, improving performance in complex and large-scale segmentation tasks. Several models are based on Unet and Transformers, like TransUnet <cit.> and TransFuse <cit.>. The Hybrid ViT component of TransUNet stacks the CNN and Transformer together, leading to high computational costs. However, this also improves accuracy and performance compared to using either model individually. TransFuse is designed to address the high computational costs of TransUNet. TransFuse utilizes a parallel architecture, which allows it to reduce the computational overhead of the Transformer and CNN components of the network. In TransFuse, the Transformer and CNN components are trained in parallel, allowing faster and more efficient training. The parallel architecture also allows for more efficient use of hardware resources, such as GPUs, which can result in faster and more efficient inference.
§.§ Lesion Segmentation for endoscopy
Gastrointestinal (GI) lesion segmentation is crucial in early detection and diagnosis of digestive tract diseases such as esophageal cancer, duodenal ulcer, and colorectal cancer. In recent years, many deep learning-related works have been published in this area. CNN models such as Unet <cit.> or Unet++ <cit.> utilize skip connections to mitigate information loss caused by stacking numerous convolutional layers. The Unet architecture enhances the overall visibility of the model and enables the integration of multi-scale information. In 2022, Manh et al. <cit.> proposed an Unet-based multi-tasking model called EndoUnet to simultaneously solve multiple upper GI tasks. Supplementary modules are also a way to enrich information representation, ASPP <cit.> expanded the field of view to a larger perceptual area, making it suitable for capturing broader contexts. PraNet <cit.> adds an RFB module <cit.> helps generate features with the different receptive fields.
Another approach to enhance the representation of information is utilizing the attention module. EncNet <cit.>, DFN <cit.> use attention in the feature map's channel dimension to consider the global context, such as the occurrence of different classes in context. CCBANet <cit.> proposed an Attention Balance module (BAM). BAM uses an attention mechanism for three distinct regions: the background, polyp, and boundary. BAM intensifies local context information when retrieving features from the encoder block.
§.§ Image classification for endoscopy
Although GI endoscopic image documentation is an economical and effective solution for endoscopic reporting, implementing computer-assisted techniques for quality control poses a challenge due to the resemblance of appearance between different anatomical sites and the considerable and inconsistent variability in site appearance across patients.
In 2018, a GoogLeNet-based diagnostic program <cit.> was developed to recognize the anatomical site from 27335 endoscopic images of four major categories (larynx, esophagus, stomach, and duodenum). The CNN is also utilized in disease detection; Lin et al. <cit.> modified Inception V3 to recognize Helicobacter Pylori infection and obtained relatively satisfactory results (accuracy of 95%). In 2019, a system called WISENSE <cit.>, which is based on deep learning and reinforcement learning, was proposed by Wu et al. for real-time blind-spot monitoring, timing the procedure, and classifying anatomical sites. In this study, the authors implemented the 27-class protocol, which includes 26 anatomical sites and a NA class for images that cannot be classified to any site. By combining data collection, automatic ROI extraction, and a CNN-based model, He et al. <cit.> presented a comprehensive workflow for EGD image classification in 2020.
§ METHOD
§.§ Overview
The architecture of our proposed network, UGCANet, is illustrated in Fig. <ref>. Our model uses the encoder-decoder design.
The encoder is capable of acquiring a shared feature representation that is proficient in serving multiple tasks, producing four intermediate feature maps, and transmitting the final three to the CGNL <cit.> and module SE <cit.> to aggregate context information then a branch will undergo the average pooling layer and three FC layer to generate the classification labels for our tasks. Additionally, it will pass through the FaPN <cit.> module to generate the prediction output for the segmentation task.
§.§ Backbone
MiT <cit.> improves upon ViT <cit.> by introducing hierarchical feature representation, overlapped patch merging, efficient self-attention, and Mix-FFN. The hierarchical feature representation generates multi-level features with different resolutions. Each hierarchical feature map F_i has a resolution of H/2^i+1×W/2^i+1× C_i, where i belongs to the set {1, 2, 3, 4} and C_i increases with each level. While overlapped patch merging preserves local continuity in image patches. Efficient self-attention reduces computational load by using a reduction ratio. Mix-FFN mitigates the impact of zero padding on location information leakage, improving accuracy.
§.§ Global Context-Aware Modules
§.§.§ Compact Generalized Non-Local
The Compact Generalized Non-Local (CGNL) <cit.> module is a computer vision technique that models how different positions of an image interact across channels. Unlike the Non-Local module <cit.>, CGNL applies distinct weights to each channel by first flattening the feature outputs after linear transformation layers. By using separate weights for each channel, it can better capture the underlying relationships between different parts of an image, leading to improved accuracy. CGNL stands out from channel attention modules due to the computational load required. Through the application of Taylor expansion, Yue et al. <cit.> are able to approximate the complex calculations required by the module with a simpler polynomial function. This makes the computation more efficient and helps to reduce the overall computational cost of the CGNL module.
§.§.§ Squeeze and Excitation
Squeeze and Excitation (SE) <cit.> is a neural network building block that improves the incantational power of CNNs by explicitly modeling interdependencies between channels of feature maps.
The SE block consists of two main operations: the squeeze and excitation operations.
In the squeeze operation, a global spatial pooling operation is applied to the feature maps, which aggregates the information across spatial locations.
In the excitation operation, the channel-wise statistics obtained from the squeeze operation are used to compute a set of learnable weights. These weights are then used to modulate the feature maps through a gating mechanism, which assigns higher weights to informative channels and lower weights to less informative ones. The gating mechanism is typically implemented as a sigmoid or ReLU activation function.
Because of its simple and efficient design, it can be stacked multiple times to form a deep network.
§.§ Decoder
In the encoder-decoder architecture, the problem of object information loss can be particularly challenging for small objects in the image. Because the encoder compresses the image features into a lower-dimensional space, which can result in the loss of fine-grained details and spatial information. This can lead to reduced segmentation performance, especially in scenarios where small objects are of great importance, such as medical image analysis.
Huang et al. introduced the Feature-aligned Pyramid Network (FaPN) <cit.>, which is a new feature-aligned pyramid network for dense image prediction. The FaPN model addresses the issue of feature misalignment in the encoder-decoder architecture by using a feature alignment module that aligns the features extracted at different levels of the network.
Feature Alignment Module: The downsampling operations in the encoder-decoder architecture result in a misalignment between the feature maps of the encoder and decoder. This misalignment can cause issues when fusing the features through addition or concatenation near object boundaries.
To address this, Feature Alignment Module (FAM) used deformable convolution and learnable offset fields to adjust the convolutional sample locations, which helps preserve accurate object boundaries.
Feature Selection Module: Rather than employing a 1 × 1 convolution to reduce the channel dimension of intricate features, FSM (Feature Selection Module) adopts a weight assignment approach for feature maps. The FSM architecture is motivated by the SE module but differs in that it includes an additional shortcut between the input and scaled feature maps. This unique shortcut enables FSM to effectively enhance multi-scale feature aggregation.
§.§ Loss
In our study, we evaluate two different types of datasets, including Upper GI and polyp data. In each task, we define a different type of loss to match the context.
§.§.§ Loss for Colonoscopy
Our loss function for this task is defined as:
ℒ =
ℒ^w_BCE +
ℒ^w_IoU
where ℒ^w_BCE is weighted binary cross-entropy and ℒ^w_IoU is weighted IoU loss. Thus, ℒ is called weighted BCE and IoU loss. In contrast to the standard binary cross-entropy loss function, the ℒ^w_BCE assigns higher weights to hard pixels rather than treating all pixels equally. As for ℒ^w_IoU, increases the weights of hard pixels to emphasize their importance.
§.§.§ Loss for upper GI tract
We collect data from various sources to train our models, which we combine to create a comprehensive training dataset. However, in this merged dataset, each sample is only relevant to a subset of the tasks. To clarify the type of sample, we use μ_i^t ∈{0, 1} as the indicator, with t ∈{pos, le, hp, seg} representing the tasks of anatomical site classification, lesion type classification, HP classification, and lesion segmentation, respectively. Suppose that 𝐲_i^t is the one-hot encoding of the label of the i-th sample in the t-th task. Assuming 𝐲̂_i^t be the probabilistic output of the i-th sample in the t-th task. If t ∈{pos, le, hp} then 𝐲_i^t and 𝐲̂_i^t are vectors whose length equals the number of classes, let C_pos represent the number of anatomical sites, C_le represent the number of lesion classes (including five lesion types and a negative one), and C_hp represent whether the HP sample is positive or not. Specifically, C_pos = 10, C_le = 6, and C_hp = 1.
If t=seg, then both 𝐲_i^t and 𝐲̂_i^t are two-dimensional matrices with dimensions equal to those of the input images.
The loss function ℒ_pos, used for the task of classifying anatomical sites, is a type of multi-class cross-entropy loss and is defined as follows:
ℒ_pos = - ∑_i=1^N( μ_i^pos∑_j=1^C_pos y_i^pos(j) * log ŷ_i^pos(j) )
where N is the number of training samples.
The loss ℒ_le for lesion type classification task is another multi-class cross-entropy loss defined as follows:
ℒ_le = - ∑_i=1^N( μ_i^le∑_j=1^C_le y_i^le(j) * log ŷ_i^le(j) )
The loss ℒ_hp for HP classification is the binary cross-entropy loss defined as follows:
ℒ_hp = - ∑_i=1^Nμ_i^hp * (y_i^hp * log ŷ_i^hp + (1 - y_i^hp) * log (1 - ŷ_i^hp) )
The total loss is a weighted combination of the above loss functions. This is defined as follows:
ℒ_total = λ_1 * ℒ_pos + λ_2 * ℒ_le + λ_3 * ℒ_hp + λ_4 * ℒ_seg
where λ_t indicates the importance level of the t-th task. In our work, we set λ_1 = λ_2 = λ_3 = λ_4 = 1.
§ EXPERIMENTS
§.§ Datasets
Our model is evaluated on two gastrointestinal datasets: one consists of images of the upper gastrointestinal tract, and the other is polyp data.
§.§.§ Colonoscopy dataset
For the task of polyp segmentation, we conducted our evaluation using a total of 5 datasets. Below, we provide detailed information about each of these datasets.
Kvasir dataset <cit.>: This dataset comprises 1000 images with varying resolutions ranging from 720 × 576 to 1920 × 1072 pixels. Kvasir data is collected using endoscopic equipment at Vestre Viken Health Trust (VV) in Norway, annotated and verified by medical doctors (experienced endoscopists).
CVC-ClinicDB dataset <cit.>: CVC-ClinicDB is an openly available collection of 612 images extracted from 31 colonoscopy sequences, with a resolution of 384×288. The dataset is commonly utilized in medical image analysis, specifically in detecting polyps in colonoscopy videos through segmentation techniques.
CVC-ColonDB dataset <cit.>: is provided by the Machine Vision Group (MVG). These images were extracted from 15 brief colonoscopy videos and consist of 380 images of 574 × 500 pixels resolution.
CVC-T dataset <cit.>: CVC-T dataset is a subset of a larger dataset named Endoscene, and it is primarily a test set. It is composed of 60 images that were obtained from 44 video sequences captured from 36 patients.
ETIS-Larib dataset <cit.>: The dataset consists of 196 images with high resolution (1226 x 996).
§.§.§ Upper GI tract dataset
There are three datasets we collected from endoscopy findings of patients at the Institute of Gastroenterology and Hepatology and Hanoi Medical University Hospital: the first for anatomical site classification, the second for lesion segmentation and classification, and the last for HP classification. They are combined into a huge dataset of the upper GI tract.
Anatomical site dataset: This dataset includes 5546 images of 10 anatomical sites, all of which are captured directly from the endoscopic machine, including four lighting modes: WLI (White Light Imaging), FICE (Flexible spectral Imaging Color Enhancement), BLI (Blue Light Imaging), and LCI (Linked Color Imaging). The images in this dataset do not contain any lesions and have labels specifying the anatomical site. Table <ref> describes the details of this dataset.
Lesion dataset: in this dataset, we have 4104 images of 5 types of lesions: reflux esophagitis, esophageal cancer, gastritis, stomach cancer, and duodenal ulcer. The images in this dataset have the annotations for both the classification and segmentation tasks. The numbers of images for reflux esophagitis, esophageal cancer, stomach cancer, and duodenal ulcer classes are 1335, 538, 1443, 538, and 250, respectively.
Figure <ref> shows some samples in the lesion dataset.
HP dataset: we have 1819 images in this dataset, including HP-positive and HP-negative images. Figure <ref> are some samples in the HP dataset.
§.§ Implementation Details
We performed two separate experiments on the upper GI and polyp datasets to validate our proposed UGCANet.
* For polyp dataset: For the polyp dataset, our approach utilizes an image size of 384x384 and employs multi-scale training with size ratios of 0.75, 1, 1.25, respectively. Regarding the optimization algorithm, we adopt Adam with an initial learning rate of 1e-4. Our data augmentation strategy comprises Flip, HueSaturation, and RandomBrightnessContrast, each with a probability of 0.5.
We perform three experiments with different dataset setups to compare with SoTA models.
Experiment 1 : The splitting method recommended in <cit.> is applied, where 90% of the Kvasir and ClinicDB datasets are allocated for training. The remaining images in Kvasir and CVC-ClinicDB datasets, along with all images from CVC-ColonDB, CVC-T, and ETIS-Larib, are used for testing.
Experiment 2: 5-fold cross-validation on the CVC-ClinicDB and Kvasir datasets.
Experiment 3: Cross-dataset evaluation with three training-testing configurations:
* CVC-ColonDB and ETIS-Larib for training, CVC-ClinicDB for testing;
* CVC-ColonDB for training, CVC-ClinicDB for testing;
* CVC-ClinicDB for training, ETIS-Larib for testing.
* For upper GI dataset:
5-fold cross-validation schema. The datasets are first divided into five subfolds, which are then merged to create larger folds. In the anatomical site dataset, each subfold consists of an equal number of images per anatomical site and lighting mode. Similarly, the lesion dataset's subfolds contain an equal number of images per lesion type, and the HP dataset's folds contain the same number of HP positive and HP negative samples. Additionally, a marker vector μ is generated to denote the sample type.
We use two variants of the MiT backbone, namely MiT-B2 and MiT-B3, to conduct training on images of size 480x480 without utilizing multi-scale techniques. The Adam optimizer is utilized with a learning rate linear warmup and cosine strategy annealing.
The following experiments are performed to evaluate the model's performance:
Experiment 4: we evaluate the impact of two MiT configurations on UGCANet, including MiT-B2 and MiT-B3, and compare their performance with backbones of EndoUNet <cit.>, including VGG19, ResNet50, and DenseNet121.
Experiment 5: in the classification tasks, we train the single-tasking instances of models, including VGG19, ResNet50, DenseNet121, and MiT-B3, each of them trained on separate data. We compare the performance of multi-tasking models and the single-tasking models.
Experiment 6: in the lesion segmentation task, we train five single-tasking instances of EndoUNet and five single-tasking instances of UGCANet, each of which trained on separate lesion data. Then, we compare the performance of multi-tasking models versus the single-tasking instances.
All training is done on a machine with a 3.7GHz AMD Ryzen 3970X CPU, 128GB RAM, and an NVIDIA GeForce GTX 3090 GPU.
§.§ Results and Discussion
§.§.§ Comparison with SoTA methods
* For polyp dataset:
Table <ref> shows the comparison result for Experiment 1. Our model UGCANet outperforms previous SoTA models in mDice and mIoU metrics. Despite the impressive performance of ColonFormer-S on the Kvasir dataset, our model UGCANet outperforms ColonFormer-S and ColonFormer-L on the ETIS-Larib dataset by 2.1% in mDice and 2.2% on mIoU. On CVC-ColonDB, UGCANet surpasses TransFuse-L* and TransUnet with 4.6% mDice and also mIoU of 5% and 4.3%, respectively. However, on our CVC-ClinicDB dataset, our model only approximates mDice and mIoU compared to TransFuse-L*.
Table <ref> shows us that UGCANet outperforms all state-of-the-art models in mDice, mIoU, precision, and even recall in the 5-fold cross-validation experiment on the ClinicDB dataset. With the Kvasir dataset, we continue to outperform other models on mDice and precision metrics. However, our model is 0.8% worse in recall and 0.1% in mIoU.
It is worth mentioning that our UGCANet model has displayed remarkable stability on both datasets, with a relatively low standard deviation.
Table <ref> compares UGCANet and other benchmark models regarding their size and computational complexity. Our model exhibits greater parameters than CNN-based models like CaraNet <cit.>, or HarDNet-MSEG <cit.>. However, compared to the corresponding transformer architectures, our model proves advantageous in terms of parameter optimization. Despite having only a slightly higher computational complexity of 2.93 GFLOPs than ColonFormer-S <cit.>, our model still outperforms it in terms of overall model performance.
* For upper GI dataset:
Our study investigated the effectiveness of various backbones in classifying lesions. According to Table <ref>, we found that model performance across datasets exhibits minimal variability. Notably, two datasets, anatomical site and lesion, demonstrated high accuracy rates. Even the lowest accuracies achieved by the single-tasking VGG19 model for anatomical site classification, lesion classification, and HP classification tasks were impressive at 97.07%, 98.51%, and 91.21%, respectively. However, our results indicate that multi-tasking models outperform single-tasking models. Particularly, SFMNet with MiT-B3 as the backbone and EndoUNet with Resnet50 as the backbone proved to be the top-performing models across all three tasks, achieving impressive accuracy rates of 98.46%, 99.63%, and 93.46%, respectively.
Table <ref> exhibits the segmentation task results for the upper digest dataset. The multi-tasking model demonstrated superior performance to its single-tasking counterpart in most tests. Our investigation indicated that multi-tasking learning was notably more effective, particularly when utilizing the UGCANet model.
§.§.§ Ablation study
Effectiveness of the FaPN: At first glance, it is apparent that both SegFormer-B3 and MiT-B3-FaPN utilize the same backbone, MiT-B3, differing only in their respective decoders. Notably, the average mDice score of MiT-B3-FaPN is higher than that of SegFormer-B3, and the parameters and GFLOPs values displayed in Table <ref> are significantly lower than those of MiT-B3-FaPN. The comparison results show that the computational cost with FaPN is much lower than that of SegFormer-B3 and the results are also slightly improved, so we choose FaPN as the decoder.
Effectiveness of the CGNL: By utilizing the CGNL module in addition, the ability to connect features by channels in groups of <cit.> is enhanced. Furthermore, the incorporation of channel attention, as opposed to MiT, yields a slight improvement in Table <ref>, particularly in the ClinicDB dataset, with an increase of approximately 1%.
Effectiveness of the SE: When replacing CGNL with SE, the results changed but not significantly. It seems that both modules add a channel attention mechanism to the MiT backbone to increase the representation of features.
Combination of CGNL and SE: After each feature group has learned
internal information, the SE module combines and enhances its channel
information. Our SFMNet performs best when the feature group and SE modules are utilized. Generally, the results of either the upper gastrointestinal dataset or the polyp dataset show a 1-2% improvement.
§ CONCLUSION
This study introduces a novel model called UGCANet that employs a Unified Global Context-Aware Transformer-based architecture. The proposed model demonstrates exceptional performance in addressing both multi-tasking and single-tasking problems. Our experimental findings indicate that UGCANet achieves state-of-the-art results in multi-tasking and exhibits competitive performance compared to prior segmentation approaches.
In future works, we plan to optimize the utilization of the FSM module to improve the performance of our method. The FSM and SE mechanisms differ only in their usage for the encoder and decoder. By potentially adjusting the FSM module or even the SE module, we can ensure that the model gains more comprehensive information before entering the decoder branch, resulting in enhanced outcomes.
unsrt
|
http://arxiv.org/abs/2307.04148v1 | 20230709104530 | Towards a RISC-V Open Platform for Next-generation Automotive ECUs | [
"Luca Cuomo",
"Claudio Scordino",
"Alessandro Ottaviano",
"Nils Wistoff",
"Robert Balas",
"Luca Benini",
"Errico Guidieri",
"Ida Maria Savino"
] | cs.AR | [
"cs.AR"
] |
Towards a RISC-V Open Platform for Next-generation Automotive ECUs
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 871669.
Luca Cuomo†,
Claudio Scordino†,
Alessandro Ottaviano⁎,
Nils Wistoff⁎,
Robert Balas⁎,
Luca Benini⁎,
Errico Guidieri†,
Ida Maria Savino†
⁎ Integrated Systems Laboratory, ETH Zurich, Switzerland
† Huawei Research Center, Pisa, Italy
{l.cuomo,c.scordino,e.guidieri,i.savino}@huawei.com
{aottaviano,nwistoff,balasr,lbenini}@ethz.ch
August 12, 2023
===============================================================================================================================================================================================================================================================================================================================================================================================================================================
The complexity of automotive systems is increasing quickly due
to the integration of novel functionalities such as assisted or autonomous driving.
However, increasing complexity poses considerable challenges to the automotive supply chain since the continuous addition of new hardware and network cabling is not considered tenable.
The availability of modern heterogeneous multi-processor chips represents a unique opportunity to reduce vehicle costs by integrating multiple functionalities into fewer Electronic Control Units (ECUs). In addition, the recent improvements in open-hardware technology allow to
further reduce costs by avoiding lock-in solutions.
This paper presents a mixed-criticality multi-OS architecture for
automotive ECUs based on open hardware and open-source technologies.
Safety-critical functionalities are executed by an AUTOSAR OS running on a RISC-V processor, while the Linux OS executes more advanced functionalities on a multi-core ARM CPU.
Besides presenting the implemented stack and the communication infrastructure, this paper provides a quantitative gap analysis between an HW/SW optimized version of the RISC-V processor and a COTS Arm Cortex-R in terms of real-time features, confirming that RISC-V is a valuable candidate for running AUTOSAR Classic stacks of next-generation automotive MCUs.
Automotive, AUTOSAR, mixed-criticality, open hardware, RISC-V, Multi-OS
§ INTRODUCTION
For decades, automotive has been a very conservative industry, with software
functionalities operated by simple Electronic Control Units (ECUs) communicating
through domain-specific networks (e.g. CAN, LIN, FlexRay).
However, the recent increase in the complexity of automotive systems due to the integration of novel functionalities, such as assisted or autonomous driving, poses considerable challenges to this industry.
Modern luxury cars already contain more than 100 different ECUs <cit.>,
and the addition of new hardware has become an untenable task due to the amount of cabling inside the vehicle <cit.> and increasing space, weight, power and cost (SWaP-C).
Thus, nowadays, a significant opportunity for this industry is represented by the availability of asymmetric multi-processor (AMP) chips, which integrate (i) high-performance, multi-core application-class CPUs running a general-purpose OS (GPOS) such as Linux, (ii) slow real-time microcontrollers (MCUs) running a real-time operating system (RTOS), and (iii) domain-specific accelerators.
The different processing units, in fact, could be used to integrate and consolidate multiple functionalities (even with different non-functional requirements) on the same ECU, reducing the amount of hardware and cabling inside the vehicle.
In parallel to this trend, another interesting opportunity for the automotive industry is represented by the open hardware initiatives, which aim at designing open instruction-set architectures (ISAs) to avoid vendor lock-in solutions and thus further reduce the recurrent costs faced by the OEMs.
In particular, the RISC-V ISA <cit.> is getting momentum across various industry domains as the future lingua franca for computing and is widely considered a promising technology with significant potential also for the transportation domain <cit.>.
While previous works combining multi-OS and open hardware architectures for the automotive and space domains have been proposed, they often rely on symmetric/heterogeneous multi-processor (SMP and HMP) chips demanding hypervisor support for multi-OS execution, closed-source hardware architectures, and bespoke software libraries for intra-OS communications (Sec. <ref>).
This paper proposes a hardware and software stack for the automotive domain that leverages both AMP and RISC-V-based hardware towards the design of an open platform for automotive that relies on typical middleware employed in automotive.
In particular, the paper provides the following contributions:
* We conceptualize a heterogeneous mixed-criticality system (MCS) with multi-OS architecture where a Linux-capable commercial multi-core system is paired with an open-source RISC-V MCU [<https://github.com/pulp-platform/cheshire>] designed around CVA6 <cit.> that runs an RTOS tailored for automotive, ERIKA Enterprise <cit.>.
* We demonstrate the MCS system on a heterogeneous FPGA, namely the Xilinx Zynq Ultrascale+, which combines a hard macro implementing an ARM-based multi-core system with programmable hardware implementing the RISC-V real-time MCU (Sec. <ref>).
To the best of the authors' knowledge, this is the first work that attempts to adopt a multi-OS open-source stack for automotive based on open hardware.
* We conduct a quantitative gap analysis of the CVA6 MCU against an Arm-based real-time MCU (Cortex-R series) available on the heterogeneous FPGA in terms of interrupt response time, showing a significant performance gap of the RISC-V interrupt support as intended in the Privileged specifications <cit.>.
* We extend the real-time capabilities of the RISC-V CVA6 MCU by coupling the core with a RISC-V fast interrupt controller (CLIC <cit.>), which allows achieving competitive real-time performance against the Arm competitor, paving the road for further development of the RISC-V ISA in the transportation domain (Sec. <ref>).
§ RELATED WORK
Automotive operating systems
In the '90s some German and French companies joined their efforts to create the OSEK/VDX consortium <cit.>, aiming at creating an open standard for the operating system and the communication stack of automotive embedded systems. Some parts of these specifications were then standardized in ISO 17356 <cit.>. Some open-source implementations have been
proposed during the years, with the most notable projects being Trampoline <cit.>
and ERIKA Enterprise <cit.>.
The AUTomotive Open System ARchitecture (AUTOSAR) consortium, started in 2004, has coordinated and driven a standardization effort in the last two decades to handle the growing complexity of the software inside vehicles.
The specification (namely, AUTOSAR Classic <cit.>) extended the original OSEK/VDX standard to design the stack for simple automotive ECUs executing tiny real-time operating systems (RTOSs) and communicating through domain-specific networks (e.g. CAN, LIN, FlexRay).
The advent of modern functionalities, like assisted or autonomous driving, has
then forced the consortium to release in 2017 an additional specification (namely, AUTOSAR Adaptive <cit.>) for a more dynamic platform based on the POSIX API <cit.>
and also capable of High-Performance Computing (HPC). The consortium has also provided an
exemplary implementation of part of the specification running on the Linux OS.
The idea of using a Multi-OS architecture for automotive is well understood in the literature.
Burgio et al. <cit.> proposed a Multi-OS architecture developed in the context of the HERCULES European project.
Despite employing the same RTOS presented in this paper (i.e., ERIKA Enterprise),
they relied on a closed-source Heterogeneous Multi-Processor (HMP) architecture based on the Arm big.Little concept, and run multiple operating systems on top of an open-source hypervisor.
Moreover, the communication between the different operating systems was achieved through ad-hoc
libraries rather than using standard middleware employed in automotive.
On the industrial side, silicon vendors have started designing AMP system-on-chips (SoC) comprising
a high-end multi-core processor (possibly in a big.Little configuration) tasked to run the GPOS, and a slower microcontroller tasked to run a safety-critical RTOS.
An example is the i.MX8 chip by NXP <cit.>, which includes Arm A52 and A72 cores along with Arm M4 cores. The intra-OS communication is delegated to bidirectional connection-less Remote Processor Messaging (RPM) interfaces.
More recently, Arm has proposed the first high-performance 64-bit real-time processor of the R-series, Cortex-R82. Despite the announced use case as a storage controller for the IoT domain, such a processor is a candidate for future dual GPOS/RTOS execution on a common hardware platform.
In fact, automotive OEMs are already transitioning from a domain architecture to a zonal architecture <cit.> similar to the one shown in Figure <ref>, where few Vehicle Computers run a multi-domain stack that includes both AUTOSAR Classic Platform (CP) and AUTOSAR Adaptive Platform (AP), along with 3rd party software (e.g., ROS2, plain Linux, other operating systems, etc.).
Albeit several architectural similarities between the available multi-OS platforms, this work distinguishes itself on multiple angles: (i) it relies on a fully open-source real-time MCU developed within the ever-growing RISC-V ecosystem to handle safety-critical tasks, (ii) the automotive software stack running on RISC-V is based on an open-source RTOS <cit.>, and (iii) it enables RISC-V as the leading actor of the transition towards zonal architectures, that is currently not taking into account open hardware processors.
The following paragraph analyzes the state-of-the-art in bringing RISC-V architectures into the automotive domain in the last few years.
Automotive RISC-V architectures
RISC-V technology has received much attention during the last years after
some vendors released high-end multi-core CPUs at 1.5 GHz
capable of running Linux.
Some recent work has also been devoted to increasing the safety levels of RISC-V architectures.
De-RISC <cit.> is an H2020 project aiming at designing a RISC-V processor and
software stack for safety-critical systems. However, the project is mainly focused
on the space industry.
Abella et al. <cit.> identified some issues of the RISC-V
ecosystem related to security and reliability and provided four contributions
to implement lock-step and system-level testing. Their work is very relevant
for the automotive domain and could be implemented on top of our proposed
architecture.
Pietzsch <cit.> presented EMSA5-FS, a 32-bit, single-issue, in-order, 5-stage RISC-V processor specifically designed for functional safety.
Cosimi et al. <cit.> proposed to mix core independent peripherals, including a Performance Monitoring Unit (PMU), an Error Management Unit, and an Execution Tracing Unit, to increase the safety integrity level of an application running on a RISC-V platform
up to the highest automotive level (i.e., ASIL-D).
Their implementation has been done on the same evaluation board used in our experiments
(i.e. Xilinx ZCU102 <cit.>) and can therefore be fully integrated with
the architecture presented in this paper.
Quite recently, Gruin et al. <cit.> presented MINOTAuR, a
timing predictable open source RISC-V core, based on the same
hardware architecture used in this work.
The experimental results have shown an overhead of 10% compared to
the unmodified core, obtained through partial speculative execution.
Although their work does not explicitly address automotive,
predictability is a non-functional requirement needed at every time- and safety-critical
domain (including automotive).
Very recently, SiFive and Renesas have announced a long-term
collaboration to design and produce RISC-V processors for the automotive
domain <cit.>.
These ISO26262-qualified processors will all have the same ISA to increase code
portability.
This work relies on the open-source 64-bit core CVA6 <cit.>. It extends its real-time capabilities to serve as a time- and safety-critical RISC-V system in a multi-OS platform, closing the gap with existing embedded COTS solutions (Sec. <ref>).
Automotive communication protocols
To ensure a reasonable and manageable complexity through composability,
the automotive industry is replacing the original
signal-oriented communication with modern service-oriented
architectures (SoA).
Using this paradigm, the various software components are decoupled from each other
and communicate by requesting and providing "services".
Each component can be designed in isolation, and the system is assembled
by composing and integrating the various functionalities.
Proposed initially by BMW, Scalable service-Oriented
MiddlewarE over IP (SOME/IP) <cit.> is a SoA protocol specifically
designed for Ethernet-based communications in automotive.
This standard specifies the serialization mechanism,
the service discovery and the integration with the AUTOSAR
stack.
More recently, Data Distribution Service (DDS) <cit.>
started attracting a growing interest from the automotive industry <cit.>.
Originally proposed in 2001, DDS became an Object Management Group (OMG)
standard in 2004, with several open-source implementations available
nowadays. The DDS specifications <cit.> describe a Data-Centric Publish-Subscribe
model for distributed application communication. This model builds on
the concept of a “global data space” contributed by publishers and
accessed by subscribers: each time a publisher posts new data into this
global data space, the DDS middleware propagates the information to all
interested subscribers.
The data-centric communication allows the decoupling of publishers from
subscribers, thus building a very scalable and flexible architecture.
The underlying data model specifies the set of data items,
identified by “topics“.
Nowadays, DDS is natively supported by most frameworks used in automotive
— namely, AUTOSAR Classic <cit.>, AUTOSAR Adaptive and ROS <cit.>.
Note that, according to some recent investigations <cit.>,
the ROS framework is already being used by about 80% of the automotive OEMs
and Tier-1s developing autonomous vehicles.
§ SYSTEM ARCHITECTURE
As shown in Figure <ref>, the proposed mixed-criticality architecture
consists of an AMP system-on-chip (SoC) comprising
a high-end multi-core processor tasked to run the GPOS, and a slower microcontroller tasked to run a safety-critical RTOS.
We design the RISC-V MCU around the 64-bit CVA6 core.
CVA6 is a 6-stage, single-issue, in-order core implementing the G and C extensions of the 64-bit RISC-V instruction set (RV64GC).
The core implements a Translation Lookaside Buffer (TLB) to accelerate address translations from the virtual to the physical domain and a classic branch predictor consisting of a branch target buffer (BTB), a branch history table (BHT), and a return address stack (RAS). The core employed in this work is configured with a 32-kiB write-through L1 data cache and a 16-kiB instruction cache.
Besides the core, the MCU hosts 128-kiB scratchpad memory (SPM), a direct memory access (DMA) engine, and low-latency peripherals (SPI, I2C, UART) for off-chip communication.
The MCU relies on an AXI4-compliant, on-chip, non-coherent interconnect system. AXI4 interfaces are exposed to the multi-core domain through a software-managed IOMMU (such as in <cit.>) consisting of an IO translation lookaside buffer (IOTLB) to efficiently translate virtual user-space application addresses from the multi-core domain to physical memory.
Fig. <ref> depicts the RISC-V MCU and its hardware interface towards the application-class host.
In the embedded domain, a general-purpose core's real-time capabilities strongly depend on its interrupt controller's design. This is a crucial and functional requirement in safety- and time-critical systems such as those operating in the automotive domain, aiming at minimizing interrupt latency and context switch time.
First, CVA6 lacks support for vectored interrupts, which store the interrupt service routine of each interrupt at a separate address.
Albeit increasing the code size as the vector table's size grows, this mechanism helps reduce the overall interrupt response time.
Furthermore, CVA6's native interrupt architecture consists of classic RISC-V PLIC and CLINT controllers from the RISC-V Privileged Specifications <cit.>. The core hosts three level sensitive interrupt signals: machine-mode timer interrupt, machine-mode software interrupt (inter-processor interrupt), and machine-supervisor-mode external interrupts, respectively.
The machine timer and machine software interrupt pending registers — and respectively — are provided by a Core Local Interruptor (CLINT) hardware Intellectual Property (IP), which generates one interrupt for each hardware thread (hart, a RISC-V execution context). While generates timer interrupts with a specific frequency, handles communications among processors by interrupting harts on writes/reads of dedicated memory-mapped registers.
The first 12 interrupts' identifiers are reserved for timer, software, and external interrupts in the machine (M), supervisor (S), and user (U) privilege modes. Other interrupt entries up to XLEN (for an RV64 processor such as CVA6, XLEN=64) are platform specific and referred to as local interrupts <cit.>.
Finally, the machine external and supervisor external interrupt pending registers bring the information from external devices to the hart.
The Platform Local Interrupt Controller (PLIC) <cit.> provides centralized interrupt prioritization and routes shared platform-level interrupts among multiple harts via the interrupt signals. The PLIC does not support interrupt preemption (nesting), nor runtime-configurable interrupt priorities and interrupt threshold control, which must be simulated in software.
As highlighted in Sec. <ref> and further detailed in Sec. <ref>, such native features are insufficient to fulfill functional real-time requirements.
An essential contribution of this work is the enhancement of CVA6's real-time capabilities in terms of interrupt response to achieve a competitive advantage against existing COTS real-time MCUs.
To ease the design and development of the AMP system, the RISC-V MCU hosting the RTOS has been implemented on a heterogeneous Xilinx Zynq Ultrascale+ FPGA <cit.> as part of the Programmable Logic (PL), thus taking advantage of the existing multi-core COTS SoC (Processing System, PS) to host the GPOS.
The PS consists of an industry-standard, quad-core, 64-bit Armv8 Cortex-A53 application-class core featuring 32 KiB L1 instruction and data cache per core and a 1 MiB L2 cache shared by all four cores and clocked at 1.2 GHz and a dual-core Cortex-R5F real time unit.
The Arm Cortex-R is employed to conduct the performance gap analysis with real-time enhanced CVA6 despite the difference in XLEN (32-bit and 64-bit, respectively).
The CVA6 MCU has been synthesized on the PL targeting 50 MHz frequency. The following describes the software stacks running on the various processors.
§.§ Real-time OS
The OSEK/VDX and AUTOSAR Classic standards specify the design of a tiny RTOS for automotive.
The programming paradigm is “run-to-completion,” and the configuration (e.g. number of tasks) is statically defined at compile time.
In this type of operating system, the Interrupt Service Routines (ISR)
are divided into two categories:
* ISR1: High-priority low-overhead routines that cannot call syscalls;
* ISR2: Priority-based routines, which could imply a rescheduling once finished.
In the proposed architecture, we have used the ERIKA Enterprise RTOS <cit.>.
This open-source RTOS supports various microcontroller architectures and
is used in several European research projects and industrial automotive products.
A fork of ERIKA Enterprise recently received ISO26262 ASIL-D qualification
(the highest safety level for automotive).
Moreover, there is an ongoing discussion with the AUTOSAR consortium to release this RTOS
within the Classic demonstrator under the name of “Open-ERIKA” <cit.>.
In the context of the AMPERE H2020 project, the ERIKA RTOS has been ported
and executed on the RISC-V architecture.
§.§ General-purpose OS
Linux is a well-known operating system implementing the POSIX API.
Its performance, open-source license, and portability made it a perfect candidate for the general-purpose operating system running on the multi-core ARM PS.
During the last decades, several attempts have been made to improve the real-time performance
of Linux systems <cit.>. From time to time, some support (e.g. preemptible kernel, priority inheritance protocol, high-resolution timers, SCHED_DEADLINE real-time scheduler <cit.>) have been merged in the official kernel.
PREEMPT_RT <cit.> is a long-term project sponsored by the Linux Foundation to improve the real-time capabilities of the operating system. The primary outcome of this project is a kernel patch that reduces the maximum latency experienced by applications and is expected to be merged in the mainline “Vanilla” codebase.
To improve the overall responsiveness of the proposed platform, we have therefore
re-compiled the Linux kernel applying the PREEMPT_RT patch and enabling the maximum preemption level.
It is worth mentioning the existence of a joint initiative, ELISA <cit.>, aiming
at easing the certification of this operating system in safety-critical environments. In the automotive scenario, the ELISA project aims to reach the ASIL-B certification of the OS. However, the project has not yet provided a process to obtain such a qualification.
§.§ Intra-OS communication
According to the latest trends in automotive <cit.>, the inter- and intra-OS communications have been entirely based on the DDS standard.
The intra-OS communication between processes running on the Linux OS has been implemented through an open-source DDS middleware (i.e., Fast-DDS, formerly known as Fast-RTPS).
The inter-OS communication between Linux and ERIKA, instead, has been based on DDS-XRCE <cit.>, a DDS protocol specifically designed by OMG
for resource-constrained systems.
As shown in Figure <ref>, in this client-server protocol, the devices (clients) communicate with an XRCE Agent (server), which provides the intermediate bridging service towards the DDS Data Global Space.
In particular, we have integrated eProsima's Micro XRCE-DDS stack <cit.>
(part of the Micro-ROS project <cit.>) on the ERIKA RTOS.
§ EVALUATION
In this section, we analyze and characterize the proposed automotive platform in terms of real-time capabilities, focusing on interrupt handling latency on both the multi-core system running the GPOS and the MCU running the RTOS, as well as the inter-domain communication time overhead:
* We analyze real-time extensions of the Linux kernel to suit the automotive domain better.
* We characterize the middleware layer for intra-OS communication.
* We optimize the real-time CVA6 MCU in hardware to boost its interrupt response capabilities with the integration of the RISC-V CLIC as the central interrupt controller for CVA6 and conduct a performance gap analysis with the COTS Arm Cortex-R5 already available in the PS of the FPGA.
§.§ Non-critical multi-core domain: Linux GPOS
For Linux, we have used an Ubuntu filesystem and the Foxy version of ROS2 on top of Fast-DDS.
The Linux kernel was version 4.19, patched with the PREEMPT_RT patch.
We have then run a set of tests
to measure the latency introduced by the OS.
The system has been stressed by creating interference through both the find command
(generating I/O traffic by scanning the filesystem on the SD memory and printing on the console) and
through the stress program generating CPU, memory and I/O interference: ./rt-test/stress -c 8 -i 8 -m 8 –vm-bytes
8000000.
The worst-case latency has been measured through the cyclictest tool provided
by the Linux kernel community developing the PREEMPT_RT patch. The tool has been run
with the following options:
./cyclictest –mlockall –smp –priority=80
–interval=200 –distance=0 –duration=5m.
The experimental results have shown a worst-case latency of 13.4 ms without PREEMPT_RT
and 159 μs with PREEMPT_RT. This means that the maximum latency experienced by user-level applications has been reduced of about 99% by simply applying the patch and recompiling the Linux kernel.
§.§ Inter-domain communication
The communication latency has been evaluated through a “ping-pong” application
that measured the round-trip time from Linux to ERIKA and back to Linux.
The involved processes on Linux (i.e., DDS Agent and ROS2 application) have been
scheduled using a real-time priority (i.e., SCHED_RR with priority 99).
Data has been exchanged through a non-cached shared memory area. We selected UART as the interrupt source, the only visible from both operating systems.
The experimental results showed a minimum, average and maximum communication time
of 2.0, 2.2, and 3.7 msec, respectively.
It is essential to highlight that the Micro-ROS framework has a periodic engine
which added some delay to the communication.
In particular, the clc_executor_spin_some function had a period of 1 msec,
while all the other interactions were event-driven.
§.§ Safety critical RISC-V MCU domain: ERIKA RTOS
When porting the ERIKA Enterprise RTOS on RISC-V, we have taken
inspiration from the previous FreeRTOS optimization <cit.>.
We have initially optimized interrupt handling by emulating the local
interrupt levels through an array statically generated by the OS tools since the core does not natively support them.
The performance of the ERIKA RTOS when running on
RISC-V has been measured through an existing benchmark <cit.>
that measures the time
needed by the RTOS for performing a set of critical scheduling
activities (e.g., task activation time, task exit time, ISR call time,
etc.).
The test suite also allows benchmarking the latency of the two types of interrupt
service routines available in AUTOSAR Classic kernels (i.e. ISR1 and ISR2)
that have been previously illustrated.
The tested functions are namely:
* act: activates a higher priority task and measures how long
it takes to start its execution.
* actl: activates a low-priority task and measures how long it
takes to return to the caller.
* intdisable: measures the time needed for disabling all interrupts.
* intenable: measures the time needed for enabling all interrupts.
* isrentry: measures the time elapsed between the occurrence
of an interrupt and the execution of the related ISR1 handler.
* isr2entry: measures the time elapsed between the occurrence
of an interrupt and the execution of the related ISR2 handler.
* isrexit : measures the time elapsed between the end of an
interrupt handler and when the task previously running resumes execution.
* istentry: measures the time elapsed between the end of an
interrupt handler and the execution of the task activated by such
interrupt handler.
* istexit: measures the time elapsed between the end of a task
handling an interrupt and when the task previously running resumes
execution.
* terml: measures the time needed for terminating a task and
switching to a lower priority one.
Execution times have been measured in processor clock cycles through the CSR register.
The same benchmark has been executed to evaluate the performance of the Cortex-R5 and the Cortex-A53 cores available on the ZCU102 board. For the Cortex-R5, the number of cycles has been measured through the PMCCNTR register.
For the Cortex-A53, instead, cycles have been measured through the cycle counter register PMCCNTR-EL0: __asm__ __volatile__
("MRC p15, 0, %0, c9, c13, 0" : "=r" (cycles));
It is essential to point out that, in the case of Cortex-A, the RTOS has been run on top of a hypervisor (namely, Jailhouse <cit.>) according to the typical configuration used when running RTOSs on Cortex-A processors.
The presence of the underlying hypervisor, however, implied some
non-negligible latency to trap and re-inject interrupts to the guest RTOS.
The possible interference from Linux on shared hardware resources has been removed by inducing the Linux kernel in panic mode through the following command: echo c > /proc/sysrq-trigger.
Since in safety-critical domains, such as automotive, we are most interested in bounding
the time needed for the various operations, we have restricted our analysis to the
worst-case number of cycles, measured over 100 consecutive runs.
The reported values in Fig. <ref> show that when CLINT/PLIC are being used, and no software optimization has been applied, the RISC-V soft-core can provide performance in the same order of magnitude of
state of the art (i.e., Cortex-R5). In particular, the worst-case cycles are higher only for handling ISR1 interrupts.
The reported values also confirm the non-negligible latency introduced by the interrupt injection mechanism on the hypervisor on Cortex-A. This strengthens the benefits of designing a mixed-criticality architecture through an AMP SoC rather than a hypervisor-based approach on an SMP SoC.
§.§ Software-driven RTOS optimization
The next step consisted in optimizing the code of the RTOS to obtain better performance in all the tested processors. The first optimization consisted in modifying the ISR2 handling by avoiding activating the ISR as a Task and directly calling the handler (i.e. not calling osEE_activate_isr2()). Moreover, similarly to <cit.>, we have used the -O3 optimization level of the GCC compiler.
Fig. <ref> reports the worst-case
number of cycles, still measured over 100 consecutive runs.
The benefits of the optimizations can be appreciated across all the architectures.
However, the proposed SW optimization of the RTOS on Cortex-R benefited more than the implementation on the RISC-V MCU, in some cases reducing the worst-case cycles to less than 25% of the original value (e.g., in the actl, intdisable and intenable tests).
From the presented values, we can see that the selected RISC-V processor still shows lower performance in terms of interrupt latency than the competing ARM Cortex-R5 architecture.
As already discussed in Sec. <ref>, we identify the bottleneck of the design in CVA6's interrupt handling support, which is not tuned for targeting fast-interrupt management and low interrupt latency, typically enabled through the following HW/SW mechanisms:
* Hardware support for fine-grained and configurable interrupt priorities
* Late-arriving interrupt behavior (preemption and nesting) <cit.>
* Context save/restore optimization with back-to-back interrupts (tail chaining)
* Banked stack pointer <cit.> (i.e. different stack pointers for different privilege levels)
* Hardware support for automatic saving of registers during the context switch.
The following section addresses the first three of the above-mentioned design items. To this aim, we extend the current CLINT interrupt controller with a Core Local Interrupt Controller (CLIC) <cit.> and evaluate the ERIKA RTOS's performance.
The remaining two design items involve implementing more advanced hardware features in the processor and will be investigated in future work.
§.§ Hardware-driven real-time optimization: RISC-V CLIC fast interrupt controller
As mentioned in Sec. <ref>, both native CVA6 and its interrupt controller architecture need adaptations to fulfill real-time needs.
We first modify CVA6's interrupt interface by replacing level-sensitive interrupts with a handshake mechanism carrying the interrupt identifier and the request to the processor that acknowledges the handshake. We then add support for vectored interrupts by implementing an interrupt identifier decoding logic to compute the jumping address of the vector table.
In the second step, we extend the CLINT with the Core Local Interrupt Controller (CLIC).
We employ an open-source implementation of the CLIC[<https://github.com/pulp-platform/clic>] that reflects the latest status of the RISC-V CLIC draft specifications. The integration process includes the addition of specific CSRs registers in the processor's micro-architecture as from specifications <cit.>.
The CLIC introduces several improvements to the standard CLINT to achieve faster interrupt handling. Among those are dedicated memory-mapped registers for software configurable interrupt priority and levels at the granularity of each interrupt line, runtime-configurable interrupt mode and trigger type, and support for interrupt preemption in the same privilege level.
Selective hardware vectoring enables the programmer to optimize each incoming interrupt for either faster response (vectored mode) or smaller code size (direct mode, when each interrupt traps to the same exception handler address).
Lastly, the CLIC introduces a novel CSR, namely <cit.> to accelerate the handling of back-to-back interrupts, a phenomenon called tail-chaining, which we have implemented in the CVA6 core.
CVA6 interrupt handling is modified as in Fig. <ref>.
In the improved design, the PLIC still arbitrates external system-level interrupts, and the legacy CLINT generates the timer interrupt. These interrupts are routed through the centralized CLIC interrupt source. Similarly, inter-processor interrupts are fired by writing to the corresponding CLIC memory-mapped registers.
Finally, local interrupts can be extended to 4096 lines instead of limited to the processor's . We implement 256 input interrupt lines arbitrated by the CLIC in this work.
Fig. <ref>
shows the experimental results on the improved hardware architecture with GCC -O3 optimization level at compile time on the selected benchmark. We notice that the worst-case overhead on RISC-V has become closer to the one measured on
the competitor MCU (i.e. ARM Cortex-R5).
The previous spikes (i.e. actl and isrentry) have been significantly reduced.
Moreover, for 4 metrics (i.e. act,
isrentry, istentry and istexit)
the number of cycles needed by the RTOS is equal to or even lower than on the Cortex-R5.
These experimental results confirm that RISC-V is
a valuable technology for running AUTOSAR Classic stacks of next-generation automotive MCUs, and can be further improved to surpass closed-source commercial solutions.
§ CONCLUSIONS
In this paper, we have illustrated some trends and challenges occurring in the automotive
domain, as well as various technologies being taken into account by the companies
operating in this industry.
We have proposed a novel mixed-criticality multi-OS architecture based on open hardware and open-source software. We have then described the optimizations done both at the software and hardware levels
to move performance closer to the commercial competitors.
The experimental results have shown performance comparable to the state-of-the-art and have also allowed identifying further room for future hardware optimizations of
the CVA6 RISC-V processor.
In the future, we plan further to evolve the proposed architecture by (i) designing advanced hardware features such as banked stack pointers and optimized context switch to improve the competitiveness of the CVA6 architecture further, (ii) leveraging the recent standardization of DDS in Classic AUTOSAR <cit.> by using plain DDS instead of DDS-XRCE for inter-domain communications, and (iii) adopting the scheduling policy <cit.> to have a more predictable timing behavior of the communication on the Linux OS.
IEEEtran
|
http://arxiv.org/abs/2307.04019v3 | 20230708173320 | GP-guided MPPI for Efficient Navigation in Complex Unknown Cluttered Environments | [
"Ihab S. Mohamed",
"Mahmoud Ali",
"Lantao Liu"
] | cs.RO | [
"cs.RO",
"cs.AI",
"cs.SY",
"eess.SY"
] |
Explicit a posteriori error representation for variational problems and application to TV-minimization
[
August 12, 2023
========================================================================================================
@topnum0
@botnum0
empty
empty
Robotic navigation in unknown, cluttered environments with limited sensing capabilities poses significant challenges in robotics. Local trajectory optimization methods, such as Model Predictive Path Intergal (MPPI), are a promising solution to this challenge. However, global guidance is required to ensure effective navigation, especially when encountering challenging environmental conditions or navigating beyond the planning horizon.
This study presents the GP-MPPI, an online learning-based control strategy that integrates MPPI with a local perception model based on Sparse Gaussian Process (SGP).
The key idea is to leverage the learning capability of SGP to construct a variance (uncertainty) surface, which enables the robot to learn about the navigable space surrounding it, identify a set of suggested subgoals, and ultimately recommend the optimal subgoal that minimizes a predefined cost function to the local MPPI planner.
Afterward, MPPI computes the optimal control sequence that satisfies
the robot and collision avoidance constraints.
Such an approach eliminates the necessity of a global map of the environment or an offline training process.
We validate the efficiency and robustness of our proposed control strategy through both simulated and real-world experiments of 2D autonomous navigation tasks in complex unknown environments, demonstrating its superiority in guiding the robot safely towards its desired goal while avoiding obstacles and escaping entrapment in local minima. The GPU implementation of GP-MPPI, including the supplementary video, is available at <https://github.com/IhabMohamed/GP-MPPI>.
Autonomous vehicle navigation, MPPI, sparse Gaussian process (SGP), occupancy grid map path planning.
§ INTRODUCTION AND RELATED WORK
Autonomous navigation of mobile robots
in unknown, cluttered, and unpredictable environments with limited sensor capabilities is a challenging task owing to the inherent uncertainty and complexity of such environments.
To tackle this challenge, a receding-horizon strategy such as Model Predictive Control (MPC) is commonly employed.
The MPC control framework allows the robot to simultaneously plan a short trajectory (sequence of actions), following which the robot executes the immediate action while planning a subsequent trajectory.
To successfully achieve receding-horizon planning, the robot must consider both safety and persistent feasibility, where safety is achieved by avoiding collisions with any obstacles while executing a planned trajectory, and persistent feasibility is maintained by always
generating a safe trajectory that does not result in dead-ends or local minima while progressing towards the desired goal.
One of the significant challenges in robot motion planning is that the desired goal is often situated beyond the planning horizon, which requires the use of local subgoals or cost-to-go heuristics
for motion safety and persistent feasibility.
A common strategy is to rely on single-query motion planning algorithms, such as A^* and RRT^X, to identify feasible paths that direct the local planner towards its desired goal <cit.>.
For instance, the RRT^X algorithm, introduced in <cit.>, incorporates replanning techniques from Dynamic
Rapidly-exploring Random Trees (DRRT) and Rapid-exploring Random Trees (RRT^*) algorithms to adjust the path during exploration based on environmental changes. However, due to its high computational demands, implementing this algorithm in real-time on a robot can be challenging.
One alternative method to achieve efficient solutions for motion planning problems
is the integration of MPC with data-driven methods, also known as learning-based MPC <cit.>.
To name a few, a subgoal planning policy using Deep Reinforcement Learning (DRL) is recently proposed to guide the local MPC planner to navigate in crowded surroundings <cit.>.
Similarly, RL was utilized to choose the next subgoal from a set of predefined possibilities <cit.>, which guides the robot through challenging environments with dead-end corridors while also prevents the MPC planner from getting trapped in local minima.
Another related work that combines learning with MPC is POLO which aims to enhance MPC performance by learning a global value function <cit.>.
Most of these approaches typically rely on either offline training or having access to the global map of the environment.
In addition, many recent studies have suggested combining Gaussian Process (GP) with MPC to learn system dynamics, leading to better control performance and robustness to uncertainty <cit.>.
Another research avenue employed gap-based techniques that identify gaps as free spaces between obstacles, enabling a robot to move through them while avoiding local minima and obstacles.
The first developed method was the Nearness Diagram (ND) <cit.>, but many of its variants exhibited undesired oscillatory motion.
To overcome these limitations, robotics researchers have developed techniques that rely on the geometry of the gap. One such technique is the Follow-the-Gap Method (FGM), which selects a gap based on its area and computes the robot's heading using the gap center's direction relative to both the robot and the final goal <cit.>.
Another approach is the sub-goal seeking method, which assigns a cost to each sub-goal based on the goal heading error with respect to the robot and the gap heading, and then selects the sub-goal with the lowest cost (error) <cit.>.
The Admissible Gap (AG) method <cit.>, an iterative algorithm that takes into account the exact shape and kinematic constraints of the robot, identifies possible admissible gaps, and selects the nearest gap as the goal.
Different from all these strategies, our proposed framework leverages a Sparse variant of Gaussian Process (SGP) which is a new perception model by “abstracting” local perception data so that the local sub-goal for navigation can be naturally extracted.
Specifically, we introduce the GP-MPPI control strategy, which enhances the state-of-the-art sampling-based MPC, Model Predictive Path Integral (MPPI) <cit.>, by incorporating the GP-subgoal recommender policy. Such a policy takes advantage of the SGP occupancy model to learn about the navigable space surrounding the robot, identifies a set of suggested subgoals, and ultimately recommends the optimal subgoal that minimizes a predefined cost function to the MPPI local planner, as demonstrated in Fig. <ref>.
Subsequently, MPPI computes the optimal control sequence that satisfies the robot and collision avoidance constraints while moving towards the recommended subgoal, followed by executing the first optimal control 𝐮_0 to the robot.
In summary, the contributions of this work can be summarized as follows:
* We propose an online learning-based control strategy that recommends subgoals solely based on local sensory information, ensuring safety and persistent feasibility; such an approach eliminates the need for a global map of the environment or an offline training process as in RL techniques, resulting in a more flexible and agile control framework that can be easily deployed in different unexplored environments, as revealed in Section <ref>.
* To the best of the authors' knowledge, this is the first attempt to utilize the SGP occupancy model in conjunction with sampling-based trajectory optimization methods, specifically MPPI, to efficiently explore the navigable space surrounding the robot.
* In Sections <ref> and <ref>, we validate our GP-MPPI control strategy for collision-free navigation in complex and unknown cluttered environments, using both simulation and experimental demonstrations; by comparing it with two baseline sampling-based approaches (namely, MPPI <cit.>, and log-MPPI <cit.>), we show its effectiveness in overcoming local minima that may arise when the sampled trajectories of MPPI are concentrated in high-cost regions or due to challenging environmental conditions.
§ PRELIMINARIES
To provide the necessary background for our proposed work, in this section, we formulate the optimal control problem and present a concise overview of the MPPI control strategy that can be utilized to address this problem, along with a brief introduction to the Sparse Gaussian Process (SGP) which is the backbone of our GP-subgoal recommender policy.
§.§ Problem Formulation
Consider a nonlinear discrete-time stochastic dynamical system
𝐱_k+1=f(𝐱_k,𝐮_k+δ𝐮_k),
with 𝐱_k ∈ℝ^n_x and 𝐮_k ∈ℝ^n_u representing the state of the system and its control input, respectively. The disturbance introduced into the control input, δ𝐮_k, is modeled as a zero-mean Gaussian noise with co-variance Σ_𝐮.
Given a finite time-horizon N, we define the control sequence 𝐔 as 𝐔 = [𝐮_0, 𝐮_1, …,𝐮_N-1]^⊤∈ℝ^n_u N and the resulting state trajectory of the system being controlled as 𝐗 = [𝐱_0, 𝐱_1, …, 𝐱_N]^⊤∈ℝ^n_x (N+1).
Furthermore, 𝒳^d is used to represent the d-dimensional space with 𝒳_rob(𝐱_k) ⊂𝒳^d and 𝒳_o b s⊂𝒳^d representing the robot's occupied area and obstacles' area, respectively.
Let 𝐱_s and 𝐱_f denote the initial and desired (goal) state of the robot, respectively.
Given 𝒳_rob(𝐱_k), 𝒳_o b s, 𝐱_s, and 𝐱_f, we aim to find the optimal control sequence, 𝐔, that allows the robot to safely and efficiently navigate from its initial state, 𝐱_s, to the desired state, 𝐱_f, by avoiding both getting stuck in local minima and collisions with obstacles, while minimizing a cost function J.
The optimization problem at hand can be approached utilizing the classical MPPI control strategy described in <cit.>.
This optimization can be mathematically expressed as in (<ref>), with the objective of minimizing the cost function, J, which is comprised of the expectation of a combination of state terminal cost ϕ(𝐱_N), running cost q(𝐱_k), and control inputs 𝐮_k, weighted by the positive-definite matrix R∈ℝ^n_u × n_u, taking into consideration the system dynamics outlined in (<ref>) and constraints such as collision avoidance and control constraints as stated in (<ref>).
min _𝐔 J = 𝔼[ϕ(𝐱_N)+∑_k=0^N-1(q(𝐱_k)+1/2𝐮_k^⊤ R 𝐮_k)],
s.t. 𝐱_k+1=f(𝐱_k, 𝐮_k+δ𝐮_k), δ𝐮_k∼𝒩(0, Σ_𝐮),
𝒳_rob(𝐱_k) ∩𝒳_obs=∅, 𝐡(𝐱_k, 𝐮_k) ≤ 0,
𝐱_0 = 𝐱_s, 𝐮_k∈𝕌, 𝐱_k∈𝕏.
§.§ Overview of MPPI Control Strategy
In order to solve the optimization control problem defined in (<ref>), MPPI leverages Monte Carlo simulation to generate a significant number of real-time simulated trajectories by propagating them from the underlying system dynamics. It then evaluates the cost-to-go of each trajectory based on a predefined cost function and updates the optimal control sequence by considering a weighted average cost from all of the simulated trajectories. More details are given in <cit.>.
Subsequently, each trajectory τ_i in the time-horizon N can have its cost-to-go evaluated as given in (<ref>), where the cost-to-go S̃(τ_i) is calculated as the sum of the terminal state cost ϕ(𝐱_N) and the instantaneous running cost q̃(𝐱_k, 𝐮_k, δ𝐮_k,i) over all time steps.
The instantaneous running cost, q̃, expressed in (<ref>), is comprised of the state-dependent running cost q(𝐱_k) and the quadratic control cost q(𝐮_k, δ𝐮_k), where γ_𝐮 = ν -1/2ν and the aggressiveness in exploring the state-space is determined by the parameter ν∈ℝ^+. Specifically,
S̃(τ_i ) =ϕ(𝐱_N) + ∑_k=0^N-1q̃(𝐱_k, 𝐮_k, δ𝐮_k,i) ∀ i ∈{0, ⋯, M-1},
q̃=
q(𝐱_k)_State-dep.+
γ_𝐮δ𝐮_k,i^⊤ R δ𝐮_k,i+ 𝐮_k^⊤ R δ𝐮_k,i+ 1/2𝐮_k^⊤ R 𝐮_k_q(𝐮_k, δ𝐮_k): Quadratic Control Cost.
As outlined in (<ref>) from <cit.>, the optimal control sequence {𝐮_k}_k=0^N-1 in the vanilla MPPI algorithm is iteratively updated by taking a weighted average cost from all simulated trajectories, where S̃(τ_m) represents the cost-to-go of the m^th trajectory, and λ∈ℝ^+ denotes the “inverse temperature”, which regulates the selectiveness of the weighted average of the trajectories.
After smoothing the resulting control sequence with a Savitzky-Galoy filter <cit.>, the first control 𝐮_0 is executed in the system, with the remaining sequence utilized as a warm-start for the next optimization step. Formally,
𝐮_k←𝐮_k +∑_m=0^M-1exp( -1/λS̃(τ_m) ) δ𝐮_k, m/∑_m=0^M-1exp( -1/λS̃(τ_m) ).
§.§ Sparse Gaussian Process
Gaussian Process (GP) is a well-established non-parametric model described by a mean function m(z) and a co-variance function k(z, z^')
(also referred to as kernel function), where z∈ℝ^n_g is the input to the GP <cit.>; it can be mathematically expressed as
f(𝐳) ∼𝒢 𝒫(m(𝐳), k(𝐳, 𝐳^')).
Let 𝒟 = {(𝐳_i, y_i)}_i=1^n denote a dataset consisting of n input-output pairs, where each output y_i ∈ℝ is assumed to be the sum of an unknown underlying function f(𝐳_i) and Gaussian noise ϵ_i with a zero-mean and variance σ^2, i.e., ϵ_i ∼𝒩(0, σ^2).
In the context of GP regression, to estimate the output y^* for a given new input z^*, the following GP prediction equation is employed
p(y^* | y) = 𝒩(y^* | m_y(z^*), k_y(z^*,z^*) + σ^2),
m_𝐲(𝐳) =K_𝐳 n(σ^2 I+K_n n)^-1𝐲,
k_𝐲(𝐳, 𝐳^') =k(𝐳, 𝐳^')-K_𝐳 n(σ^2 I+K_n n)^-1 K_n 𝐳^',
where m_𝐲(𝐳) and k_𝐲(z,z^') are the GP posterior mean and co-variance functions, respectively, while K_nn∈ℝ^n × n refers to the n × n co-variance matrix of the training inputs and K_𝐳n∈ℝ^n is n-dimensional row vector of kernel function values between 𝐳 and the training inputs, with K_n𝐳 = K_𝐳n^⊤.
Achieving a more accurate GP prediction requires the optimization of hyper-parameters, such as kernel parameters Θ and noise variance σ^2, by maximizing the log marginal likelihood
log p(𝐲)=log[𝒩(𝐲|0, σ^2 I+K_n n)].
The standard GP can be computationally intensive due to its complexity of 𝒪(n^3), where n represents the number of training instances. To mitigate this issue, various approximation methods, collectively known as Sparse Gaussian Process (SGP), have been developed as an alternative approach. Instead of using the complete training data, SGP employs a smaller set of m_s training points, called inducing points Z_m_s, resulting in a more efficient process and a lower computation complexity of 𝒪(n m_s^2) <cit.>.
Our present work leverages the variational SGP method, proposed in <cit.>, to approximate the true posterior of a GP p(f|𝐲) using an approximated variational posterior distribution q(f,f_m_s), where f_m_s are the values of the underlying function f at the inducing points Z_m_s.
This approximation is done by augmenting the true posterior with the variable f_m_s such as p(f,f_m_s|𝐲) = p(f|f_m_s) p(f_m_s|y).
Then, the approximated variational distribution q(f,f_m_s) can be factorized in the same manner as the augmented true posterior, as follows
q(f,f_m_s) = p(f|f_m_s)ϕ(f_m_s),
where ϕ(f_m_s) is an unconstrained variational distribution over f
_m_s and p(f|f_m_s) is the conditional GP prior.
By minimizing the Kullback-Leibler (KL) divergence between the approximated and true posteriors, 𝕂𝕃[q(f, f_m_s)||p(f|𝐲)], the variational SGP obtains estimates of the inducing inputs Z_m_s and hyperparameters (Θ, σ^2).
§ GP-MPPI CONTROL STRATEGY
The goal of our present research, as outlined in (<ref>), is to determine the optimal control sequence 𝐔={𝐮_k}_k=0^N-1 that enables safe and efficient navigation of the mobile robots through complex and unknown cluttered environments, while avoiding collisions with obstacles and getting trapped in local minima.
Although the MPPI control framework, as summarized in <cit.>, has many positive attributes, it is prone to generating infeasible control sequences or trajectories, particularly when the distribution of all sampled trajectories are concentrated within high-cost regions.
To mitigate this issue, new sampling strategies proposed in <cit.> have enabled more efficient exploration of the state-space, allowing the algorithm to find better solutions and potentially reduce the risk of trapping in local minima.
Nevertheless, for specific tasks such as the one depicted in Fig. <ref>, eliminating the local minima remains a potential challenge that needs to be tackled.
One solution could be incorporating MPPI with a global planner, such as the solution presented in <cit.>, which utilizes the RRT algorithm to guide MPPI.
Instead, we introduce the GP-MPPI control strategy, a new online navigation technique
that leverages the SGP occupancy model to learn about the navigable space surrounding the robot. Specifically, we introduce the GP-subgoal recommender policy, which identifies a set of recommended subgoals and subsequently suggests the optimal subgoal that minimizes a predefined cost function to the MPPI local planner, as depicted in Fig. <ref> and explained in detail in Section <ref>.
Unlike conventional methods, a distinctive aspect of the proposed control strategy is that it does not require either a global map for long-term planning or an offline training process.
§.§ SGP Occupancy Surface Representation
Our proposed GP-subgoal recommendation policy relies on our earlier work presented in <cit.>, where we transformed pointcloud data into an occupancy surface and modeled it using a Sparse Gaussian Process (SGP).
Within this approach, the occupancy surface takes the form of a 2D circular surface centered around the sensor origin and has a predefined radius of r_oc.
This surface serves as the projection space for all observed points, which are represented in spherical coordinates (θ_i, α_i, r_i), where (θ_i, α_i, r_i) correspond to the azimuth, elevation, and radius values of each observed point, respectively.
Each point 𝐳_i on the occupancy surface is defined by two attributes: the azimuth and elevation angles 𝐳_i= (θ_i, α_i), and assigned an occupancy value f(𝐳_i) that is a function of the point radius r_i, such as f(𝐳_i)=r_oc-r_i. Afterward, the probability of occupancy f(𝐳) over the occupancy surface is modeled by an SGP occupancy model, as follows
f(𝐳) ∼𝒮𝒢𝒫(m(𝐳), k(𝐳, 𝐳^')),
k(𝐳, 𝐳^') =σ_f^2(1+(𝐳-𝐳^')^2/2 αℓ^2)^-α,
where σ_f^2 is the signal variance, l is the length-scale, and α is the relative weighting factor that manipulates large and small scale variations.
In our SGP model, the point's occupancy to radius relation is encoded as a zero-mean function, m(𝐳)=0, where the occupancy value of the non-observed points is set to zero.
The Rational Quadratic (RQ) kernel, k(𝐳, 𝐳^'), is selected as the SGP kernel due to its ability to model functions that vary across different length-scale <cit.>. This characteristic makes the RQ kernel well-suited for modeling the occupancy surface.
In Fig. <ref>, we present a concrete example of the SGP occupancy model applied to our Jackal robot, which is equipped with a Velodyne VLP-16 LiDAR and located in an unknown cluttered environment, as depicted in Fig <ref>. The figure also illustrates the raw pointcloud generated by the onboard sensor (Fig <ref>), as well as the original occupancy surface, which represents the projection of the point clouds onto the 2D circular surface with radius r_oc, where warmer colors indicate areas of lower occupancy (Fig <ref>). Furthermore, Fig <ref> exhibits the SGP occupancy surface reconstructed by the SGP occupancy model, as previously expressed in (<ref>).
The precision of the SGP occupancy model is intensively evaluated in our previous work <cit.>, where the results showed that an SGP occupancy model comprising of 400 inducing points generates a reconstructed point cloud with an average error of approximately 12.
§.§ GP-Subgoal Recommender Policy
The primary advantage of GP and its variants, compared to other modeling techniques, is their ability to provide a measure of variance, which indicates the level of uncertainty, along with a function estimate (i.e., mean).
More precisely, in the context of the occupancy surface, the SGP occupancy model prediction, as defined in (<ref>), provides both mean μ_oc_i and variance σ_oc_i values for each point on the surface, where the mean represents the expected occupancy while the variance reflects the uncertainty associated with the predicted occupancy.
Consequently, constructing the SGP occupancy surface is accompanied by an SGP variance surface that captures the uncertainty in the occupancy estimate, as depicted in Fig. <ref>.
Within this research, we have opened up a new avenue for effectively utilizing the SGP variance surface as a reliable indicator for distinguishing between occupied and free spaces around the robot, where regions with variances higher than a certain threshold V_th correspond to free space, while low-variance regions indicate occupied space.
In fact, the variance surface changes across observations due to variations in the number and distribution of observed points employed in the training of the SGP model.
As a result, the variance threshold V_th is considered to be a variable that relies on the distribution of the variance across the surface and can be calculated as V_th=K_m v_m, where K_m ∈ℝ^+ is a tuning parameter
and v_m represents the mean of the variance distribution.
To identify free navigable spaces, we define a Gaussian Process frontier (namely, GP frontier) as the centroid point (θ_i, α_i) of each high variance region. These GP frontiers {f_i}_i=1^ℱ serve as local recommended subgoals (see colored circles in Fig. <ref>).
Unlike the well-known frontier concept introduced in <cit.>, it is worth noting that our GP frontier does not rely on a global occupancy map; instead, it is extracted from the uncertainty of the current observation.
Following the identification of the GP frontiers by the SGP model, a cost function J_gp is utilized to determine the optimal GP frontier f^* that guides the local planner (in our case, MPPI) towards the desired state 𝐱_f.
Our cost function J_gp, given in (<ref>), has been established with two distinct terms. The first term, as introduced in <cit.>, calculates the distance d_fs between a GP frontier f_i and the desired state 𝐱_f. This distance criterion is used to identify the GP frontier closest to 𝐱_f. The second term, inspired by the direction criterion proposed in <cit.>, evaluates the direction θ_f_i of a GP frontier with respect to the robot heading. This criterion prioritizes a GP frontier that aligns better with the robot heading.
J_gp(f_i) = k_dst d_fs + k_dirθ_fi^2 ,
f^* =argmin _f_i∈ℱ(J_gp(f_i)),
where k_dst, k_dir are weighting factors. The GP frontier direction θ_f_i is squared to indicate the absolute direction.
Finally, the local planner receives the optimal subgoal g^*, obtained by acquiring the Cartesian coordinate of the optimal GP frontier f^*, which leads the robot to its desired state 𝐱_f.
§.§ Real-Time GP-MPPI Control Algorithm
Algorithm <ref> summarizes the real-time control cycle of the GP-MPPI algorithm, which includes two primary components: the local MPPI motion planner (described earlier in Section <ref>) and the GP-subgoal recommender (explained in Section <ref>).
Each time-step Δ t, the GP policy recommends the optimal subgoal g^*, the current state is estimated, and a M × N random control variations δ𝐮 are generated (lines 2:4). Then,
M trajectories are simulated in parallel, propagated from the system dynamics defined in (<ref>), and evaluated using (<ref>) (lines 5:13). It is noteworthy that the minimum sampled cost trajectory, denoted as S̃_min, among all simulated trajectories prevents numerical overflow or underflow without affecting the optimality of the algorithm <cit.>.
After that, the optimal control sequence {𝐮_k}_k=0^N-1 is updated, smoothed with a Savitzky-Galoy filter, and the first control 𝐮_0 is applied to the system (lines 14:18), while the remaining sequence of length N - 1 is slid down to be utilized at next time-step (lines 19:22).
In lines 25 to 38, the function known as GP-SubgoalRecommender is described, which takes a pointcloud input (PCL) and returns the optimal subgoal g^* for the local planner.
To optimize the hyper-parameters Θ and inducing points Z_m_s of the SGP occupancy model, the pointcloud data is transformed into training data 𝒟 (lines 26:29).
The mean occupancy μ_oc and variance σ_oc are then estimated over the surface Z^*, and the GP frontiers are defined as those with σ_oc > V_th, where the centroids of these frontiers are converted to Cartesian coordinates (lines 30:34). Finally, the cost function J_gp in (<ref>) is used to select the optimal subgoal g^* (lines 35:37).
In this study, we introduce two operating modes for the GP-MPPI algorithm: the simple mode (SM) and the recovery mode (RM). Under the simple mode, MPPI consistently leverages the optimal subgoal 𝐠^* suggested by the GP policy. In contrast, in the recovery mode, MPPI generates the optimal control sequence that steers the robot towards its desired state 𝐱_f, adhering to the recommended subgoal only when the robot is at risk of encountering local minima. Such local minima occur when the robot's linear velocity is zero (v=0) and its current state 𝐱_k does not match 𝐱_f (i.e., 𝐱_k ≠𝐱_f).
Thanks to the optimal control sequence {𝐮_k}_k=0^N-1 obtained by MPPI, we can efficiently anticipate the occurrence of local minima by imposing a condition on the mean of the predicted linear velocities over the time-horizon N, expressed as follows:
μ_𝐮 = 1/N∑_i=0^N-1 |v_i| < 𝐮_th,
where 𝐮_th∈ℝ^+ represents a control switching threshold set based on N.
If this condition is fulfilled, then MPPI will follow the subgoal recommended by the GP rather than navigating directly towards its desired state 𝐱_f.
§ SIMULATION-BASED EVALUATION
In this section, the effectiveness of our proposed control strategy is assessed and compared with both vanilla MPPI and log-MPPI control strategies in a goal-oriented autonomous ground vehicle (AGV) navigation task conducted in 2D cluttered environments of unknown nature.
§.§ Simulation Setup:
In this study, we consider the kinematics model of a differential wheeled robot presented in <cit.>, specifically the fully autonomous ClearPath Jackal robot, where the robot's position and orientation in the world frame are given by 𝐱 = [x, y, θ]^⊤∈ℝ^3, and the control input 𝐮 = [v,ω]^⊤∈ℝ^2 denotes the robot's linear and angular velocities.
Our autonomous AGV platform is equipped with a 16-beam Velodyne LiDAR sensor utilized for two key functions: (i) constructing the SGP variance surface, and (ii) generating the local costmap.
The simulations for all proposed control schemes were conducted with the following parameters: a prediction time of 6, a control frequency of 30 (i.e., N=180), sampling 2528 rollouts per time-step Δ t, and an exploration variance ν of 1200. Additionally, a control weighting matrix R, expressed as λΣ_n^-1/2, is utilized.
In the case of MPPI and GP-MPPI, the inverse temperature λ and the control noise co-variance Σ_𝐮 = Σ_n = Diag(σ_v^2, σ_w^2) are both set to 0.572 and Diag(0.023, 0.028), respectively. However, for log-MPPI, different values of 0.169 and Diag(0.017, 0.019) are used for these parameters, along with a normal distribution that has a co-variance of Σ_n = Diag(0.002, 0.0022) (For more details, refer to <cit.>).
The Savitzky-Galoy (SG) convolutional filter is utilized with a quadratic polynomial function, i.e., n_sg=2, and a window length l_sg of 51.
The occupancy surface was constructed with an occupancy radius r_oc of 5 meters, a full azimuth range of -180^o to 180^o,
and elevation height of 0^o to 15^o. The SGP occupancy model was designed with 400 inducing points (Z_m = 400), where the GP frontiers were identified based on a variance threshold of V_th= K_m v_m, where K_m was set to 0.4.
For the distance and direction factors K_dst and K_dir of the cost function J_gp, we assigned weighting factors of 5 and 4, respectively.
To enable the recovery mode of the GP-MPPI, we have set the control threshold, 𝐮_th, to 0.55[].
All the proposed control schemes, which are written in Python and integrated with the Robot Operating System (ROS) framework, are executed in real-time on an NVIDIA GeForce GTX 1660 Ti laptop GPU, with the GP-subgoal recommender built on GPflow<cit.>.
To accomplish the 2D navigation task, we adopt a state-dependent cost function described in (<ref>), which comprises two terms. The first term, with Q = Diag(2.5,2.5,5), aims to steer the robot towards its desired state, whereas the second term incorporates a Boolean variable 𝕀_crash to heavily penalizes collisions with obstacles.
q(𝐱_k)= (𝐱_k-𝐱_f)^⊤ Q (𝐱_k-𝐱_f) + 10^3 𝕀_crash.
Since the robot is operating in unknown environments, it relies on a 2D costmap to maintain a record of obstacles in its vicinity. This costmap is generated by analyzing sensor data from the environment and constructing a 2D occupancy grid, with each cell typically categorized as occupied, free, or unknown <cit.>.
The generated occupancy grid is subsequently employed as a 2D local costmap, feeding directly into the sampling-based MPC algorithm, enabling safe and collision-free navigation.
The robot-centered 2D local costmap, which is built by the on-board Velodyne VLP-16 LiDAR sensor, has a size of 200×200 and a grid resolution of 0.05/.
Finally, throughout the simulations, the maximum linear velocity v_max of the robot is set to 1.5/.
§.§ Simulation Scenarios and Performance Metrics:
The benchmark evaluation utilizes two types of Gazebo simulation environments, as depicted in Fig. <ref>. The first type, referred to as Forest #1, is a 50×50 forest-like environment characterized by tree-shaped obstacles with a density of 0.2/□;
The other type, named Maze #1, is a 20×20 maze-like environment with three U-shaped rooms (i.e., U_1, U_2, and U_3), as well as various other obstacles (highlighted in red in Fig. <ref>)[To evaluate the local planner's obstacle avoidance capability, the red obstacles are intentionally made undetectable as occupied space by the GP-subgoal recommender, as occupancy elevation height is set to a higher value.].
In the first scenario, denoted as Forest #1, the robot is directed to navigate from an initial pose 𝐱_s = [-5,-8,0]^⊤ to a desired pose 𝐱_f = [20,20,45]^⊤ in ([], [], []).
Meanwhile, in Maze #1, we conducted two separate control missions to (i) evaluate the robustness of our proposed control strategy, and (ii) examine its performance under the two different operating modes, previously described in Section <ref>.
The first mission, MU_1, requires the robot to navigate from 𝐱_s = [-5,-8,60]^⊤ to a desired pose 𝐱_f = [4,4,45]^⊤ located inside U_1; while, in the second mission, named MU_2, the robot starts at 𝐱_s = [-6,8,0]^⊤, crosses U_2, and reaches a desired pose of 𝐱_f = [8,-8,170]^⊤.
To ensure a fair and comprehensive comparison of the three control schemes, we have established a set of performance metrics, including the task completion percentage 𝒯_c, the average distance traveled by the robot d_av to reach 𝐱_f from 𝐱_s, the average linear velocity v_av of the robot within the cluttered environment, and the percentage of assistance 𝒜_gp provided by the GP-subgoal recommender policy to MPPI when the recovery mode is utilized. The successful task completion entails the robot reaching the target position without encountering obstacles or getting trapped in local minima ℛ_lm.
§.§ Simulation Results:
We evaluated the effectiveness of the proposed control strategies in Forest #1 and Maze #1 (i.e., MU_1 & MU_2) through 10 trials each, and the resulting performance statistics are summarized in Table <ref>.
The performance results demonstrate that, as expected, the proposed GP-MPPI control strategy outperforms both the vanilla MPPI and log-MPPI as the autonomous vehicle successfully accomplished all control missions (with 𝒯_c=100%) without getting stuck in local minima or colliding with obstacles (i.e., ℛ_lm =0), despite having limited perception range and incomplete knowledge of the environment.
In contrast, in Forest #1, log-MPPI achieved a task completion rate 𝒯_c of 95.72% over 10 trials, compared to 86.87% when MPPI was utilized. Additionally, log-MPPI encountered local minima only twice, while MPPI was trapped six times.
Nevertheless, both control methods were unable to complete any of the trials in MU_1 and MU_2 due to the challenging environmental conditions (refer to the robot trajectories generated by log-MPPI in Fig. <ref>).
Additionally, our proposed approach in Forest #1 provided a shorter route towards the desired state 𝐱_f, especially when the recovery mode (RM) is activated, similar to the optimal trajectory of the baselines, with an average linear velocity v_av of 1.30/, which approaches the maximum specified velocity of 1.5/.
Concerning the two modes of GP-MPPI, it is observed that activating the recovery mode (RM) during Forest #1 and MU_1 missions improves the average distance traveled d_av by the robot. For instance, in MU_1, d_av was approximately 32.74 with RM, whereas with the simple mode (SM), which consistently relies on the subgoal recommended by GP, d_av was roughly 34.48. On the other hand, during the MU_2 mission, the RM produced a slightly longer robot trajectory than the SM since operating our proposed GP-MPPI in the RM strikes a balance between the state-dependent cost function that directs the robot to follow a direct route towards the desired state and the optimal subgoal recommended by the GP policy that forces the robot to avoid the dead-ends associated with rooms U_2 and U_3 on its way to 𝐱_f, as illustrated in Fig. <ref>.
We can also see that, due to the presence of U-shaped rooms in Maze #1, the GP provides more assistance, represented by 𝒜_gp, than in Forest #1.
In Fig. <ref>, we illustrate through an example from the conducted trials the robot trajectories generated by GP-MPPI under the two operating modes in Maze #1. We can clearly observe that our proposed control strategy successfully achieves collision-free navigation in both modes, without getting stuck in local minima.
As an example, Fig. <ref> displays the velocity profile of the robot during the MU_1 mission shown in Fig. <ref>, while using GP-MPPI with RM, along with its corresponding mean of the predicted linear velocities μ_𝐮 over the given time-horizon N (see Fig. <ref>). The mean values that fall below the switching threshold 𝐮_th, set at 0.55[], denote the intervals where the RM is active, and are visually emphasized in yellow in Fig. <ref>.
§ REAL-WORLD DEMONSTRATION
In this section, we experimentally demonstrate the applicability of our proposed control strategy in achieving a safe 2D grid-based collision-free navigation in a complex and unknown indoor cluttered environment.
§.§.§ Experimental Setup and Validation Environment:
To conduct our experimental validation, we used the simulation setup previously outlined in Section <ref>, except for (i) setting the maximum speed v_max to 1.0/ to avoid the robot localization error associated with using the RealSense camera as a source of localization, (ii) setting the occupancy radius r_oc to 3.0,
and (iii) decreasing the size of the 2D grid map to 120×120.
r0.25
< g r a p h i c s >
Panoramic photo of our L-shaped indoor environment.
We also decreased the recovery mode switching threshold 𝐮_th to 0.3/ to be compatible with the updated v_max.
Additionally, to ensure real-time execution of the GP-subgoal recommender policy, we decrease the resolution of the SGP variance surface to one-third of its original value along the azimuth axis while keeping the original resolution along the elevation axis.
We employed an L-shaped indoor corridor environment measuring 9×14 for experimental validation. The environment has a varying width between 1.8 and 2.8 and contains randomly placed boxes-like obstacles, as depicted in Fig. <ref>.
The assigned control mission of the robot is to navigate from 𝐱_s = [0,0,0]^⊤ and arrive at 𝐱_f = [7.5,13,90]^⊤.
§.§.§ Experimental Results:
The performance statistics of our proposed GP-MPPI control scheme, gathered from four trials conducted in our indoor environment, are summarized in Table <ref> for the two operating modes.
From all trials, we can conclude that both operating modes provide collision-free navigation in the cluttered environment with an average linear velocity of 0.80, without the risk of being trapped in local minima (as ℛ_lm = 0) while moving towards its desired state. This ensures the safety and consistent feasibility of the receding-horizon planning.
In contrast, it is observed that the vanilla MPPI and log-MPPI consistently failed to complete any of the trials due to being trapped in the first edge of the L-shaped environment.
However, MPPI managed to avoid such traps with the aid of the GP-subgoal recommender policy in the recovery mode (RM), which provides an average assistance percentage 𝒜_gp of roughly 31.36%.
More details about the simulation and experimental results, including the behavior of the baselines, are provided in the supplementary video: <https://youtu.be/et9t8X1wHKI>.
§ CONCLUSION
In this work, we proposed the GP-MPPI control strategy, which comprises two primary components: the GP-subgoal recommender policy and the local planner, the MPPI. First, the GP-subgoal recommender utilized the learning capacity of SGP to create a reliable SGP variance surface, which served as an indicator for differentiating between occupied and free spaces around the robot. Consequently, a set of suggested subgoals was identified, and the optimal subgoal that minimizes a predefined cost function was recommended to the local MPPI planner. Based on the recommended subgoal, MPPI computes the optimal control input that enables the robot to navigate towards the goal efficiently and safely while accounting for its dynamics and avoiding collisions.
By conducting a combination of simulated and real-world experiments, we have shown that our proposed control strategy is superior to the vanilla MPPI and log-MPPI methods in achieving efficient and safe navigation in unknown and complex environments, thereby avoiding the risk of getting stuck in local minima.
IEEEtran
|
http://arxiv.org/abs/2307.05290v1 | 20230711143540 | Correlating the CDF $W$-mass shift with the muon $g-2$ and the $b \to s \ell^+ \ell^-$ transitions | [
"Xin-Qiang Li",
"Ze-Jun Xie",
"Ya-Dong Yang",
"Xing-Bo Yuan"
] | hep-ph | [
"hep-ph"
] |
An update of the catalog of radial velocity standard stars from the APOGEE DR17
[
===============================================================================
§ INTRODUCTION
The Standard Model (SM) of particle physics has proven incredibly successful in describing most phenomena observed in experiments <cit.>. At present, the major focus of the Large Hadron Colliders (LHC) is the direct searches for new particles and new interactions beyond the SM. While no confirmed direct signals for new physics (NP) beyond the SM have been observed at the LHC so far, several interesting hints of NP have been emerging from the precision measurements.
The long-standing anomaly of the muon g-2 provides an intriguing hint of NP. The latest measurement by the Muon g-2 collaboration at Fermilab <cit.>, after combined with the previous measurement by the Brookhaven E821 experiment <cit.>, shows a 4.2 σ discrepancy with the SM prediction <cit.>:
Δ a_μ=a_μ^ exp-a_μ^=(251±59)×10^-11,
with a_μ=(g-2)_μ/2. On the theoretical side, further detailed studies are presently on going to improve the precision of the hadronic contribution to the muon g-2 <cit.> (see also ref. <cit.> for a recent review).
Another interesting hint of NP comes from the updated measurement of the W-boson mass by the Collider Detector at Fermilab (CDF) collaboration <cit.>. Using the complete dataset collected by the CDF II detector, the CDF collaboration reported a value <cit.>
m_W^ CDF=80.4335 ± 0.0064_ stat± 0.0069_ syst.
Such a high precision measurement deviates from the average of the previous measurements from LEP, CDF, D0 and ATLAS, m_W^ PDG=80.379± 0.012 <cit.>, as well as from the LHCb measurement m_W^ LHCb = 80.354 ± 0.031 <cit.>, and the recent ATLAS measurement m_W^ ATLAS=80.360± 0.016 <cit.>. Furthermore, it shows a 7σ deviation from the SM expectation obtained through a global electroweak (EW) fit, m_W^ SM=80.357 ± 0.006 <cit.>. If confirmed by future precision measurements, this anomaly could imply another sign of NP beyond the SM.
Finally, several interesting discrepancies with the SM predictions have been observed in the b → s ℓ^+ ℓ^- (with ℓ=e, μ) processes over the last decade. Notably, the experimental picture has changed dramatically at the end of 2022: the previous R_K^(*) anomalies were not confirmed by the updated LHCb measurements <cit.>, and the recent CMS measurement of the branching fraction (B_s →μ^+ μ^-) <cit.> made the current world average <cit.> in excellent agreement with the SM prediction <cit.>. However, several observables in the b → s ℓ^+ ℓ^- transitions, especially the angular observable P_5^' in B → K^∗μ^+ μ^- <cit.> as well as the branching ratios of B → K^(∗)μ^+ μ^- <cit.> and B_s →ϕμ^+ μ^- <cit.>, still show 2∼ 3 σ deviations from the corresponding SM predictions. Furthermore, the recent global fits show that the overall consistency of the current b → s ℓ^+ ℓ^- data with the theoretical predictions can be significantly improved by adding NP to the short-distance Wilson coefficient _9 <cit.>.
It is noted that, within the SM, the (g-2)_μ, the W-boson mass and the b → s ℓ^+ ℓ^- transition all receive significant contributions from the loop diagrams with quarks or leptons. These loop diagrams could also be mediated by the NP fermions that have the same SM quantum numbers as of the SM ones. Therefore, these NP effects could simultaneously explain the data mentioned above. In this paper, in order to investigate such a possibility, we construct a NP model in which the SM is extended with the vector-like fermion partners that are featured by additional U(1)^' gauge symmetry. We assume that the fermion partners are SU(2)_L singlets and have the same SM quantum numbers as of the right-handed SM fermions. After the EW and the U(1)^' symmetry are spontaneously broken, possible mixings between the fermion partners and the SM right-handed fermions are obtained, which can result in desirable loop-level corrections to the (g-2)_μ, m_W, and _9. Our model can be regarded as an extension of the one introduced in refs. <cit.>. We will also consider the various constraints coming from the Z →μ^+ μ^- decay, the neutrino trident production and the LHC direct searches for the vector-like quarks and leptons.
The paper is organized as follows: In section <ref>, we introduce the NP model based on a new U(1)^' symmetry. In section <ref>, we recapitulate the theoretical framework for the various processes and investigate the NP effects on them. Our detailed numerical results and discussions are presented in section <ref>. We conclude in section <ref>. Details of the one-loop corrections to the Zμμ coupling and the global fit of the b → s ℓ^+ ℓ^- processes are presented in appendices <ref> and <ref>, respectively. The relevant loop functions are collected in appendix <ref>.
§ MODEL
As discussed in ref. <cit.>, in order to simultaneously accommodate the m_W^ CDF, the (g-2)_μ anomaly and the b → s ℓ^+ ℓ^- discrepancies, one can introduce the new fermions that are characterized by additional U(1)^' gauge symmetry and have the same SM quantum numbers as of the SM ones. To this end, let us firstly introduce our model based on the new U(1)^' symmetry, where the SM and NP fields as well as their charges under the SU(3)_C ⊗ SU(2)_L ⊗ U(1)_Y ⊗ U(1)^' gauge symmetry are given in table <ref>.
§.§ Quark sector
The quark sector of the model is identical to that introduced in refs. <cit.>. All the SM quarks do not carry the U(1)^' charge, while a vector-like top partner with
U_L/R^'=(3 , 1 , 2/3 , q_t),
and a complex scalar with
=(1 , 1 , 0 , q_t),
are introduced, where the quantum numbers in brackets dictate the transformations under the SU(3)_C ⊗ SU(2)_L ⊗ U(1)_Y ⊗ U(1)^' gauge symmetry.
The general Lagrangian involving the top partner U_L/R^', the SM Higgs doublet H and the scalar can be written as
ℒ⊃ q_t g^'(U̅_L^'γ^μ U_L^'+U̅_R^'γ^μ U_R^') Z_μ^'
-(∑_iλ_iiQ̅_i LH̃ u_i R+ λ_4iU̅_L^' u_i R +μU̅_L^' U_R^'+h.c.),
where g^' denotes the U(1)^' gauge coupling, and Q_iL=(u_iL,d_iL)^T and u_iR (i=1, 2, 3) stand for the i-th generation of the left-handed quark doublet and the right-handed quark singlet of the SM, respectively. As discussed in refs. <cit.>, mixings of U_L/R^' with the first two generations suffer from severe experimental constraints. As a consequence, such mixings are assumed to be small compared to its mixing with the third generation and can be therefore neglected, as done in refs. <cit.>; this leads us to set λ_41=λ_42=0. In the following, we will use the abbreviations λ_H=λ_33 and λ_=λ_43.
The fermions present in eq. (<ref>) are all given in the interaction eigenbasis. Without loss of generality, we have chosen the basis where the up-type Yukawa matrix is diagonal in the 3× 3 SM flavor space. After the EW and the U(1)^' symmetry breaking, the mass matrix for the up-type quarks takes the form
[ λ_11v_H 0 0 0; 0 λ_22 v_H 0 0; 0 0 λ_H v_H 0; 0 0 λ_ v_ √(2)μ ],
where the vacuum expectation values (vevs) of H and fields are defined by ⟨ H ⟩=v_H/√(2) and ⟨⟩=v_/√(2), respectively. When diagonalizing the mass matrix, only the rotation between u_3 and U^' is needed in the up sector, and the CKM matrix is defined by the rotation among the down-type quarks. We refer the readers to refs. <cit.> for more details. As a result, the physical top quarks (t_L, t_R) and their partners (T_L, T_R) are related to the fermions present in eq. (<ref>) through <cit.>
[ t_L; T_L ]
=R(θ_L)
[ u_3 L; U_L^' ]
, [ t_R; T_R ]
=
R(θ_R)
[ u_3 R; U_R^' ]
,
with the rotation matrix given by
R(θ)=
[ cosθ -sinθ; sinθ cosθ ]
,
where the mixing angles θ_L and θ_R parameterize the rotation matrices of the left- and right-handed quarks, respectively. In terms of the physical parameters, the top-quark mass m_t, the top-partner mass m_T and the two mixing angles are related to each other through
tanθ_L=m_t/m_Ttanθ_R,
with m_t and m_T determined by
λ_H =cosθ_L/cosθ_R√(2) m_t/v_H,
λ_ = cosθ_Lsinθ_R
√(2) m_T/v_(1- m_t^2/ m_T^2).
In the fermion mass eigenbasis, the gauge interactions involving the top quark and the top partner take the form <cit.>
ℒ_γ^t = 2/3 e t̅A t+2/3 e T̅A T,
ℒ_W^t = g/√(2)V_td_i(c_L t̅W P_L d_i+s_L T̅W P_L d_i)+ h.c. ,
ℒ_Z^t = g/c_W(t̅_L, T̅_L)
[ 1/2 c_L^2-2/3 s_W^2 1/2 s_Lc_L; 1/2 s_Lc_L 1/2 s_L^2-2/3 s_W^2 ] Z[ t_L; T_L ]
+g/c_W(t̅_R, T̅_R) (-2/3s_W^2 ) Z[ t_R; T_R ],
ℒ_^t= g_t(t̅_L, T̅_L)
[ s_L^2 -s_Lc_L; -s_Lc_L c_L^2 ]Z^'[ t_L; T_L ]
+ (L → R),
where g is the SM weak coupling constant, = q_t, c_L,R=cosθ_L,R, s_L,R=sinθ_L,R, and s_W=sinθ_W with θ_W being the Weinberg angle; d_i ∈{ d,s,b } denote the down-type quarks and V_td_i the CKM matrix elements. In eq. (<ref>), V_td_i arise from the rotation among the left-handed down-type quarks, while c_L and s_L from the rotation specified by eq. (<ref>). The scalar interactions involving the top quark and the top partner can be written as
ℒ_h^t =-m_t/v_H(t̅_L, T̅_L) ([ c_L^2 c_L^2 tanθ_R; s_L c_L s_L c_L tanθ_R ]) h[ t_R; T_R ]+h.c. ,
ℒ_ = -λ_Φ_t/√(2)(t̅_L, T̅_L) ([ -s_L c_R s_L s_R; c_L c_R c_L s_R ]) h[ t_R; T_R ]+h.c. .
Except for the interactions involving and , the quark sector of this model is similar to that of the generic vector-like quark models, which have been extensively studies in the literature <cit.>.
§.§ Lepton sector
In the lepton sector, the model is based on the U(1)_L_μ-L_τ gauge symmetry <cit.>. Both the second- and the third-generation leptons are charged under the U(1)^' gauge group. Explicitly, their quantum numbers under the SU(3)_C⊗ SU(2)_L ⊗ U(1)_Y⊗ U(1)^' gauge symmetry are given by
L_2L =(1 , 2 , -1/2 , +q_ℓ), e_2R =(1 , 1 , -1 , +q_ℓ),
L_3L =(1 , 2 , -1/2 , -q_ℓ), e_3R =(1 , 1 , -1 , -q_ℓ),
where L_2L and e_2R (L_3L and e_3R) denote the second-generation (third-generation) left-handed lepton doublet and right-handed lepton singlet of the SM, respectively. q_ℓ denotes the charge of the U(1)^' gauge symmetry. We also introduce a vector-like muon partner
E_L/R=(1 , 1 , -1 , 0),
as well as two complex scalar fields
ϕ=(1 , 1 , 0 , 0), Φ_ℓ=(1 , 1 , 0 , -q_ℓ).
After spontaneous symmetry breaking, the field ϕ provides mass to the muon partner, while the field Φ_ℓ induces mixing between the muon lepton and the muon partner.
The general Lagrangian involving the boson and the scalars H, and ϕ is given by
ℒ⊃ q_ℓ g^'(L̅_2Lγ^μ L_2L+e̅_2Rγ^μ e_2R-L̅_3Lγ^μ L_3L-e̅_3Rγ^μ e_3R) Z_μ^'
-[ ∑ _iλ_ii^ℓL̅_iL H e_i R+ λ_42^ℓE̅_L e_2 R + λ_43^ℓE̅_L e_3RΦ_ℓ^*
+(λ_41^ℓE̅_L e_1R+λ_44^ℓE̅_L E_R ) ϕ+h.c.],
where L_iL=(ν_iL, e_iL)^T and e_iR (i=1, 2, 3) are the i-th generation of the SM left-handed lepton doublet and right-handed lepton singlet, respectively. Here, for simplicity, we assume that the mixings of E_L/R with the first and third generations are small and can be therefore neglected, i.e., λ_43^ℓ=λ_41^ℓ=0. The abbreviations η_H=λ_22^ℓ, λ_=λ_42^ℓ and λ_ϕ=λ_44^ℓ will be used in the following.
In eq. (<ref>), the fermions are all given in the interaction eigenbasis. After the EW and the U(1)^' symmetry breaking, where the scalars H, and ϕ acquire their vevs, the resulting mass matrix for the charged leptons is similar to eq. (<ref>). After diagonalization, the physical muons (μ_L, μ_R) and their partners (M_L, M_R) can be expressed as
[ μ_L; M_L ]
=
R(δ_L)
[ e_2 L; E_L ]
,
[ μ_R; M_R ]
=
R(δ_R)
[ e_2 R; E_R ]
,
where the mixing angles δ_L and δ_R parameterize the rotation matrices of the left- and right-handed leptons, respectively. The two mixing angles are related to each other through
tanδ_L=m_μ/m_Mtanδ_R,
where m_μ and m_M denote the muon and the muon-partner mass, respectively. Together with the mixing angles and the vevs of H, Φ_ℓ and ϕ, the physical masses m_μ and m_M can be expressed in terms of the Yukawa couplings η_H, λ_Φ_ℓ and λ_ϕ as
m_μ =1/√(2)(cosδ_L cosδ_R v_H η_H-sinδ_L cosδ_R v_Φ_ℓλ_Φ_ℓ+sinδ_L sinδ_Rv_ϕλ_ϕ),
m_M =1/√(2)(sinδ_L sinδ_R v_H η_H+cosδ_L sinδ_R v_Φ_ℓλ_Φ_ℓ+cosδ_L cosδ_Rv_ϕλ_ϕ),
and vice versa as
η_H =cosδ_L/cosδ_R√(2) m_μ/v_H,
λ_Φ_ℓ = cosδ_Lsinδ_R√(2) m_M/v_Φ_ℓ(1- m_μ^2/ m_M^2),
λ_ϕ =cosδ_R/cosδ_L√(2) m_M/v_ϕ,
where v_Φ_ℓ=√(2)⟨Φ_ℓ⟩, v_ϕ=√(2)⟨ϕ⟩, and v_H is the vev of the SM Higgs doublet.
In the fermion mass eigenbasis, explicit expressions of the gauge interactions involving the muon and the muon partner can be written as
ℒ_γ^ℓ = - e μ̅Aμ- e M̅A M,
ℒ_W^ℓ = g/√(2)(ĉ_L μ̅W P_Lν_μ+ŝ_L M̅W P_Lν_μ)+ h.c. ,
ℒ_Z^ℓ = g/c_W(μ̅_L, M̅_L)
[ -1/2ĉ_L^2+ s_W^2 -1/2ŝ_Lĉ_L; -1/2ŝ_Lĉ_L -1/2ŝ_L^2+s_W^2 ] Z[ μ_L; M_L ]
+g/c_W s_W^2(μ̅_R, M̅_R) Z[ μ_R; M_R ],
ℒ_^ℓ= [g_ℓ(μ̅_L, M̅_L)
[ ĉ_L^2 ŝ_L ĉ_L; ŝ_L ĉ_L ŝ_L^2 ]Z^'[ μ_L; M_L ]
-g_ℓτ̅_L τ_L
+ (L → R)]
+ g_ℓ(ν̅_μZ^' P_Lν_μ-ν̅_τZ^' P_Lν_τ),
where = q_ℓ, ŝ_L,R=sinδ_L,R and ĉ_L,R=cosδ_L,R. The Yukawa interactions of the scalars with the muon and the muon partner are given by
ℒ_h =-m_μ/v_H(μ̅_L, M̅_L) ([ ĉ_L^2 ĉ_L^2 tanδ_R; ŝ_L ĉ_L ŝ_L ĉ_L tanδ_R ]) h[ μ_R; M_R ]+h.c. ,
ℒ_Φ_ℓ =-λ_Φ_ℓ/√(2)(μ̅_L, M̅_L) ([ -ŝ_L ĉ_R -ŝ_L ŝ_R; ĉ_L ĉ_R ĉ_L ŝ_R ])Φ_ℓ[ μ_R; M_R ]+h.c. ,
ℒ_ϕ =-λ_ϕ/√(2)(μ̅_L, M̅_L) ([ ŝ_L ŝ_R -ŝ_L ĉ_R; -ĉ_L ŝ_R ĉ_L ĉ_R ])ϕ[ μ_R; M_R ]+h.c. ,
where h, and ϕ denote the physical scalar fields after spontaneous symmetry breaking.
§.§ Choice of the model parameters
To complete our model, we also need to specify the scalar potential, which is given by
𝒱 = ∑_S [ μ_S^2 |S|^2 + Re(λ_S^(3)ϕ) |S|^2 - λ_S^(4) |S|^4]
+( λ_Hϕ|ϕ|^2 + λ_H ||^2 + λ_H|Φ_ℓ|^2 ) H^† H
+ ( λ_ϕ ||^2 + λ_ϕ |Φ_ℓ|^2 )|ϕ|^2 + λ_ ||^2|Φ_ℓ|^2,
where S∈{H, ϕ, , }. The potential is assumed to be such that all the scalar fields acquire only real vevs. The scalar fields and are responsible for the spontaneous breaking of the U(1)^' gauge symmetry, and give mass to the boson, with m_^2=g^'^2 (v_^2+v_^2).
In this paper, we focus on the parameter region with m_T>m_t and m_M > m_μ. In the quark sector, without loss of generality, the parameters λ_H, v_H and μ are chosen to be positive. After taking into account the relations in eqs. (<ref>) and (<ref>), the cases with v_ >0 and v_<0 correspond to the ranges of 0⩽θ_L<θ_R<π/2 and π/2<θ_R<θ_L⩽π, respectively. Similarly, in the lepton sector, the parameters η_H, λ_Φ_ℓ and λ_ϕ are also chosen to be positive. Considering the relations in eqs. (<ref>)–(<ref>), we find that v_ϕ>0, and the cases with v_ >0 and v_<0 correspond to the ranges of 0⩽δ_L<δ_R<π/2 and π/2<δ_R<δ_L⩽π, respectively. In the following analysis, we will focus on the case where all the vevs are positive and the mixing angles are restricted within the regions of 0⩽θ_L<θ_R<π/2 and 0⩽δ_L<δ_R<π/2.
§ THEORETICAL FRAMEWORK
In this section, we investigate the relevant observables affected by our model, including the W-boson mass, the (g-2)_μ, the b → s ℓ^+ ℓ^- processes, as well as the Zμ^+μ^- couplings. The NP contributions to most of them arise at the loop level. During our evaluations, all the loop diagrams are calculated in the unitary gauge. As in ref. <cit.>, the computations are implemented in two independent methods by using different packages including <cit.>, <cit.>, <cit.>, <cit.>, as well as some in-house routines.
§.§ W-mass shift and oblique parameters
The global fit to the EW precision data, known as the global EW fit <cit.>, is a powerful tool to test the SM as well as to probe possible NP effects <cit.>. As an important EW precision observable, the W-boson mass in the SM is determined from the global EW fit. Recently, the CDF m_W measurement <cit.> shows large deviation from the SM prediction, which could be explained by NP contributions; see e.g. refs. <cit.> and the references therein. In our model, the W-boson mass shift can be divided into the following three parts:
Δ m_W = Δ m_W^Q+Δ m_W^L + Δ m_W^Wμν,
where Δ m_W^Q, Δ m_W^L and Δ m_W^Wμν denote the contributions from the top-partner, the muon-partner and the modified Wμν vertex, respectively. We show in figure <ref> the relevant one-loop Feynman diagrams for these NP contributions.
The oblique parameters S, T and U encode most of the NP effects on the SM EW sector <cit.>. To be specific, these parameters capture the NP contributions from the gauge-boson vacuum-polarization corrections and can be generically written as <cit.>
S =4 s_W^2c_W^2/α_e [Π_Z Z(m_Z^2)- Π_Z Z(0)/m_Z^2-c_W^2-s_W^2/s_W c_WΠ_Z γ^'(0)- Π_γγ^'(0)],
T =1/α_e[Π_W W(0)/ m_W^2-Π_Z Z(0)/m_Z^2],
U =4 s_W^2/α_e [Π_W W(m_W^2)- Π_W W(0)/m_W^2-c_W/s_WΠ_Zγ^'(0)- Π_γγ^'(0)]-S,
where Π_XY denotes the vacuum polarization of the gauge fields with X,Y=W,Z,γ, c_W=cosθ_W, and α_e is the fine structure constant. In terms of these oblique parameters, the W-boson mass shift can be written as <cit.>
Δ m_W^2=α_e c_W^2 m_Z^2/c_W^2-s_W^2[-S/2+c_W^2 T+c_W^2-s_W^2/4 s_W^2 U].
Therefore, to explain the discrepancy of the CDF m_W measurement from the SM expectation, we need a global EW fit to the oblique parameters S, T and U, which could be affected by contributions from both the quark and the lepton sector in our model. Let us discuss them in turn.
In the model introduced in section <ref>, extra contributions to the vacuum polarizations of gauge fields arise from the diagrams shown in figure <ref>. Figures <ref> and <ref> encode the contributions from the modified quark-gauge couplings that are characterized by the mixing angle θ_L and the loops involving the top partner, respectively. Their contributions to the oblique parameters read <cit.>
S_Q = s_L^2/12π[ K_1(y_t, y_T) + 3 c_L^2 K_2 (y_t, y_T) ],
T_Q = 3s_L^2/16π s_W^2[x_T-x_t - c_L^2(x_T+x_t +2x_tx_T/x_T-x_tlnx_t/x_T)],
U_Q = s_L^2/12π[ K_3(x_t,y_t)-K_3(x_T,y_T)] -S_Q,
where x_q=m_q^2/m_W^2 and y_q=m_q^2/m_Z^2 for q=t,T. Explicit expressions of the loop functions K_1,2,3(x,y) are recapitulated in appendix <ref>.
Similar to the quark sector, contributions from the lepton sector arise from figures <ref> and <ref>. Their contributions to the oblique parameters can be written as
S_L =ŝ_L^2/12π[ K_4(y_μ, y_M) + ĉ_L^2 K_5 (y_μ, y_M) ],
T_L =ŝ_L^2/16π s_W^2[x_M-x_μ - ĉ_L^2(x_M+x_μ +2x_μx_M/x_M-x_μlnx_μ/x_M) ],
U_L = ŝ_L^2/12π[ K_6(x_μ,y_μ)-K_6(x_M,y_M)] -S_L,
where x_ℓ=m_ℓ^2/m_W^2 and y_ℓ=m_ℓ^2/m_Z^2 for ℓ=μ,M, ŝ_L=sinδ_L and ĉ_L=cosδ_L. Explicit expressions of the loop functions K_4,5,6(x,y) are listed in appendix <ref>. From eqs. (<ref>)–(<ref>) (eqs. (<ref>)–(<ref>)), one can see that the contributions to the oblique parameters S, T and U from the lepton (quark) sector are solely determined by the two NP parameters δ_L and m_M (θ_L and m_T) and are proportional to sin^2δ_L (sin^2θ_L).
Finally, let us discuss the last term of eq. (<ref>), Δ m_W^Wμν, which is induced by the modified Wμν coupling and characterized by the mixing angle δ_L, as given in eq. (<ref>). The modified Wμν coupling can affect the muon lifetime, from which the Fermi constant G_F is extracted. During the global EW fit, the prediction for m_W is obtained from its relation to G_F. This implies that the input G_F used for calculating m_W should be the modified one rather than the one given by the Particle Data Group (PDG) <cit.>. Therefore, the NP correction to the Wμν coupling indirectly translates to a shift of the W-boson mass, Δ m_W^Wμν, which can be written as <cit.>
(m_W^SM+Δ m_W^Wμν)^2/m_Z^2=1/2+√(1/4-πα_e/√(2) G_F m_Z^2|1+Δ r|),
where Δ r encodes the NP correction to the muon lifetime. Specific to our model, the modified Wμν coupling results in Δ r=cosδ_L-1.
§.§ b → s ℓ^+ ℓ^- transitions
Our model can also be efficiently explored through the quark-level b → s ℓ^+ ℓ^- transitions, such as the B → K^(∗)ℓ^+ ℓ^-, B_s →ℓ^+ ℓ^- and B_s →ϕℓ^+ ℓ^- decays. Here the NP contributions arise firstly at the one-loop level, with the relevant Feynman diagrams shown in figure <ref>. As found already in ref. <cit.>, the NP contributes only to the short-distance Wilson coefficients _9 and _10 of the low-energy effective Hamiltonian governing the b → s ℓ^+ ℓ^- transitions <cit.>
ℋ_eff⊃-4 G_F/√(2) V_t b V_t s^*α_e/4 π(_9ℓ𝒪_9ℓ+ _10ℓ𝒪_10ℓ)+h.c.,
with the two semi-leptonic operators given by 𝒪_9ℓ=(s̅γ^μ P_L b)(ℓ̅γ_μℓ) and 𝒪_10ℓ=(s̅γ^μ P_L b)(ℓ̅γ_μγ_5 ℓ), respectively. The NP contributions to _9ℓ and _10ℓ can be divided into the Lepton-Flavor Universal (LFU) and the Lepton-Flavor Violating (LFV) parts. The former arise from the diagrams with the SM gauge bosons, and are given by
_9ℓ^ NP= s_L^2 I_1+ s_L^2 (1-1/4s_W^2) (I_2 + c_L^2 I_3 ),
_10ℓ^ NP= s_L^2/4s_W^2(I_2 + c_L^2 I_3 ),
with s_L,R=sinθ_L,R and c_L,R = cosθ_L,R. They are proportional to sin^2θ_L. The LFV contributions can be written as
_9μ^ NP=Δ_+^W,Z+Δ_+^,
_10μ^ NP=Δ_-^W,Z+Δ_-^,
with
Δ_±^W,Z= ±ŝ_L^2 s_L^2/8s_W^2 (I_6+c_L^2 I_7),
Δ_±^ = (ĉ_R^2±ĉ_L^2)g_ℓ g_t /e^2m_W^2/m_^2c_L^2s_R^2 ( I_4 -c_L^2/c_R^2 I_5 ),
where = q_ℓ, = q_t, ŝ_L,R = sinδ_L,R and ĉ_L,R = cosδ_L,R. Here the contributions Δ_±^W,Z arise from the diagrams involving the W and Z bosons, while Δ_±^ from the -penguin diagrams. In our case of small mixing angles θ_L and δ_L, as will be demonstrated in section <ref>, Δ_±^W,Z will be highly suppressed because they are proportional to ŝ_L^2 s_L^2=sin^2δ_Lsin^2θ_L. Thus, we can safely take the approximations _9e^ NP≈_10e^ NP≈0 and (_9μ^, _10μ^) ≈(Δ_+^, Δ_-^). The loop integrals I_1-7 are functions of m_t,T and m_μ,M, whose explicit expressions are given in appendix <ref>. Keeping only the leading terms in sinθ_R and m_W^2/m_T^2, our results are in agreement with that obtained in ref. <cit.>.
§.§ Muon g-2
The muon anomalous magnetic moment a_μ≡
(g-2)_μ/2 can provide very promising probes of potential NP effects <cit.>. Specific to our model, the observable a_μ is affected by the one-loop Feynman diagrams shown in figure <ref>, which involve the , ϕ and bosons, as well as the SM diagrams but with the modified Zμμ, hμμ and Wμν couplings. Their total contributions to Δ a_μ can be written as
Δ a_μ=Δ a_μ^+Δ a_μ^ϕ+Δ a_μ^+Δ a_μ^W +Δ a_μ^Z +Δ a_μ^h,
where Δ a_μ^i=Δ a_μ^i, μ+Δ a_μ^i, M (i=Z, , ϕ,), and Δ a^i, j_μ (j=μ, M) denotes the contribution from the diagrams involving the particles i and j. Their explicit expressions are given, respectively, by
Δ a_μ^Z, μ= -g^2 w_μ(ĉ_L^4-2ŝ_L^2s_W^2-1)/48π^2c_W^2,
Δ a_μ^Z, M= -ĉ_L^2ŝ_L^2w_μg^2(5w_M^4-14w_M^3+39w_M^2-38w_M+8-18w_M^2 ln w_M)/384π^2c_W^2(w_M-1)^4,
Δ a_μ^, μ= - g_ℓ^2 x_μ(ĉ_L^4-3 ĉ_L^2 ĉ_R^2+ĉ_R^4)/12 π^2,
Δ a_μ^, M= ŝ_L^2 ĉ_R^2 g_ℓ^2 x_M (x _M^3+3 x_M-4-6 x_M ln x_M)/16 π^2(x_M-1)^3,
Δ a_μ^ϕ, μ= ŝ_L^2 ŝ_R^2 λ_ϕ^2 /32π^2 f_s(z_μ),
Δ a_μ^ϕ, M= ŝ_L^2 ĉ_R^2 λ_ϕ^2 z_M(z_M^2-4z_M+3+2 ln z_M)/32π^2(z_M-1)^3
+z_μ(ĉ_R^2 ŝ_L^2+ĉ_L^2 ŝ_R^2)λ_ϕ^2( z_μ^3-6z_μ^2+3z_μ+2+6x_M ln z_μ)/192π^2(z_M-1)^4,
Δ a_μ^Φ_ℓ, μ= ŝ_L^2 ĉ_R^2 λ_Φ_ℓ^2 /32π^2f_s(y_μ),
Δ a_μ^Φ_ℓ, M= -ŝ_L^2 ĉ_R^2 λ_Φ_ℓ^2 y_M (y_M^2-4y_M+3+2 lny_M) /32π^2(y_M-1)^3 ,
Δ a_μ^W= -5ŝ_L^2g^2w_μ/96π^2c_W^2,
with
f_s(x)= 3x -2/x +(3x-1)/x^2ln x +2 √(1-4 x)(x-1)/x^2ln1+√(1-4x)/2√(x),
where w_i=m_i^2/m_Z^2, x_i=m_i^2/m_^2, y_i=m_i^2/m_Φ_ℓ^2 and z_i=m_i^2/m_ϕ^2 (i=μ, M). From these results, one can see that the NP contributions to a_μ will vanish in the limit of δ_L→ 0 or δ_R → 0.
§.§ Neutrino trident production
The couplings to muons are significantly constrained by the rare process of neutrino trident production, i.e., the production of a μ^+μ^- pair from the scattering of a muon-neutrino with heavy nucleus <cit.>. In our model, the boson contributes to this rare process through the tree-level Feynman diagram shown in figure <ref>. The ratio of the cross section in our model to that in the SM, r_ν TP≡σ_/σ_, is calculated to be <cit.>
r_ν TP=|g_ℓ^2 ( ĉ_L^2+ ĉ_R^2)/m_^2+√(2)(1+4 s_W^2) G_F|^2+|g_ℓ^2(ĉ_L^2- ĉ_R^2)/m_^2+√(2) G_F|^2/2 G_F^2[1+(1+4 s_W^2)^2],
where =g_ℓ and ĉ_L,R=cosδ_L,R.
§.§ Zμμ and Wμν couplings
Any modification of the Z couplings to leptons must receive stringent constraints from the LEP measurements at the Z pole <cit.>. In our model, the effective Z couplings to leptons are given by
ℒ=g/ c_Wℓ̅Z(g_LℓP_L+g_RℓP_R)ℓ ,
with ℓ=μ or e. The effective couplings g_Lμ/Rμ include both the tree-level couplings specified by eq. (<ref>) and the one-loop vertex corrections shown in figure <ref>. The latter can be written as
Δ g_Γμ = Δ g_Γμ^ + Δ g_Γμ^ + Δ g_Γμ^ϕ + Δ g_Γμ^W + Δ g_Γμ^Z + Δ g_Γμ^γ + Δ g_Γμ^h + Δ g_Γμ^Z- ,
with Γ=L or R. Here Δ g_Γμ^i denotes the correction from the diagram involving the particle i for i=, , ϕ, W, Z, γ and h. As done in ref. <cit.>, these corrections are calculated in the on-shell renormalization scheme. Furthermore, we find that the renormalization of the mixing angle δ_L has to be taken into account. Details of the calculation are given in appendix <ref>. The vertex corrections Δ g_Γμ^, ϕ, depend on , λ_Φ_ℓ, λ_ϕ, m_μ/M, m_, m_Φ_ℓ, m_ϕ, and their explicit expressions are provided as an ancillary notebook file. In addition, the Z- mixing via the top (top-partner) loops could also affect the Zμμ couplings <cit.>. Such an effect, denoted by Δ g_Γμ^Z- in eq. (<ref>), can be written as
Δ g_Γμ^Z-= 3 s_L^2 ĉ_Γ ^2/32 π^2 (m_Z^2-m_^2) (c_L^2I_L + c_R^2I_R) ,
where the loop functions I_L,R depend on m_t and m_T, and their explicit expressions can be found in appendix <ref>. Finally, the contributions involving the SM particles are all proportional to sin^2δ_L. In our numerical analysis, we will consider the case of small mixing angle δ_L to avoid large modification of the Zμμ vertex at the tree level. Therefore, these contributions can be safely neglected, which means that Δ g_Γμ^W,Z,γ,h≈ 0.
To constrain the Zμμ couplings, we follow refs. <cit.> and consider the following two observables. The first one is the LFU ratio, R^Z_μ/e≡Γ(Z→μ^+μ^-)/Γ(Z→ e^+ e^-), measured at LEP-I <cit.>. Using the effective couplings defined in eq. (<ref>), we can write the decay width of Z →ℓ^+ ℓ^- as
Γ(Z→ℓ^+ ℓ^-) = m_Z λ^1/2_Z6π v^2[ (|g_Lℓ|^2+|g_Rℓ|^2)(1- m_ℓ^2m_Z^2)+ 6 m_ℓ^2m_Z^2 Re(g_Lℓ g_Rℓ^*) ] ,
with λ_Z≡ m_Z^2(m_Z^2-4m_ℓ^2). The second observable is the leptonic asymmetry parameter defined by <cit.>
𝒜_μ=Γ(Z→μ^+_Lμ^-_L)-Γ(Z→μ^+_Rμ^-_R)/Γ(Z→μ^+ μ^-),
where μ_L/R corresponds to the left/right-handed muon. Specific to our model, we obtain
Γ(Z→ℓ^+_Γℓ^-_Γ )= m_Z λ^1/2_Z6π v^2|g_Γℓ|^2 (1- m_ℓ^2m_Z^2 ) ,
for Γ =L or R.
From eq. (<ref>), one can see that the mixing angle δ_L can change the Wμν coupling, and thus affects the branching ratio of the W^+→μ^+ ν decay. Explicitly, we have the relation
(W^+→μ^+ν)/_(W^+→μ^+ν)=cos^2 δ_L,
which is valid at the tree level.
§ NUMERICAL ANALYSIS
In this section, we proceed to present our numerical results and discussions. Firstly we list in table <ref> the main input parameters used in our numerical analysis.
As discussed in section <ref>, the relevant independent parameters in our model contain, besides the masses of the NP particles, the mixing angles and the couplings g_t, g_ℓ, λ_ϕ, λ_, λ_, where the re-definitions = q_ℓ and = q_t have been taken. In our numerical analysis, we consider the following parameter space:
0 < g_t, g_ℓ, λ_ϕ, λ_, λ_ < 2.
Taking into account the relations specified by eqs. (<ref>) and (<ref>), we will take θ_L and δ_L as the two independent mixing angles in the quark and lepton sectors, respectively. Their values are varied within the following ranges:
0<sinθ_L<0.5, 0<sinδ_L<0.01,
to avoid large tree-level modifications to the W, Z and h couplings to fermions, as will be discussed in detail in the next subsection.
§.§ Zμμ and Wμν couplings
As discussed in section <ref>, the Zμμ coupling is stringently constrained by the LEP measurements <cit.>, especially by the observables R^Z_μ/e and 𝒜_μ. In our model, the Zμμ coupling is affected by the mixing angle δ_L at the tree level, and also receives contributions from the Z- mixing and the vertex corrections at the one-loop level.
In our numerical analysis, we take ==1, λ_ϕ=λ_=1,[We have checked that varying λ_ϕ and λ_ within the range of eq. (<ref>) only changes our results slightly.] m_ϕ=1 and m_=2, which are all consistent with the constraints discussed in the following subsections. The parameter space of sinθ_L is chosen within the range allowed by the CDF m_W measurement, as will be derived in section <ref>. Then, the remaining relevant parameters are (m_ , m_T, m_M, sinδ_L). Taking the LEP measurements R^Z_μ/e= 1.0001 ± 0.0024 <cit.> and 𝒜_μ=0.1456± 0.0091 <cit.> as constraints, we can finally obtain the allowed values of these parameters. As an illustration, we show in figure <ref> the 2 σ upper bounds on sinδ_L as a function of m_ from the measured Zμμ coupling, for m_M=110, 300 and 500 as well as m_T=1000, 1500 and 2000. It can be seen that a lighter or a heavier T implies a stronger upper bound on sinδ_L, while the upper bound is not sensitive to m_M. Numerically, we obtain sinδ_L < 0.05 in the 2 σ allowed parameter space. Furthermore, we have checked that, in the 2 σ allowed parameter space, the NP contributions to the Zμμ coupling from the Z- mixing and the one-loop vertex corrections are both less than 1% of the SM prediction. This in turn means that the fine-tuning among the different NP contributions is small.
On the other hand, the mixing angle δ_L receives also constraint from the measured branching ratio of W^+→μ^+ν decay, as indicated by eq. (<ref>). Taking as input the LEP measurement (W^+→μ^+ν)_ exp=0.1063(15) <cit.> and the SM prediction ℬ(W^+→μ^+ν)_ SM=0.1083 <cit.>, we obtain the upper bound sinδ_L<0.43, which is much weaker than that from the Zμμ coupling.
In the following, we will consider the upper bound sinδ_L < 0.01, which definitely satisfies the constraint from the Zμμ coupling. In addition, since the Zμμ coupling is not sensitive to m_M, a value of m_M=300 will be taken for simplicity. As will be demonstrated later, such a choice of sinδ_L and m_M is enough to explain the muon g-2 anomaly and the b→ s ℓ^+ℓ^- discrepancies, while satisfying most of the relevant constraints mentioned in section <ref>.
§.§ W-boson mass and global EW fit
As the NP effects could affect all the three oblique parameters S, T and U, a global EW fit is necessary to see if our model could explain the CDF m_W measurement <cit.>. Recently, the global EW fit including the latest CDF m_W measurement has been performed by several groups <cit.>. Here we will adopt the result obtained in ref. <cit.>, which is based on the package <cit.>.[Itis found that using the global EW fit results <cit.> based on the package <cit.> does not substantially change our numerical results.] The resulting values of the oblique parameters, together with their correlations, read <cit.>
[ S=0.06 ± 0.10,; T=0.11 ± 0.12,; U=0.14 ± 0.09, ] cor=[ 1.00 0.90 -0.59; 1.00 -0.85; 1.00 ],
where “cor" denotes the correlation matrix.
As discussed in section <ref>, the NP contributions arise mainly from the diagrams involving the top-partner, the muon-partner and the modified Wμν coupling. In the following, we will consider the region of sinδ_L < 0.01 obtained in the last subsection, and investigate these three contributions one by one.
* By using the input parameters in table <ref> and considering sinδ_L < 0.01, we find that the W-boson mass shift from the modified Wμν coupling (cf. eq. (<ref>)) is given numerically by Δ m_W^Wμν < 0.007. Therefore, such an effect is too small to explain the latest CDF m_W measurement <cit.>, and can be safely neglected.
* From eqs. (<ref>)–(<ref>), we can see that the muon-partner contributions to the oblique parameters S, T and U are proportional to sin^2 δ_L. After considering sinδ_L < 0.01, the resulting shift Δ m_W^M is highly suppressed and found to be (10^-3) smaller than from the top-partner contribution. Therefore, we could also safely neglect the muon-partner contributions.
* The top-partner effects on the S, T and U parameters depend only on the mixing angle θ_L and the top-partner mass m_T (cf. eqs. (<ref>)–(<ref>)). Here we consider sinθ_L<0.5 to avoid large modification to the Wtb coupling. After taking into account the global EW fit results in eq. (<ref>), we find that there still exist allowed parameter regions at the 2 σ level, as shown in figure <ref>. It can be inferred that sinθ_L=0.14∼0.20 within the mass range 500<m_T<2000.
Consequently, the latest CDF W-boson mass shift can be explained in our model, and the allowed parameter regions shown in figure <ref> will be used in the following numerical analysis. When the parameters vary within the regions, deviations of the top-Higgs coupling (cf. eq. (<ref>)) from its SM value are less than 10%, which are also consistent with the current Higgs measurements at the LHC <cit.>. In addition, for comparison, we show in figure <ref> the allowed parameter regions by taking as constraint the average of m_W <cit.> including both m_W^ CDF and all the previous measurements <cit.>. In this case, the allowed values of sinθ_L become smaller.
§.§ Muon g-2 and neutrino trident production
The U(1)_L_μ-L_τ extension of the SM can explain the (g-2)_μ anomaly, but receives significant constraint from the rare process of neutrino trident production <cit.>. In this subsection, we will investigate the possibility of explaining the (g-2)_μ anomaly in our model, while satisfying the constraint from neutrino trident production. This rare process has been searched for in several neutrino beam experiments, including CHARM-II <cit.>, CCFR <cit.> and NuTeV <cit.>. Combining the data from these collaborators, the ratio of the measured cross section to that in the SM is given by σ_exp/σ_SM=0.95 ± 0.25 <cit.>, which should be confronted with the theoretical result in eq. (<ref>).
The NP contributions to (g-2)_μ can be divided into three parts (cf. eq. (<ref>)), arising from the penguin diagrams involving , Φ_ℓ and ϕ, respectively. Let us now discuss them one by one.
* From eqs. (<ref>)–(<ref>), one can see that the contribution Δ a_μ^ is proportional to λ_Φ_ℓ^2 and depends on the parameters (m_M, sinδ_L, m_Φ_ℓ). Taking λ_Φ_ℓ=1 for simplicity, we show in figure <ref> the resulting Δ a^Φ_ℓ_μ as a function of m_Φ_ℓ for various values of m_M and sinδ_L. It can be seen that the Φ_ℓ contribution is always negative, which increases the discrepancy of (g-2)_μ. However, such a negative contribution will be highly suppressed by heavy Φ_ℓ or small λ_. In addition, by comparing the left and right plots in figure <ref>, one finds that |Δ a _μ^| becomes smaller for smaller sinδ_L. Especially, in the case of m_>2, λ_<0.1 and sinδ_L<0.01, we obtain |Δ a_μ^|< 7.5× 10^-13, which can be safely neglected. In the following, we will consider such a special case.
* The ϕ contribution Δ a_μ^ϕ depends on the parameters (sinδ_L, m_M, m_ϕ, λ_ϕ). We take m_M=300 as discussed in section <ref>, and show in figure <ref> the 2 σ allowed regions of (m_ϕ, λ_ϕ) for sinδ_L=(1.0, 0.5, 0.1)× 10^-2. It can be seen that the ϕ boson can explain the (g-2)_μ anomaly at the 2 σ level. Furthermore, larger λ_ϕ is required for larger m_ϕ and the lower limits on λ_ϕ depend only marginally on sinδ_L for large m_ϕ. Taking λ_ϕ=2, we can then derive the 2 σ allowed region of (m_ϕ, sinδ_L), which is shown in the left panel of figure <ref>. As can be seen from figures <ref> and <ref>, smaller m_ϕ corresponds to a larger range of sinδ_L and λ_ϕ. Hence, we take m_ϕ=1 as a benchmark value and derive the allowed region of (λ_ϕ, sinδ_L), which is shown in the right panel of figure <ref>.
* The contribution Δ a_μ^ depends on the parameters (m_, m_M, , sinδ_L) and is positive. However, different from the ϕ-boson case, the can also affect the neutrino trident production. After considering the constraint from this rare process, the contribution alone is not sufficient to explain the (g-2)_μ anomaly, as observed in the minimal U(1)_L_μ-L_τ model <cit.>.
As discussed above, in the case of m_>2 and λ_<0.1, the relevant contributions to the (g-2)_μ arise from both the and the ϕ boson. Thus, we consider them together, and take m_M=300, m_=1000 and =2 for simplicity. After taking into account the constraints from the (g-2)_μ anomaly and the neutrino trident production, we can derive the 2 σ allowed regions of (sinδ_L, m_ϕ, λ_ϕ), which are also shown in figures <ref> and <ref>. One can see that the allowed regions become much larger compared to that obtained by including only the ϕ contribution. We take m_ϕ=1 as a benchmark value, which approximately provides the largest parameter space for sinδ_L and λ_ϕ, and show in figure <ref> the allowed region of (λ_ϕ, sinδ_L). In this case, the lower bound on sinδ_L is 3.2× 10^-4 and, after combining the bound derived in section <ref>, we obtain
3.2 × 10^-4 <sinδ_L < 1.0 × 10^-2 .
It is noted that the and the ϕ contribution are both positive, and the latter alone can account for the (g-2)_μ anomaly while providing no influence on other processes discussed in section <ref>. Therefore, we will take m_ϕ=1 and the bound in eq. (<ref>) in the following analysis. This in turn implies that we do not need to consider the (g-2)_μ anomaly anymore when investigating the contributions to other processes, as the anomaly can be definitely resolved in our model.
§.§ b → s ℓ^+ ℓ^- processes
The most relevant observables to our model include the b → s ℓ^+ ℓ^- transitions,[The top partner can also affect the radiative b → s γ decays. However, the NP contribution is proportional to sin^2θ_L and thus highly suppressed for small θ_L.] the W-boson mass, the (g-2)_μ, the neutrino trident production, as well as the Zμμ coupling. As discussed in the last subsection, the ϕ contribution alone can explain the (g-2)_μ anomaly and, at the same time, does not bring any significant effect on other observables. Therefore, the remaining question is to see if the parameter space required to account for the latest CDF m_W measurement can also explain the b → s ℓ^+ ℓ^- discrepancies, while satisfying the constraints from the neutrino trident production and the Zμμ coupling. This will be explored in this subsection.
For the b → s ℓ^+ ℓ^- transitions, only the short-distance Wilson coefficients _9 and _10 are affected in our model. Therefore, we will perform a global fit of _9μ^ and _10μ^ by considering the various measurements of the b → s ℓ^+ ℓ^- processes, including the recent measurements of R_K^(*) <cit.> and (B_s →μ^+ μ^-) <cit.>. Details of the fit are given in appendix <ref>. The final allowed regions of (_9μ^, _10μ^) obtained through such a global fit are shown in figure <ref>, which are very similar to that of the latest global fits by other groups <cit.>.
Since a small mixing angle θ_L is required to explain the latest CDF m_W measurement, the NP contributions to _9 and _10 from the W-box, γ- and Z-penguin diagrams, which are all proportional to sin^2 θ_L, are highly suppressed. Therefore, the dominant contributions arise from the -penguin diagrams, which depend on the parameters (m_T, sinθ_L, m_M, sinδ_L, m_, , ). In the following, we take m_M=300 as discussed in section <ref>, and choose (m_, m_T)=(1.0, 1.0), (1.0, 1.5), (1.0, 2.0), (1.5, 1.5), (1.5, 2.0), (2.0, 2.0) as several benchmark values. Then, the remaining relevant parameters are (sinθ_L,sinδ_L, , ). Considering the 2 σ allowed regions obtained from the global EW fit as well as the constraints from the b → s ℓ^+ ℓ^- global fit, the neutrino trident production and the Zμμ coupling,[For the constraint from the Zμμ coupling, we take λ_ϕ=λ_=1, m_ϕ=1 and m_=2, as discussed in the section <ref>.] we can finally derive the allowed regions of these parameters. As an illustration, we show in figure <ref> the allowed regions in the (, ) plane for sinδ_L=(0.1, 0.4, 0.7, 1.0)× 10^-2. It can be seen that the allowed values of the parameters and for TeV-scale and T are both of 𝒪(1) simultaneously, therefore being safely in the perturbative region. For heavier T or lighter , smaller and are required. In addition, for larger sinδ_L, the allowed regions of and are reduced.
As a conclusion, our model can accommodate the (g-2)_μ anomaly, the latest CDF W-mass shift and the b → s ℓ^+ ℓ^- discrepancies, while satisfying the other constraints like the neutrino trident production and the Z→μ^+μ^- processes. With the model parameters varied within the allowed regions, the predicted Wilson coefficients _9μ^ and _10μ^ are also shown in figure <ref>. From figure <ref>, we can also see that the departure of the ratio / from unity is more stringently constrained for heavier or lighter T. Furthermore, for different values of (m_, m_T), = (or equivalently q_ℓ = q_t) is always allowed. In the case of =≡ g_, we show in figure <ref> the allowed parameter space in the (g_, sinδ_L) plane. Notably, this in turn implies that the possibility of = Φ_ℓ^* is allowed in our model.[The possibility of = is, however, not possible, since q_ℓ=-q_t makes the sign of _9μ and _10μ (cf. eq. (<ref>)) opposite to that shown in figure <ref>.]
§.§ Collider phenomenology
The vector-like top partner, being colored, can be efficiently produced at the hadron colliders <cit.>. Searches for single and pair productions of the vector-like top partner have been performed at the LHC, and strong constraints on its mass and mixing angle have been obtained <cit.>. However, in most of these searches, it is assumed that the top partner decays exclusively into the SM particles, i.e., T → bW/tZ/th, and the top partner has been excluded for masses below 1.3. In our model, this lower bound applies only for the case of m_T < m_ + m_t. In the case of m_T > m_ + m_t, on the other hand, the T → t channel is open, and the bounds from these direct searches could be therefore relaxed <cit.>. In this case, as shown in figure <ref>, the cascade decay T → t (→μ^+ μ^-) makes the dimuon resonance searches at the LHC <cit.> sensitive to our model. In order to derive the collider constraints, we take m_ϕ = 1, m_=m_=2 and m_M = 300, and consider the allowed parameter space of (sinθ_L, sinδ_L, g_t, g_ℓ) derived in the last subsection, which corresponds to that shown in figure <ref>. For the dimuon resonance searches, the cross section is estimated by σ(p p → T T̅)· 2 ·(T → t )·(→μ^+ μ^-) and the result of σ(p p → T T̅) in ref. <cit.> is used. The total width of the top partner is calculated by considering all the tree-level two-body decay modes, which consist of T→ t Z^('), b W and th. Here the main decay channel is T → t for small q_ℓ/q_t and T → t h for large q_ℓ/q_t. For the boson, its total width can be estimated by including all the tree-level two-body decay modes, which consist of → M^+ M^-, M^+ μ^-, μ^+ M^-, μ^+ μ^-, τ^+ τ^-, ν_μν̅_μ, ν_τν̅_τ and t t̅. The main decay channel is → t t̅ for small q_ℓ /q _t. For large q_ℓ /q_t, on the other hand, the decay is dominated by →τ^+ τ^- and →ν_μ,τν̅_μ,τ, while the branching ratio (→μ^+ μ^-) can at most reach about 20%. The cross sections corresponding to the allowed parameter space can then be derived, which are shown as a function of q_ℓ/q_t in figure <ref>. It can be seen that the cross sections for various values of m_ and m_T are all below the current CMS bound <cit.>. The cross section becomes larger for a lighter ; especially for m_T=1.0, the maximum cross section is close to the CMS bound. Furthermore, in most of the parameter space, the cross sections are higher than the sensitivities expected at the High-Luminosity LHC (HL-LHC) corresponding to 3000^-1 at √(s) = 14 <cit.>.
Searches for the multi-top final states <cit.> can also provide evidence of the boson <cit.>. As shown in figure <ref>, the boson in this case can be produced in association with top pairs, and the decay → t t̅ leads to the four-top final state t t̅ t t̅. Similar to the analysis of the dimuon channel, we consider the allowed parameter space corresponding to figure <ref>. Predictions of the NP contributions to the t t̅ t t̅ cross section are shown in figure <ref>, which are estimated by σ_(p p → t t̅ t t̅) ≈σ(pp → t t̅)·(→ t t̅), with the result of σ(p p → t t̅) taken from ref. <cit.>. It can be seen that our predictions for various values of m_ and m_T are all well below the current CMS bound.[The recent ATLAS measurement <cit.> is roughly 1.8 σ higher than the SM prediction <cit.>. For small /, σ(p p → t t̅ t t̅) can be enhanced by 10∼15 % compared to that in the SM. Thus, the tension is relaxed in our model.] Furthermore, the dimuon channel is enhanced by the NP contribution for large q_ℓ/q_t, while the four-top channel enhanced by small q_ℓ/q_t. Therefore, the two processes are complementary to each other in searching for the boson.
Similar to the top partner, the vector-like lepton partner can also be pair-produced at the LHC. However, the production occurs only through the s channel involving the EW vector bosons, leading to much smaller cross sections <cit.>. Searches for vector-like leptons have been performed at the LEP <cit.> and the LHC experiment <cit.>, and most of these studies focus on the case of vector-like tau partner. For example, by using 138^-1 of pp collisions at √(s)=13, the SU(2)_L doublet and singlet vector-like tau partner are already excluded for masses below 1045 and in the mass range of 125-150, respectively <cit.>. In our model, due to the small mixing angle δ_L, the particle M can be approximated as a singlet vector-like muon partner. Similar to the case of the singlet vector-like tau partner, searches for the singlet vector-like muon partner is also very challenging for the LHC, due to its small production cross section. Based on the ATLAS searches for the anomalous productions of multi-lepton events <cit.>, the vector-like muon partner has been investigated in ref. <cit.> (see ref. <cit.> for similar study by using the CMS data <cit.>). However, no limits on the vector-like muon partner are found for masses below 105 or above 300. In addition, the LEP limit on additional heavy leptons <cit.> places a lower bound of around 100 on the mass of the vector-like muon partner. Confronted with these experimental status, the benchmark value m_M=300 used in our numerical analysis is therefore reasonable. In addition, the muon partner lighter than around 300 could produce a signal at the future proton-proton colliders <cit.>. Detailed analysis of the prospects for its discovery will be presented in a future work.
Besides the top partner, the muon partner and the boson, signals of the scalars , and ϕ can also be searched for at the high-energy colliders. For example, as shown in figure <ref>, the scalar ϕ can be produced in association with a muon pair, and the decay ϕ→μ^+ μ^- leads to a four-muon final state, which has been measured at the LHC <cit.> and the B factories <cit.>. For m_ϕ=1, which has been used in our numerical analysis, future measurements at the Belle II <cit.> and the Super Tau-Charm Facility (STCF) <cit.> are expected to provide sensitive probes of this scalar. Detailed analysis of the current constraints on the scalars in our model and the future prospects at the LHC, Belle II and STCF is, however, beyond the scope of this paper, but will be explored in our future work.
§ CONCLUSION
In this paper, we have constructed a NP model that successfully addresses the latest CDF W-boson mass shift, the (g-2)_μ anomaly and the b → s ℓ^+ ℓ^- discrepancies. In our setup, the SM is extended by the SU(2)_L-singlet vector-like top and muon partners that are featured by additional U(1)^' gauge symmetry. The top and the muon partner have also the same SM quantum numbers as of the right-handed top and muon respectively, and can therefore mix with the latter after the EW and the U(1)^' symmetry breaking.
Similar to the SM case, the loop diagrams involving these fermion partners can contribute to the (g-2)_μ, the W-boson mass m_W and the b → s μ^+ μ^- transitions. After considering the most relevant constraints, such as the Z →μ^+ μ^- decay and the neutrino trident production, both the (g-2)_μ anomaly and the latest CDF W-boson mass shift can be explained in our model. This requires that the mixing angles between the fermions and the fermion partners should be small. At the same time, the Z^'-penguin diagrams involving the top partner can affect the short-distance Wilson coefficients _9μ and _10μ in the b → s μ^+ μ^- transitions. Furthermore, the small lepton mixing angle δ_L makes the μμ interaction almost of a vector-type. Therefore, _9μ^≪_10μ^≈ 0 is obtained in our setup. This is also favored by the b → s ℓ^+ ℓ^- global fit after including the recent measurements of R_K^(*) <cit.> and (B_s →μ^+ μ^-) <cit.>. It is also found that both the boson and the top partner can be as light as of 1 ∼ 2, and thus may be accessible at the LHC Run 3 and its upgrade. Searches for the dimuon resonances and the top partner could also provide evidences of the boson and the top partner. Especially, the boson can enhance the p p → t t̅ t t̅ production cross section. Finally, the scalar ϕ with a mass of around 1 can be produced in association with a muon pair, and the subsequent decay ϕ→μ^+ μ^- leads to a four-muon final state, which can be searched for at the Belle II and STCF experiments.
As a final comment, our model can be further explored in several phenomenological directions. By allowing for nonzero mixings of the top (muon) partner with the first and second (third) generations of quarks (leptons), several other interesting observables could be affected. In particular, the CKM matrix should be extended in this case, which may be responsible for the Cabibbo angle anomaly <cit.>. Our model can also be extended by right-handed neutrinos or a dark sector. These points will be investigated in our future works.
This work is supported by the National Natural Science Foundation of China under Grant Nos. 12135006, 12075097 and 11805077, as well as by the Fundamental Research Funds for the Central Universities under Grant Nos. CCNU19TD012, CCNU20TS007 and CCNU22LJ004. XY is also supported in part by the Startup Research Funding from CCNU.
§ ONE-LOOP CORRECTIONS TO Z →ℓ^+ ℓ^-
At the one-loop level, the NP contributions to the Z→μ^+μ^- decay arise from the diagrams shown in figure <ref>. In our calculation, the on-shell renormalization scheme specified in ref. <cit.> is adopted. Furthermore, in order to cancel the ultraviolet (UV) divergences, the renormalization of the mixing angle δ_L should be performed. In the following, details of the renormalizations of the lepton fields and the mixing angle are given, respectively.
In the mass eigenbasis, the fermion field renormalization is performed through
f^L/R_i,0 =(Z^L/R_ij)^1/2 f^L/R_j≡(δ_ij+1/2δ Z^L/R_ij)f^L/R_j ,
where f^L/R_i and f^L/R_i,0 denote the renormalized and bare left/right-handed lepton fields, respectively. Z^L/R_ij represent the fermion field renormalization matrices, and can be fixed by the following on-shell renormalization conditions:
Γ̂_i j(p) u_j(p) |_p^2=m_j^2=0,
lim _p^2 → m_i^2 p+m_i/p^2-m_i^2Γ̂_i i(p) u_i(p)=i u_i(p),
where u_i(p) denotes the spinor of the external fermion fields. The renormalized one-particle irreducible two-point function Γ̂_i j is defined by
Γ̂_i j(p) = [c]2cm
< g r a p h i c s >
,
= i δ_i j(p-m_i) + i [p P_L Σ_i j^L(p^2)+p P_R Σ_i j^R(p^2)+P_L Σ_i j^SL(p^2)+P_R Σ_i j^SR(p^2)] ,
where the scalar functions Σ_i j^L, Σ_i j^R, Σ_i j^SL and Σ_i j^SR are functions of p^2. Then, the on-shell renormalization conditions yield
δ Z_i j^L= 2/m_i^2-m_j^2Re[m_j^2 Σ_i j^L(m_j^2)+m_i m_jΣ_i j^R(m_j^2)+m_iΣ_i j^SL(m_j^2)+m_jΣ_i j^SR(m_j^2)], ,
δ Z_i j^R= 2/m_i^2-m_j^2Re[m_j^2 Σ_i j^R(m_j^2)+m_i m_jΣ_i j^L(m_j^2)+m_jΣ_i j^SL(m_j^2)+m_iΣ_i j^SR(m_j^2)], ,
δ Z_i i^L/R= -ReΣ_i i^L/R(m_i^2)-.m_i^2 ∂/∂ p^2Re[Σ_i i^L/R(p^2)+Σ_i i^R/L(p^2)+2/m_iΣ_i i^SL/SR(p^2)]|_p^2=m_i^2.
Specific to our model, their explicit expressions are also given in the ancillary notebook file.
After performing the field renormalization, the one-loop contribution to g_Rμ in eq. (<ref>) is already finite. However, the one-loop contribution to g_Lμ is still divergent. Here the renormalization of the mixing angle δ_L is required to cancel the remaining divergence. This is similar to the case in the Two-Higgs-Doublet Model, where the mixing angle relating the light and heavy neutral scalars should also be renormalized <cit.>. The renormalization of the mixing angle δ_L reads
δ_L,0=δ_L+Δδ_L,
where δ_L,0 and δ_L denote the bare and renormalized mixing angles, respectively. Then, in terms of the renormalized quantities, the Z and W interactions with the left-handed leptons in eqs. (<ref>) and (<ref>) become
Δℒ_W^ℓ = g/√(2)[(-ŝ_L Δδ_L+ĉ_L/2δ Z_μμ^L+ŝ_L/2δ Z_Mμ^L) μ̅W P_Lν_μ.
. +(ĉ_L Δδ_L+ŝ_L/2δ Z_MM^L+ĉ_L/2δ Z_μ M) M̅W P_Lν_μ]+ h.c. ,
Δℒ_Z^ℓ = g/2c_W(μ̅, M̅) [ (
[ 2 ŝ_L ĉ_L ŝ_L^2 - ĉ_L^2; ŝ_L^2 - ĉ_L^2 - 2 ŝ_L ĉ_L ]Δδ_L
+ĈẐ_L + Ẑ_L^T Ĉ )
Z P_L
+ s_W^2 ( Ẑ_R + Ẑ_R^T )
Z P_R
]
[ μ; M ],
with
Ĉ =
[ -1/2ĉ_L^2+ s_W^2 -1/2ŝ_Lĉ_L; -1/2ŝ_Lĉ_L -1/2ŝ_L^2+s_W^2 ],
Ẑ_L/R =
[ Z_μμ^L/R Z_μ M^L/R; Z_M μ^L/R Z_MM^L/R ].
In order to fix the renormalization of the mixing angle δ_L, we require the one-loop renormalized matrix element for the Z→μ^+μ^- decay is finite, i.e.,
ℳ(Z →μ^+ μ^-)|_UV=0,
By calculating the vertex-correction diagrams shown in figure <ref>, we find that this renormalization condition leads to the following renormalization constant:
Δδ_L=-1/2ĉ_L/ŝ_Lδ Z_μμ^L |_UV-1/2δ Z_μ M^L |_UV.
It is also found that, with these prescriptions, the one-loop renormalized matrix elements for the decays Z→ℓ^+_iℓ^-_j and W^+→ℓ^+_iν (ℓ_i,j=e or μ) are all finite after including the above renormalization constants.
§ GLOBAL FIT OF B → S ℓ^+ ℓ^- TRANSITIONS
To perform a global fit of the b → s ℓ^+ ℓ^- transitions, as done in refs. <cit.>, we have considered the following experimental data: 1) the branching ratios of B → K μ^+μ^- <cit.>, B → K^*μ^+μ^- <cit.>, B_s →ϕμ^+μ^- <cit.>, Λ_b →Λμ^+ μ^- <cit.>, B → X_s μ^+μ^- <cit.>, and B_s →μ^+μ^- <cit.>; 2) the angular distributions in B → K μ^+μ^- <cit.>, B → K^*μ^+μ^- <cit.>, B_s →ϕμ^+μ^- <cit.>, and Λ_b →Λμ^+ μ^- <cit.>; 3) the LFU ratios R_K and R_K^* <cit.>.
During the global fit, we firstly construct a likelihood function that depends only on the short-distance Wilson coefficients <cit.>,
-2ln L() = x^T () [ V_ exp + V_ th(θ)]^-1x(),
where x_i()=_i^ fexp - _i^ th(, θ), with _i^ th and _i^ exp representing the central values of the theoretical predictions and the experimental measurements, respectively. Their values depend on the input parameters θ and the Wilson coefficients . Here V_ exp and V_ th denote the covariance matrices of the experimental measurements and the theoretical predictions, respectively. All the theoretical uncertainties and their correlations are included in V_ th. Approximately, V_ th can be obtained by fixing the Wilson coefficients to their default values within the SM. Furthermore, the theoretical uncertainties are approximated as Gaussian, which can be obtained by random samplings of the probability density functions of the input parameters θ. Finally, the Δχ^2 function can be written as Δχ^2()=-2ln L()/L_ max, where L_ max represents the maximum value of the likelihood function L for different values of the Wilson coefficients . More details about the fitting procedures can be found in refs. <cit.>. Here we have used an extended version of the package <cit.> when performing such a global fit.
§ LOOP FUNCTIONS
Explicit expressions of the loop functions present in the oblique parameters S, T, U in eqs. (<ref>)–(<ref>) are listed below:
K_1(x,y) = -44x + 2ln x + f_-9(x,x) - (x → y),
K_2(x,y) = (x-y)^2 -3x -[(x-y)^3 - 3(x-y) - 6xy/x-y]ln x -f_3(x,x) + f_3(x,y) + (x ↔ y),
K_3(x,y) = -3 x - 6 x^2 +(8-18x+6x^3)ln x - (12 - 18 x+ 6x^3) ln(x-1) -32 y + 4 f_0(y, y),
K_4(x,y) = -20x + 2ln x + f_-3(x,x) - (x → y),
K_5(x,y) = 1/x-y[-3x^2+2 x^3-6 x^2 y ]-[(x-y)^3+3(x+y)+6 x^2/-x+y] ln x
-f_3(x,x)+f_3(x,y)+ (x ↔ y),
K_6(x,y) = -x - 2 x^2 +2(2-3x+x^3)ln[x/x-1] -16 y + 2 f_0(y, y),
with the function f_n(x,y) defined by
f_n(x,y) = -[(x+y-1)^2-4xy-3+n(x+y)] √(4xy-(x+y-1)^2)cos^-1(x+y-1/√(4xy)).
Explicit expressions of the loop integrals I_1-7 present in the short-distance Wilson coefficients _9^ NP and _10^ NP in section <ref> are given, respectively, by
I_1 = f_t (-176,-724,14 ) +[-14+f_t (3,3,16,-14)]ln x_t - (t → T),
I_2 = 34x_t-f_t ( 94)+ [ 94+f_t (92,94 ) ]ln x_t - (t → T),
I_3 = 3(x_t+x_T)+3/2x_tx_T/x_T-x_tlnx_t/x_T,
I_4 = 3x_t/4[ 1/3-f_t(1) +2g(x_t,x_T)ln x_t + (t ↔ T) ],
I_5 = x_t^2/2[1/x_t +2ln x_t/x_T-x_t + (t ↔ T) ],
I_6 =-(x_t-x_T)(x_t x_T-x_T-x_t+4)/2(x_t-1)(x_T-1)-3x_t^2f_t(1)lnx_t-(t ↔ T),
I_7 =-(x_t+x_T)+2x_tx_T/x_t-x_Tlnx_t/x_T.
where x_t=m_t^2 / m_W^2, x_T=m_T^2 / m_W^2, and the functions g(x,y) and f_q(a_1, a_2, a_3, …, a_n) are defined, respectively, as
g(x,y)=y(4-8y+y^2)+x(4-2y+y^2)/6(x-y)(y-1)^2,
f_q(a_1, a_2, a_3, …, a_n) = ∑_i=1^na_i/(x_q-1)^i.
The loop functions I_L and I_R introduced in eq. (<ref>) are given, respectively, by
I_L = - (m_t^2+m_T^2) + 2m_t^2 m_T^2/m_t^2-m_T^2lnm_t^2/m_T^2,
I_R = 4m_T^2 - 2(m_t^2+m_T^2)m_T^2/m_t^2-m_T^2lnm_t^2/m_T^2.
JHEP
|
http://arxiv.org/abs/2307.07228v1 | 20230714084753 | Alleviating Cosmological Tensions with a Coupled Scalar Fields Model | [
"Gang Liu",
"Zhihuan Zhou",
"Yuhao Mu",
"Lixin Xu"
] | astro-ph.CO | [
"astro-ph.CO"
] |
APS/123-QED
[email protected]
[email protected]
Institute of Theoretical Physics
School of Physics
Dalian University of Technology
Dalian 116024, People's Republic of China
In this paper, we investigate the interaction between early dark energy (EDE) and scalar
field dark matter, proposing a coupled scalar fields model to
address the Hubble tension and S_8 tension. While the EDE model successfully alleviates
the Hubble tension, it exacerbates
the S_8 tension. To mitigate the negative impact of EDE,
we introduce the interaction between EDE and dark matter. Specifically, we replace cold
dark matter with scalar field dark matter, given its capability to suppress structure
growth on small scales.
We constrained the new model using cosmological observations including
the temperature and polarization anisotropy power spectra data of cosmic microwave background
radiation (CMB) from Planck 2018 results,
baryon acoustic oscillations (BAO) measurements extracted from 6dFGS, SDSS and BOSS,
the Pantheon sample of type Ia supernovae (SNIa),
the local distance-ladder data (SH0ES),
and the Dark Energy Survey Year-3 data.
Employing Markov Chain Monte Carlo method, we find that this novel model yields
best-fit values of H_0 and S_8 equal to 71.13 km/s/Mpc and 0.8256, respectively.
Compared to the ΛCDM model, the new model alleviates the Hubble tension but
still fails to resolve the S_8 tension. However, we obtain a smaller value of S_8
compared to the result of 0.8316 obtained for EDE model, which
mitigates to some extent the shortcoming of the EDE model.
Alleviating Cosmological Tensions with a Coupled Scalar Fields Model
Lixin Xu
August 12, 2023
====================================================================
§ INTRODUCTION
Over the past few years, the ΛCDM model has encountered numerous challenges as a
result of the growing quantity and quality of observations.
The emergence of the Hubble tension and the S_8 tension has garnered significant attention.
The Hubble tension <cit.> pertains to the discrepancy between the
H_0 value obtained from model-independent local measurements such as Type Ia supernovae
(SNIa) <cit.>, and the H_0 value derived from the cosmic
microwave background (CMB) <cit.> and the large-scale structure (LSS)
<cit.>.
More precisely, the Planck 2018 CMB data estimates the value of H_0 to be
67.37±0.54 km/s/Mpc <cit.>, while the cosmic distance ladder measurement
(SH0ES) yields H_0=73.04±1.04 km/s/Mpc <cit.>, with a statistical error
of 4.8σ.
The S_8 tension characterizes the inconsistency between CMB and LSS observations
<cit.>.
The Planck best-fit ΛCDM model estimates the value of S_8 to be
0.834±0.016 <cit.>, while LSS observations yield
0.759^+0.024_-0.021 for KiDS-1000 <cit.>, 0.800^+0.029_-0.028 for
HSC-Y1 <cit.>, and 0.776±0.017 for Dark Energy Survey Year-3 (DES-Y3)
<cit.>.
Numerous models have been proposed to address the Hubble tension, as reviewed recently
by <cit.>. These models incorporate modifications to the late universe, such as
the Phenomenologically Emergent Dark Energy model <cit.>, the Phantom Transition
<cit.>, and the early universe, including the Early Dark Energy (EDE) model
<cit.>, the Acoustic Dark Energy model
<cit.>, and the New Early Dark Energy model <cit.>, etc.
Despite the proposed models, they still encounter several issues. For instance, the
late-time solutions that do not alter the sound horizon are
generally unable to account for the SH0ES measurement. Conversely, the early-time
solutions that introduce a new component before recombination to decrease the scale
of the acoustic horizon on the final scattering surface, increase the value of H_0,
and maintain the angular scale of the acoustic horizon in CMB observations, but they
exacerbate the S_8 tension <cit.>.
This paper focuses on the EDE model and aims to address the associated concerns.
EDE is characterized by an ultra-light axion scalar field <cit.>.
In this model, z_c denotes the redshift at the apex of the EDE component contribution,
while f_EDE represents the proportion of EDE energy density relative to
the total energy density at that time. The evolution of the EDE fraction with the redshift
is depicted in Fig. <ref>, where the red vertical dashed line denotes the redshift
at recombination, and the apex of the EDE component occurs before recombination. The
Hubble tension can be resolved when the EDE ratio f_EDE reaches approximately
10% <cit.>.
During its contribution period, the EDE component marginally diminishes the perturbed
growth of the structure. To align with the CMB data, the cold dark matter (CDM) density
must be augmented to offset these losses. Moreover, some other cosmological parameters,
such as the scalar spectral index n_s, the baryon density
ω_b, and the amplitude of density perturbations σ_8, will also undergo
changes <cit.>. Consequently, the EDE model will invariably exacerbate the
CMB-LSS inconsistency <cit.>.
To reduce the S_8 tension, it is common to investigate the interaction of dark matter
(DM) and dark energy (DE), which can inhibit the growth of structure through the drag of
DE on DM <cit.>. In addition, since the nature of dark
matter is not yet comprehensively understood, alternative descriptions can be developed
to substitute cold dark matter, and alleviate the S_8 tension.
The ΛCDM model posits that dark matter is comprised of non-baryonic, pressureless,
and non-relativistic particles <cit.>. This model has been successful in
explaining large-scale observations from the cosmic microwave background (CMB) and
large-scale structure (LSS). However, despite its achievements, the microscopic
properties of dark matter remain unknown <cit.>. The aforementioned
assumptions have led to a number of unresolved issues, such as the unexpected behavior
of central densities in galactic halos and the overpopulation of secondary structures
on small scales. These observations suggest that the cold dark matter (CDM) may not be an
adequate description of dark matter, particularly on smaller scales <cit.>.
Scalar field dark matter (SFDM) presents an alternative to CDM, which is composed of a
light scalar field with a mass of approximately 10^-22 eV
<cit.>. In this model, the scalar field forms a Bose-Einstein
condensate (BEC) at the galactic scale, which modifies the dynamics of dark matter on
small scales while maintaining the success of CDM on large scales. This condensation
leads to the suppression of structure growth on small scales, which could potentially
alleviate the S_8 tension.
The behavior of the scalar field dark matter is similar to that of the cosmological
constant in the early universe, followed by oscillations, and ultimately similar to
CDM. Fig. <ref> displays the evolution of the SFDM equation of
state with the redshift. The initiation time of the oscillations is determined
by the field mass. A smaller mass results in later oscillations.
This paper proposes a coupled scalar fields (CSF) model that explores the interaction
between early dark energy (EDE) and scalar field dark matter (SFDM). The coupling
between the two fields is inspired by the Swampland Distance Conjecture (SDC)
<cit.>, which has previously been applied to quintessence
models <cit.> and the EDE model <cit.>.
According to the SDC, a low-energy effective field theory is deemed valid only within
a region of field space constrained by the Planck scale. Moreover, any breakdown of
the effective field theory that arises due to Planckian field excursions can be
expressed as an exponential sensitivity reflected in the mass spectrum of the
effective theory. Specifically,
the CSF model posits that the mass of dark matter is exponentially dependent on
the EDE scalar:
m_DM(ϕ)=m_0e^βϕ/M_pl,
where, m_0 represents the present-day mass of dark matter, ϕ denotes the EDE
scalar, β∼𝒪(1) is a constant, and M_pl=2.435×10^27 eV
denotes the reduced Planck mass.
In this study, we have conducted a comprehensive investigation into the evolutionary
equations of the coupled model at both the background and perturbation levels. We employed
a Markov Chain Monte Carlo (MCMC) analysis of three
cosmological models, namely the ΛCDM, EDE, and CSF models. We utilized
various datasets, including the Planck 2018 primary CMB data and CMB lensing
data <cit.>, BAO measurements from the BOSS DR12,
the 6dF galaxy survey, and SDSS DR7 <cit.>,
the Pantheon supernovae Ia data <cit.>, the SH0ES measurement
<cit.>, and the Dark Energy Survey Year-3 data <cit.>.
Based on the entire datasets, we found that the H_0 values obtained by the EDE and
CSF models are 72.46± 0.86 km/s/Mpc and 72.20± 0.81 km/s/Mpc at a 68%
C.L., respectively, both exceeding the result of 68.71^+0.35_-0.41 km/s/Mpc
obtained by the ΛCDM model. Therefore, both models can alleviate the Hubble
tension. The S_8 value for the EDE model is 0.822^+0.011_-0.0093, while the
result for CSF is 0.820^+0.014_-0.008. Furthermore, the obtained coupling
constant is constrained to be -0.014± 0.016, indicating an interaction between
dark matter and dark energy.
Despite the failure of the coupled model to resolve the S_8
tension, it has yielded a smaller S_8 and χ^2_tot compared to the EDE
model, thereby mitigating the adverse effect associated with EDE.
The paper is organized as follows. Section <ref> presents an introduction to
the CSF model, including the dynamics of background and perturbation.
In Section <ref>, we present numerical results illustrating the impact of the
coupled model on the large-scale structures.
In Section <ref>, we discuss the datasets utilized in our analysis and present the
corresponding results. Finally, we summarize our findings in Section
<ref>.
§ COUPLED SCALAR FIELDS
We examine the coupling between SFDM and EDE. The Lagrangian
is defined as follows:
ℒ=-1/2∂^μχ∂_μχ-1/2m_χ(ϕ)^2χ^2-1/2∂^μϕ∂_μϕ-V(ϕ),
where ϕ is the EDE scalar with the potential <cit.>
V(ϕ)=m_ϕ^2f_ϕ^2[1-cos(ϕ/f_ϕ)]^3+V_Λ,
and χ is SFDM scalar with a ϕ-dependent mass m_χ(ϕ), V_Λ
in Eq.(<ref>) serves as the cosmological constant.
The subscript ϕ is used to denote dark energy and χ is used to denote dark matter.
Numerous potentials of SFDM have been investigated in previous studies
<cit.>, but the common features of them can be represented
by 1/2m_χ^2χ^2 <cit.>. The specific form of m_χ(ϕ) is
m_χ(ϕ)=m_0e^βϕ/M_pl,
which is given by the Swampland Distance Conjecture as Eq.(<ref>), and m_0
represents the present-day mass of SFDM.
§.§ Background Equations
The motion equations of the scalar field dark matter (SFDM) in a flat
Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology can be expressed as follows:
3M_pl^2H^2=∑_Iρ_I,
-2M_pl^2Ḣ=∑_Iρ_I+p_I,
χ̈=-3Hχ̇-m_χ^2χ,
where the dot denotes the derivative with respect to cosmic time, and H is the
Hubble parameter, ρ_I and p_I are the energy density and pressure for each
component respectively.
The expressions for the energy density and pressure of SFDM are as follows:
ρ_χ=1/2χ̇^2+1/2m_χ^2χ^2,
p_χ=1/2χ̇^2-1/2m_χ^2χ^2.
We define a new set of variables to transform the Klein-Gordon equation (<ref>)
<cit.>,
x=χ̇/√(6)M_plH,
y=-m_χχ/√(6)M_plH,
y_1=2m_χ/H.
We utilize the polar coordinate variable transformation form as proposed in previous
works <cit.>:
x=√(Ω_χ)sinθ/2,
y=√(Ω_χ)cosθ/2,
where Ω_χ=ρ_χ/3M_pl^2H^2 is the density parameter
of the dark matter.
The Friedman equations (<ref>) and (<ref>) are reformulated as follows:
Ḣ/H^2=-3/2(1+w_t),
1=∑_IΩ_I+Ω_χ,
where
w_t=p_t/ρ_t represents the total equation of state,
which is the ratio of total pressure p_t to total energy density ρ_t,
and Ω_I=ρ_I/3M_pl^2H^2
is the density parameter of each components.
The Klein-Gordon equation (<ref>) becomes:
Ω̇_χ/Ω_χ=3H(w_t+cosθ)+βϕ̇/M_pl(1+cosθ),
θ̇=H(y_1-3sinθ)-βϕ̇/M_plsinθ,
ẏ_̇1̇=3/2H(1+w_t)y_1+βϕ̇/M_ply_1.
The equations of motion for the EDE is given by the variation of
the action expanded to linear order in δϕ,
ϕ̈+3Hϕ̇+dV/dϕ=-3β M_plH^2Ω_χ(1+cosθ) .
The left panel of Fig. <ref> depicts the evolution of the EDE scalar,
while the right panel shows the EDE energy density fraction as a function of the
redshift across various coupling constants. The cosmological parameters utilized in this
analysis are derived from the best-fit values listed in Tab. <ref>.
The amplitude and phase of the EDE scalar will be altered by varying coupling constants.
The sign of the coupling constant determines the direction of conversion between dark
matter and dark energy components. A negative coupling constant results in a source
term on the right-hand side of Eq.(<ref>), causing the conversion of dark matter
into dark energy and leading to an increase in the energy density fraction of EDE.
Conversely, a positive coupling constant causes the conversion of dark energy into
dark matter.
The energy density and pressure of the EDE are,
ρ_ϕ=1/2ϕ̇^2+V(ϕ),
p_ϕ=1/2ϕ̇^2-V(ϕ).
The equations of continuity for SFDM and EDE can be derived from the Klein-Gordon
equations presented in Eq.(<ref>) and Eq.(<ref>), respectively,
ρ̇_̇χ̇=-3H(ρ_χ+p_χ)+βϕ̇/M_pl(1+cosθ)ρ_χ,
ρ̇_̇ϕ̇=-3H(ρ_ϕ+p_ϕ)-βϕ̇/M_pl(1+cosθ)ρ_χ.
Many coupled dark energy models have this common form <cit.>, which ensures covariant
conservation for the total stress tensor.
§.§ Perturbution Equations
We calculated the perturbation equation using the synchronous gauge, where the metric
is defined as follows:
ds^2=-dt^2+a^2(t)(δ_ij+h_ij)dx^idx^j.
The scalars of SFDM and EDE are given by
χ(x,t)=χ(t)+δχ(x,t),
ϕ(x,t)=ϕ(t)+δϕ(x,t),
with χ(t), ϕ(t) the background parts, and δχ(x,t),
δϕ(x,t) the linear perturbations respectively.
The perturbed Klein-Gordon equation for a Fourier mode of δχ(x,t) is
δ̈χ̈=-3Hδ̇χ̇-(k^2/a^2+m_χ^2)δχ-1/2χ̇ḣ.
where h representes the trace of scalar metric perturbations.
The density perturbations δρ(χ), pressure perturbations δ p(χ),
and velocity divergence Θ(χ) can be expressed as provided in
<cit.>,
δρ_χ=χ̇δ̇χ̇+∂_χV(χ)δχ,
δ p_χ=χ̇δ̇χ̇-∂_χV(χ)δχ,
(ρ_χ+p_χ)Θ_χ=k^2/aχ̇δχ.
As previously done in the background section, we introduce new variables to derive
the perturbation equation <cit.>:
u=√(2/3)δ̇χ̇/M_plH=-√(Ω_χ)e^αcosϑ/2
v=√(2/3)m_χδχ/M_plH=-√(Ω_χ)e^αsinϑ/2
Once more, we introduce a new set of variables:
δ_0=-e^αsin(θ/2-ϑ/2),
δ_1=-e^αcos(θ/2-ϑ/2).
The equations of motion for the new variables are:
δ̇_̇0̇=δ_0Hωsinθ-δ_1[3Hsinθ+Hω(1-cosθ)]-ḣ/2(1-cosθ)-βϕ̇/M_plδ_1sinθ,
δ̇_̇1̇=δ_0Hω(1+cosθ)-δ_1(3Hcosθ+Hωsinθ)-ḣ/2sinθ-βϕ̇/M_plδ_1cosθ,
where
ω=k^2/2a^2m_χH=k^2/a^2H^2y_1.
The relationship between the new variables and density, pressure, and velocity
divergence can be established by referring to the definition given in Eq.(<ref>),
δρ_χ=ρ_χδ_0,
δ p_χ=ρ_χ(δ_1sinθ-δ_0cosθ),
(ρ_χ+p_χ)Θ_χ=k^2/aHy_1ρ_χ[δ_1(1-cosθ)-δ_0sinθ].
Expanding the action to the second order and varying with respect to δϕ, we
obtain the equation of motion for EDE perturbation is,
δ̈ϕ̈+3Hδ̇ϕ̇+1/2ḣϕ̇+(k^2/a^2+d^2V/dϕ^2)δϕ
=β/M_plρ_χ[δ_1sinθ-δ_0(1+cosθ)]-2(β/M_pl)^2ρ_χ(1+cosθ)δϕ.
§.§ Initial Conditions
In the early universe, Hubble friction induces the effective freezing of the scalar
fields at their initial value, leading to a slow-roll process. The
initial value of ϕ̇ can be set to 0. And the energy
density of dark matter can be approximated to
be negligible at that time.
As a result, the equations of the EDE and SFDM simplify to an uncoupled form.
We treat the ratio between the initial value of EDE scalar and the axion decay constant,
α_i=ϕ_i/f_ϕ as the model parameter <cit.>.
We employ the attractor solution initial conditions for SFDM, namely,
θ_i=2/5m_0e^βϕ_i/M_pl/H_0√(Ω_r a_i^-4),
y_1i=5θ_i,
where Ω_r represents the energy density fraction of the present radiation
component, and a_i denotes the initial value of the scale factor.
The initial value of Ω_χ is calculated using the widely employed shooting
algorithm in the Boltzmann code <cit.>, based on
the current value of the energy density of dark matter.
The specific deductive process can be found in reference
<cit.>, for further details.
It should be noted that in the new model, the mass of SFDM is ϕ-dependent.
Therefore, the value of θ_i need to be adjusted accordingly.
We adopt adiabatic initial conditions for the perturbation equations of EDE and SFDM,
for a detailed description, please refer to <cit.> and <cit.>.
§ NUMERICAL RESULTS
The publicly available Boltzmann code <cit.>
was modified as described in Sec.<ref>. We replace cold dark matter with
SFDM as the constituent of dark matter. In order to compute various perturbation
equations using the synchronous gauge in , we set the energy density
fraction of cold dark matter, Ω_cdm,0 to a value of 10^-6
<cit.>. We performed computations of the CMB power spectrum
and the matter power spectrum by employing the existing spectrum module of
.
We investigate the impact of the new model on the tension of large-scale structures.
Fig. <ref> displays the evolution of fσ_8(z) with the redshift
for three models, each with the corresponding best-fit values
taken from Tab. <ref>. The ΛCDM, EDE, and CSF models are depicted by
dashed black, dash-dotted blue, and solid orange lines, respectively.
The 63 observed Redshift Space Distortion fσ_8(z) data points are collected
from <cit.>.
Compared to the ΛCDM model, both the EDE and CSF models yield larger values of
fσ_8, exacerbating the S_8 tension. However, the results of the CSF model are
slightly smaller than those of the EDE model. This discrepancy primarily stems from
the inhibitory influence of SFDM on structure growth on small scales. We can more
distinctly observe this characteristic in the matter power spectrum.
Fig. <ref> presents the linear matter power spectra (upper panel) and their
relative differences compared to the ΛCDM model (lower panel) for three
models. All parameters are obtained from the best-fit values in Tab. <ref>.
Due to the interplay between dark matter and dark energy, as well as
the condensation effect of SFDM, the matter power spectrum obtained from the CSF
model is smaller than that of the EDE model on small scales, indicating suppressed
growth of structures and thus alleviating the S_8 tension caused by EDE.
It is worth noting that the CSF model still obtains a larger power spectrum on small
scales compared to the ΛCDM model, thus we have not resolved the
S_8 tension completely.
§ CONSTRAINTS AND RESULTS
The Markov Chain Monte Carlo (MCMC)
analysis was performed using <cit.>,
and the MCMC chains were analyzed using <cit.>.
We conducted the analysis using the following datasets:
1. CMB: The temperature and polarization power spectra
from Planck 2018 low-ℓ, high-ℓ and CMB lensing power spectrum
<cit.>.
2. BAO: The measurements from BOSS-DR12 fσ_8 sample,
namely, the combined LOWZ and CMASS galaxy samples <cit.>
and the small-z measurements from 6dFGS and the SDSS DR7 <cit.>.
3. Supernovae: The Pantheon sample, composed of 1048 supernovae Ia
in the redshift range 0.01 < z < 2.3 <cit.>.
4. SH0ES: The recent SH0ES measurement with
H_0 = 73.04 ± 1.04 km/s/Mpc <cit.>.
5. DES-Y3: The S_8 = 0.776 ± 0.017 from
Dark Energy Survey Year-3 weak lensing and galaxy clustering data <cit.>.
The results of the parameter constraints are shown in Tab. <ref>.
The upper part of the table enlists the cosmological parameters that underwent
sampling in the Markov chain Monte Carlo (MCMC) method. Meanwhile, the lower
section exhibits the derived parameters.
We use the complete dataset to ensure convergence for all models, with each parameter
achieving the Gelman-Rubin statistic value of R-1 < 0.05 <cit.>.
According to the results presented in Tab. <ref>, the EDE and CSF models obtained
H_0 values of 72.46± 0.86 km/s/Mpc and 72.20± 0.81 km/s/Mpc at a
68% confidence level, respectively, which are higher than the value of
68.71^+0.35_-0.41 km/s/Mpc obtained by the ΛCDM model.
This suggests that both the EDE and CSF models can alleviate the Hubble tension.
However, the EDE model and CSF model resulted in larger values of S_8,
which further exacerbated the tension with the LSS.
We obtained a non-zero coupling constant, β, with a value of
-0.014± 0.016 at a 68% C.L., indicating the interaction between dark
components through the conversion of dark matter into dark energy.
Combined with the condensation of SFDM on small scales, it is clear
from Fig. <ref> that the CSF model yields smaller density fluctuation amplitude
σ_8 compared to the EDE model, thereby
alleviating the S_8 tension caused by the EDE model.
The penultimate row of Tab. <ref> displays the Δχ^2_tot values for the EDE and CSF models
relative to the ΛCDM model, which are -11.74 and -13.78, respectively,
which is primarily attributed to the data from SH0ES. This
indicates that both models fit the data better than the standard model.
Furthermore, the χ^2_tot obtained by our new model is smaller than that of the
EDE model. This is attributed to the CSF model obtaining a smaller S_8 compared to
the EDE model, resulting in a closer alignment with the data from DES-Y3.
Thus, from a χ^2_tot perspective, our novel model exhibits the performance.
We also compared the models by calculating the Akaike Information Criterion (AIC)
<cit.>,
AIC=χ^2_tot+2k,
where k represents the number of fitted parameters.
The smaller the AIC value of a model, the higher its goodness of fit.
The results are presented in the last row of Tab. <ref>. The ΔAIC values for the
EDE and CSF models relative to the ΛCDM model are -5.74 and -3.78, respectively,
which indicates that the EDE model has the best fit.
Despite the CSF model demonstrating a smaller χ^2_tot value, its performance is
slightly inferior to that of the EDE model from the perspective of AIC, primarily
due to the incorporation of additional parameters.
§ CONCLUSION
This study examines the interplay between early dark energy (EDE) and scalar field dark
matter (SFDM), proposing a coupled scalar fields (CSF) model to reconcile the
discrepancies in H_0 and S_8 measurements. The CSF model leverages the EDE
component to enhance H_0 without compromising the cosmic microwave background (CMB)
observations, additionally, the suppression of SFDM on small-scale structure growth and
the drag of dark energy on dark matter can alleviate the extra S_8 tension caused by EDE.
We investigated the evolutionary equations of the coupled model, encompassing both the
background and perturbation levels, and explored their impact on the growth of
structures and the power spectrum of matter.
We then constrain the parameters of the ΛCDM, EDE, and CSF models using the full
data including CMB, BAO, SNIa, SH0ES, and S_8 from DES-Y3.
We constrain the coupling constant to be -0.014± 0.016 at a 68% C.L.,
indicating the interaction between dark matter and dark energy.
The EDE and CSF models yield H_0 values of 72.46± 0.86 km/s/Mpc and
72.20± 0.81 km/s/Mpc at a 68% C.L., respectively, which are higher than the
ΛCDM value of 68.71^+0.35_-0.41 km/s/Mpc, thus alleviating the Hubble
tension.
In addition, the EDE model and CSF model yield S_8 best-fit values of 0.8316 and
0.82 56 respectively, both of which exceed the result of the ΛCDM model at
0.7985, further exacerbating the existing S_8 tension. However, it is notable
that the S_8 for the CSF model is lower than that of the EDE model, and the
χ^2_tot obtained from fitting the data in the former is also smaller
than that in the latter, indicating the potential
of the new model to alleviate the negative effect associated with the EDE model.
We have also computed the AIC for model comparison. Despite the smaller
χ^2_tot of the CSF model, its weaker fit compared to the EDE model
can be attributed to the introduction of additional parameters.
This work is supported in part by National Natural Science Foundation of China
under Grant No.12075042, Grant No.11675032 (People's Republic of China).
*
§ THE FULL MCMC POSTERIORS
|
http://arxiv.org/abs/2307.07656v1 | 20230714232205 | On weighted two-mode network projections | [
"Vladimir Batagelj"
] | cs.SI | [
"cs.SI",
"math.CO",
"91D30 (Primary) 05C76, 94A16, 01A90 (Secondary)"
] |
[
[
=====
The standard and fractional projections are extended from binary two-mode networks to weighted two-mode networks.
Some interesting properties of the extended projections are proved.
Keywords: weighted two-mode network, projection, fractional approach, strict collaboration, bibliometrics.
§ INTRODUCTION
In the paper <cit.> we studied the collaboration (co-authorship) between scientists from different post-Soviet countries was studied. We decided to repeat the study on the European countries. It turned out that there are different ways how we can define a network describing the co-authorship collaboration between countries. Some options are discussed in this paper.
Most of the bibliometric networks are obtained by a projection of a non-weighted network represented by a binary matrix. For example from the authorship network WA describing the authorship relation of the set of works (papers, books, reports, etc.) W by the authors from the sets A. It is represented by a matrix 𝐖𝐀 = [wa[w,a]] where wa[w,a] = 1 iff a is an author of the work w and 0 otherwise. We get the co-authorship (counting) network Co_A determined by the projection
𝐂𝐨_𝐀 = 𝐖𝐀^T ·𝐖𝐀
As we know <cit.>
* For a b, co_A[a,b] = number of works co-authored by authors a and b.
* co_A[a,a] = number of works from W written by the author a.
* The works with a large number of coauthors are "overrepresented" in the network Co_A – for example, the co-authorship of authors of a paper with 2 authors counts the same as the co-authorship between any pair of authors of the paper with 1000 co-authors; a paper with 1000 co-authors adds 1000000 links to projection network; while a single author paper only a loop. For this reason, the number co_A[a,b] is not the best measure for measuring the collaboration intensity.
The case of collaboration between countries is slightly different because the two-mode network WC is weighted. Actually, we could get it as 𝐖𝐂 = 𝐖𝐀·𝐀𝐂 where AC is the author-to-country affiliation network. This view opens a possibility to deal with authors affiliated to different countries provided that ∑_c ac[a,c] = 1. If the affiliations are changing through time the temporal quantities can be used <cit.>.
To obtain a collaboration network between a set of countries C based on a set of works W, we start with a two-mode network WC described by a matrix 𝐖𝐂 = [wc[w,c]] where
wc[w,c] =
In the network WC we can consider all authors of selected works W by adding to the set of countries C also the "country" Others. Instead of countries other partitions of the set of authors can be used, for example institutions.
We will use T(N) = ∑_e ∈ L w(e) to denote the total sum of weights of all links of the network N=(V,L,w).
§ COLLABORATION COUNTING NETWORK
The authors counting collaboration network Co_C described by the matrix 𝐂𝐨_𝐂 is obtained by projection
𝐂𝐨_𝐂 = 𝐖𝐂^T ·(𝐖𝐂)
where (𝐖𝐂) = [wc[w,c]], and wc[w,c] = 1 iff wc[w,c] 0 and 0 otherwise.
What are the meaning of the entry co_C[a,b] and their properties?
* For a b, co_C[a,b] = ∑_w wc[w,a] ·wc[w,b] – number of appearances of authors from the country a in works co-authored also by some author from the country b. We will denote this number _WC(a/b).
* co_C[a,a] = ∑_w wc[w,a] ·wc[w,a] = _WC(a) – number of appearances of authors from the country a in works from W; a column sum for country a in the matrix 𝐖𝐂.
* From a simple example
𝐖𝐂 = c_1 c_2 c_3
w_1 0 2 1
w_2 2 1 0
w_3 1 3 1
w_4 3 0 2
w_5 2 3 1
w_6 1 0 3
𝐂𝐨_𝐂 = c_1 c_2 c_3
c_1 9 5 7
c_2 7 9 8
c_3 7 3 8
we see that the matrix 𝐂𝐨_𝐂 is in general not symmetric – there can exist pairs a, b such that co_C[a,b] co_C[b,a].
* Consider a row sum R(a) for the country a in the matrix 𝐂𝐨_𝐂. We get
R(a) = ∑_b co_C[a,b] = ∑_w wc[w,a] ·∑_b wc[w,b] = ∑_w wc[w,a] ·_WC(w)
Since in the network WC only works W with co-authors from at least 2 countries are considered, we have _WC(w) ≥ 2 and we can continue
R(a) ≥ 2 ∑_w wc[w,a] = 2 _WC(a)
Now, combined with b, we finally get
∑_b: b a co_C[a,b] ≥_WC(a) = co_C[a,a]
The sum of the out-diagonal entries in the a row of the matrix 𝐂𝐨_𝐂 is larger or equal to its diagonal entry.
From the example in c we see that this property does not hold for columns – see the column c_2.
* For the diagonal values of the network Co_C it holds co_C[c,c] = _WC(c)
co_C[c,c] = ∑_w wc[w,c] ·wc[w,c] = ∑_w wc[w,c] = _WC(c)
Therefore ∑_c co_C[c,c] = T(WC).
* In the case when also the matrix 𝐖𝐂 is binary, (𝐖𝐂) = 𝐖𝐂, we deal with the standard projection mentioned in the introduction 𝐂𝐨_𝐂 = 𝐖𝐂^T ·𝐖𝐂. In the works counting collaboration network 𝐂𝐨_𝐛 = (𝐖𝐂)^T ·(𝐖𝐂) its weight co_b[a,b] counts works:
co_b[a,b] = number of works from W co-authored by authors from countries a and b, and co_b[a,a] = number of works from W co-authored by authors from the country a.
Note that the inequality from d still holds (and also for columns).
§ FRACTIONAL APPROACH
For binary networks, we define their normalized versions: standard n(𝐖𝐀) = [ wan[w,a]]
wan[w,a] = wa[w,a]/max(1,_WA(w))
and strict (or Newman's) N(𝐖𝐀) = [ waN[w,a]]
waN[w,a] = wa[w,a]/max(1,_WA(w)-1)
Using the normalized networks we define the standard fractional projection
𝐂𝐨_𝐧 = n(𝐖𝐀)^T · n(𝐖𝐀)
and the strict fractional projection
𝐂𝐨_𝐍 = D_0(n(𝐖𝐀)^T · N(𝐖𝐀))
where the function D_0(𝐌) sets the diagonal of a square matrix 𝐌 to 0.
We know <cit.> that if _WA(w) > 0, each work w ∈ W contributes equally, a unit 1, to the total weight of links in 𝐂𝐨_𝐧. The same holds for 𝐂𝐨_𝐍 if _WA(w) > 1.
To extend the fractional projections to weighted two-mode networks we define for the standard case n(𝐖𝐂) = [ wcn[w,c]]
wcn[w,c] = wc[w,c]/max(1,_WC(w))
Again we have T(Co_n) = |W| for 𝐂𝐨_𝐧 = n(𝐖𝐂)^T · n(𝐖𝐂).
𝐂𝐨_𝐛 = c_1 c_2 c_3
c_1 5 3 4
c_2 3 4 3
c_3 4 3 5
𝐂𝐨_𝐧 = c_1 c_2 c_3
c_1 1.0180556 0.5088889 0.5230556
c_2 0.5088889 1.1655556 0.4255556
c_3 0.5230556 0.4255556 0.9013889
There is no obvious way how to define the strict normalization for weighted networks.
There is another possible view on fractional projections. The definition of matrix n(𝐖𝐀) can be written as n(𝐖𝐀) = 𝐝_𝐧·𝐖𝐀 and similarly N(𝐖𝐀) = 𝐝_𝐍·𝐖𝐀 where 𝐝_𝐧 is a diagonal W × W matrix with d_n[w,w] = 1/max(1,_WA(w)) and 𝐝_𝐍 with d_N[w,w] = 1/max(1,_WA(w)-1).
In both cases we get (𝐝^T = 𝐝)
𝐂𝐨_𝐧 = n(𝐖𝐀)^T · n(𝐖𝐀) = 𝐖𝐀^T ·𝐝_𝐧·𝐝_𝐧·𝐖𝐀
𝐂𝐨_𝐍 = n(𝐖𝐀)^T · N(𝐖𝐀) = 𝐖𝐀^T ·𝐝_𝐧·𝐝_𝐍·𝐖𝐀
Because a product of diagonal matrices is a diagonal matrix, (a_w) ·(b_w) = (a_w · b_w), both cases have a common form 𝐖𝐀^T ·𝐝·𝐖𝐀. It can be related to the weighted scalar product. Maybe this form can lead also to the extension of strict projection for weighted two-mode networks.
§ STRICT FRACTIONAL COLLABORATION
Let us look at a simple example. Assume, that a work w has authors from three countries a, b, and c. Then, since the co-authors inside the same country do not count, its contribution T(w) to the total weight, see the contribution matrix
𝐂𝐨_𝐂(w) = a b c
a 0 wc[w,a] · wc[w,b] wc[w,a] · wc[w,c]
b wc[w,b] · wc[w,a] 0 wc[w,b] · wc[w,c]
c wc[w,c] · wc[w,a] wc[w,c] · wc[w,b] 0,
is T(w) = ∑_e,f ∈{a,b.c} e f wc[w,e] · wc[w,f]. By the rule of product and the rule of sum from basic combinatorics <cit.>, T(W) is equal to twice the number of all co-authorships of authors from different countries – pairs (a,b) and (b,a) are representing co-authorship of authors a and b.
To make T_N(w) = 1 we must set the entry d_N[w,w] of
the diagonal matrix 𝐝_𝐍 for the weighted network 𝐖𝐂 to d_N[w,w] = 1/T(w) = 1/(_WC(w)^2 - ∑_c wc[w,c]^2). Note that ∑_c wc[w,c] = _WC(w) and
_WC(w)^2 - ∑_c wc[w,c]^2 = ∑_e,f : e f wc[w,e] · wc[w,f]
The left side of this equality is computationally more convenient.
It is easy to see that we made a good guess – in the corresponding projection 𝐂𝐨_𝐍 = D_0(𝐖𝐂^T ·𝐝_𝐍·𝐖𝐂) each work contributes equally, a unit 1, to the total of link weights.
T_N(w) = d_N[w,w] ·∑_e,f : e f wc[w,e] · wc[w,f] = 1
Therefore
T(Co_N) = ∑_w T_N(w) = |W|
For our example from Section 2 c we get
𝐂𝐨_𝐍 = c_1 c_2 c_3
c_1 0.000000 0.9870130 1.1623377
c_2 0.987013 0.0000000 0.8506494
c_3 1.162338 0.8506494 0.0000000
with T(Co_N) = 6.
§ COMPUTING
C is a set of countries of our interest.
𝒲 is a list of metadata about the works from the selected bibliographic data source, co-authored by authors from at least two different countries from C. All four projection matrices 𝐂𝐨_𝐛, 𝐂𝐨_𝐧, 𝐂𝐨_𝐂, and 𝐂𝐨_𝐍 can be constructed in a single run through the list using the Algorithm <ref>.
Notes on the implementation of the algorithm:
* If we do not need the network WC we essentially need in line 4 the current list of pairs (c,wc(c)) for wc(c) > 0.
* Networks Co_b, Co_n, and Co_N are symmetric. They can be represented by an undirected network with the weight of an edge (e:f) equal to twice the computed value, except for loops. The computation can be restricted to pairs (e,f) for which e ≤ f.
§ CONCLUSIONS
In the paper, we derived the results in terms of the binary authorship network WA and the weighted network WC. The results hold in general for similar weighted two-mode networks such as (journals, universities, number of published articles of authors from the university u in the journal j in the selected time interval), (territorial units, universities, number of students from the territorial unit t studying this year at the university u), (web resources (movies or music tracks), types of resource, number of times the resource r of type t was downloaded in the selected time interval), (retail chain customers (chain card owners), (types of) products, the value of the product p bought by the customer c in the selected time interval), etc.
An application of the proposed projections in an analysis of a large real-life data set will be published in a separate paper(s).
§ ACKNOWLEDGMENTS
This work is supported in part by the Slovenian Research Agency
(research program P1-0294 and research projects J5-2557, J1-2481 and J5-4596),
and prepared within the framework of the COST action CA21163 (HiTEc).
99
fraca
Batagelj, V: On fractional approach to analysis of linked networks. Scientometrics 123 (2020) 2: 621-633
onbib
Batagelj, V, Cerinšek, M: On bibliographic networks. Scientometrics 96 (2013) 3, 845-864.
tempbib
Batagelj, V, Maltseva, D: Temporal bibliographic networks. Journal of Informetrics, Volume 14, Issue 1, February 2020, 101006.
soviet
Matveeva, N., Batagelj, V., Ferligoj, A.: Scientific collaboration of post-Soviet countries: the effects of different network normalizations. Scientometrics 128 (2023), 4219–4242.
rules Wikipedia: Combinatorial principles.
https://en.wikipedia.org/wiki/Combinatorial_principleshttps://en.wikipedia.org/wiki/Combinatorial_principles
|
http://arxiv.org/abs/2307.06117v1 | 20230708094335 | A qubit regularization of asymptotic freedom at the BKT transition without fine-tuning | [
"Sandip Maiti",
"Debasish Banerjee",
"Shailesh Chandrasekharan",
"Marina K. Marinkovic"
] | hep-lat | [
"hep-lat",
"cond-mat.str-el",
"hep-th",
"quant-ph"
] |
[email protected]
Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India
Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India
[email protected]
Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India
Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India
[email protected]
Department of Physics, Box 90305, Duke University, Durham, North Carolina 27708, USA
[email protected]
Institut für Theoretische Physik, Wolfgang-Pauli-Straße 27, ETH Zürich, 8093 Zürich, Switzerland
We propose a two-dimensional hard core loop-gas model as a way to regularize the asymptotically free
massive continuum quantum field theory that emerges at the BKT transition. Without fine-tuning, our
model can reproduce the universal step-scaling function of the classical lattice XY model in the massive
phase as we approach the phase transition. This is achieved by lowering the fugacity of Fock-vacuum
sites in the loop-gas configuration space to zero in the thermodynamic limit. Some of the universal
quantities at the BKT transition show smaller finite size effects in our model as compared to the
traditional XY model. Our model is a prime example of qubit regularization of an asymptotically free
massive quantum field theory in Euclidean space-time and helps understand how asymptotic freedom can
arise as a relevant perturbation at a decoupled fixed point without fine-tuning.
A qubit regularization of asymptotic freedom at the BKT transition without fine-tuning
Marina K. Marinkovic 0000-0002-9883-7866
August 12, 2023
======================================================================================
The success of the Standard Model of particle physics shows that at a fundamental level, nature is well described by a continuum QFT. Understanding QFT non-perturbatively continues to be an exciting area of research, since defining them in a mathematically unambiguous way can be challenging. Most definitions require some form of short-distance (UV) regularization, which ultimately needs to be removed. Wilson has argued that continuum QFT arise near fixed points of renormalization group flows <cit.>. This has led to the concept of universality, which says that different regularization schemes can lead to the same QFT. Following Wilson, traditional continuum quantum field theories are usually regulated non-perturbatively on a space-time lattice by replacing the continuum quantum fields by lattice quantum fields and constructing a lattice Hamiltonian with a quantum critical point where the long distance lattice physics can be argued to be the desired continuum QFT. However, universality suggests that there is a lot of freedom in choosing the microscopic lattice model to study a particular QFT of interest.
Motivated by this freedom and to study continuum quantum field theories in real time using a quantum computer, the idea of qubit regularization has gained popularity recently <cit.>. Unlike traditional lattice regularization, qubit regularization explores lattice models with a strictly finite local Hilbert space to reproduce the continuum QFT of interest. Euclidean qubit regularization can be viewed as constructing a Euclidean lattice field theory with a discrete and finite local configuration space, that reproduces the continuum Euclidean QFT of interest at a critical point. If the target continuum theory is relativistic, it would be natural to explore Euclidean qubit regularized models that are also symmetric under space-time rotations. However, this is not necessary, since such symmetries can emerge at the appropriate critical point. Lattice models with a finite dimensional Hilbert space that can reproduce continuum QFT of interest were introduced several years ago through the D-theory formalism <cit.> and has been proposed for quantum simulations <cit.>. In contrast to qubit regularization, the D-theory approach allows the local Hilbert space to grow through an additional dimension when necessary. In this sense, qubit regularization can be viewed as the D-theory approach for those QFT where a strictly finite Hilbert space is sufficient to reproduce the desired QFT.
Examples of using qubit regularization to reproduce continuum QFT in the IR are well known. Quantum spin models with a finite local Hilbert space are known to reproduce the physics of classical spin models with an infinite local Hilbert space near Wilson-Fisher fixed points <cit.>. They can also reproduce QFT with topological terms like the Wess-Zumino-Witten theories <cit.>. Gauge fields have been proposed to emerge dynamically at some quantum critical points of simple quantum spin systems <cit.>. From the perspective of Euclidean qubit regularization, recently it was shown that Wilson-Fisher fixed points with O(N) symmetries can be recovered using simple qubit regularized space-time loop models with N+1 degrees of freedom per lattice site <cit.>. Similar loop models have also been shown to produce other interesting critical behavior <cit.>. Loop models are extensions of dimer models, which are also known to describe interesting critical phenomena in the IR <cit.>. All this evidence shows that Euclidean qubit regularization is a natural way to recover continuum QFT that emerge via IR fixed points of lattice models.
A non-trivial question is whether we can also recover the physics of ultraviolet fixed points (UV-FPs), using qubit regularization. In particular, can we recover massive continuum QFT which are free in the UV but contain a marginally relevant coupling? Examples of such AF theories include two-dimensional spin models and four dimensional non-Abelian gauge theories. In the D-theory approach, there is strong evidence that the physics at the UV scale can indeed be recovered exponentially quickly as one increases the extent of the additional dimension <cit.>. Can the Gaussian nature of the UV theory emerge from just a few discrete and finite local lattice degrees of freedom, while the same theory then goes on to reproduce the massive physics in the IR? For this we will need a special type of quantum criticality where three length scales, as sketched in <ref>, emerge. There is a short lattice length scale a, where the non-universal physics depends on the details of the qubit regularization, followed by an intermediate length scale ≫ a, where the continuum UV physics sets in and the required Gaussian theory emerges. Finally, at long length scales ≫, the non-perturbative massive continuum quantum field theory emerges due to the presence of a marginally relevant coupling in the UV theory. The qubit regularized theory thus reproduces the universal continuum QFT in the whole region ℓ_ UV to ℓ_ IR. The special quantum critical point must be such that ℓ_ UV/a →∞.
Recently, a quantum critical point with these features was discovered in an attempt to find a qubit regularization of the asymptotically free massive non-linear O(3) sigma model in two space-time dimensions in the Hamiltonian formulation <cit.>. Using finite size scaling techniques, it was shown that the qubit regularized model recovers all the three scales. In this paper, we report the discovery of yet another example of a quantum critical point with similar features. In the current case, it is a Euclidean qubit regularization of the asymptotically free massive continuum quantum field theory that arises as one approaches the BKT transition from the massive phase <cit.>. In both these examples, the qubit regularized model is constructed using two decoupled theories and the AF-QFT emerges as a relevant perturbation at a decoupled quantum critical point. The coupling between the theories plays the role of the perturbation that creates the three scales, as illustrated in the RG flow shown in <ref>. An interesting feature of this discovery is that there is no need for fine-tuning to observe some of the universal features of the BKT transition that have been unattainable in practice with other traditional regularizations <cit.>.
The BKT transition is one of the most widely studied classical phase transitions, since it plays an important role in understanding the finite temperature superfluid phase transition of two-dimensional systems <cit.>. One simple lattice model that captures the universal behavior of the physics close to the phase transition is the
classical two-dimensional XY model on a square lattice given by the classical action,
S = -β∑_⟨ ij⟩cos(θ_i-θ_j),
where the lattice field 0≤θ_i < 2π is an angle associated to every space-time lattice site i and ⟨ ij⟩ refers to the nearest neighbor bonds with sites i and j. The lattice field naturally lives in an infinite dimensional Hilbert space of the corresponding one dimensional quantum model. Using high precision Monte Carlo calculations, the BKT transition has been determined to occur at the fine-tuned coupling of β_c ≈ 1.1199(1) <cit.>. The Villain model is another lattice model which is friendlier for analytic calculations and has been used to uncover the role of topological defects in driving the phase transition <cit.>. More recently, topological lattice actions which seem to suppress vortices and anti-vortices but still drive the BKT transition have also been explored <cit.>.
As one approaches the BKT transition from the massive phase, the long distance physics of the <ref> is known to be captured by the sine-Gordon model whose Euclidean action is given by<cit.>,
S = ∫ dx dt [ 1/2t (∂_μθ_1)^2 + t/8π^2 (∂_μθ_2)^2 -
A t/4π^2cosθ_2 ]
where t ≥π/2. The field θ_1(x,t) captures the spin-wave physics while the vortex dynamics is captured by the field θ_2(x,t). The BKT transition in this field theory language occurs at t = π/2 where the cosθ_2 term becomes marginal as one approaches the critical point and the physics is governed by a free Gaussian theory. In this sense, the long distance physics of the lattice XY model, as β is tuned to β_c from smaller values, is an asymptotically free massive Euclidean continuum QFT.
Qubit regularizations of the classical XY-model have been explored recently using various quantum spin formulations <cit.>. Lattice models based on the spin-1 Hilbert space are known to contain rich phase diagrams <cit.>, and quantum field theories that arise at some of the critical points can be different from those that arise at the BKT transition. Also, the presence of a marginally relevant operator at the BKT transition can make the analysis difficult, especially if the location of the critical point is not known. In these cases, it becomes a fitting parameter in the analysis, increasing the difficulty. Since in our model the location of the critical point is known, our model can be analyzed more easily.
The model we consider in this work is a variant of the qubit regularized XY model introduced in Euclidean space recently <cit.>. The model can be viewed as a certain limiting case of the classical lattice XY-model <ref> written in the world-line representation <cit.>, where the bosons are assumed to be hard-core. The partition function of our model is a sum of weights associated with configurations of oriented self-avoiding loops on a square lattice with Fock-vacuum sites. An illustration of the loop configuration is shown as the left figure in <ref>. The main difference between our model in this work and the one introduced previously is that closed loops on a single bond are now allowed. Such loops seemed unnatural in the Hamiltonian framework that motivated the previous study, but seem to have profoundly different features in two dimensions <cit.>. It is also possible to view the loop configurations of our model as a configuration of closed packed oriented dimers on two layers of square lattices. The dimer configuration corresponding to the loop configuration is shown on the right in <ref>. The dimer picture of the partition function arises as a limiting case of a model involving two flavors of staggered fermions, introduced to study the physics of symmetric mass generation <cit.>. In this view point the inter-layer dimers (or Fock vacuum sites) resemble t'Hooft vertices (or instantons) in the fermionic theory. Using this connection, the partition function of our model can be compactly written as the Grassmann integral
Z = ∫ [d d] [d d] exp(λ ∑_i _i _i _i_i)
× exp( ∑_⟨ ij⟩( _i _i _j_j + _i _i _j _j))
where on each site i of the square lattice we define four Grassmann variables _i, _i, _i and _i. We consider periodic lattices with L sites in each direction. Using the fermion bag approach <cit.>, we can integrate the Grassmann variables and write the partition function as a sum over dimer configurations whose weight is given by λ^N_I where N_I is the number of instantons (or Fock-vacuum sites). Thus, λ plays the role of the fugacity of Fock-vacuum sites. It is easy to verify that the action of our model is invariant under _j _j → e^iσ_jθ_j _j and _j _j → e^-iσ_jθ_j _j where σ_j = ± tracks the parity of the site j. This U(1) symmetry is connected to the BKT transition and in order to track it, the dimers are given an orientation as explained in <ref>.
Using worm algorithms (see <cit.>) we study our model for various values of L and λ.
At λ = 0, one gets two decoupled layers of closed packed dimer models, which is known to be critical <cit.>. The effect of λ≠ 0 was studied several years ago, and it was recognized that there is a massive phase for sufficiently large values of λ <cit.>. However, the scaling of quantities as λ→ 0 was not carefully explored. Recently, the subject was reconsidered, and a crossover phenomenon was observed for small λ as a function of L. An understanding of this crossover was largely left unresolved as a puzzle <cit.>. In this paper, we demonstrate that the observed crossover phenomena captures the asymptotic freedom of <ref>. We do this by comparing the universal behavior of <ref> with the traditional XY model <ref> near the massive phase of the BKT transition <cit.>.
To compare universal behaviors of <ref> and <ref> we compute the second moment finite size correlation length ξ(L) defined as ξ(L) = √((χ/F)-1)/(2sin(π/L)) (see <cit.>), where χ = G(0) and F = G(2π/L) are defined through the two point correlation function
G(p) = ∑_j e^i p x⟨ O^+_(x,t) O^-_(0,0)⟩.
In the above relation j is the space-time lattice site with coordinates (x,t) and O^+_j, O^-_j are appropriate lattice fields in the two models. In the XY model O^+_j = e^iθ_j, O^-_j = e^-iθ_j, while in the dimer model O^+_j = O^-_j = _j _j. We demonstrate that the step-scaling function (SSF) (i.e., the dependence of ξ(2L)/ξ(L) on ξ(L)/L) of the two lattice models show excellent agreement with each other in the scaling regime ℓ_UV≫ a, in <ref>.
Another interesting universal result at the BKT transition is the value of the helicity modulus, which can be defined using the relation, Υ = ⟨ Q_w^2⟩ where Q_w is the spatial winding number of bosonic worldlines. In the XY model <ref>, it is usually defined using a susceptibility of a twist parameter in the boundary conditions <cit.>. In our model, we can easily compute the winding charge Q_w in each loop configuration illustrated in <ref>. The universal result in the massive phase as we approach the BKT transition is that Υ≈ 2/π in the UV up to exponentially small corrections <cit.>, although in the IR Υ = 0. While it is difficult to obtain the UV value in lattice calculations using the traditional model <ref>, in our model, we can see it emerge nicely at λ=0.01. We demonstrate this in <ref>. Again, as expected, the value of Υ when λ=0 is very different, since it is a theory of free bosons but at a different coupling. Using the different value of the coupling gives Υ ≈ 0.606 <cit.>. Our results provide strong evidence that the AF-QFT at the BKT transition emerges from our dimer model when we take the limit L→∞ followed by λ→ 0. The opposite limit leads to the critical theory of the decoupled dimer model.
Acknowledgments: We are grateful to J. Pinto Barros, S. Bhattacharjee, T. Bhattacharya, H. Liu, A. Sen, H. Singh and U.-J. Wiese for inspiring discussions. We acknowledge use of the computing clusters at SINP, and the access to Piz Daint at the Swiss National Supercomputing Centre, Switzerland under the ETHZ’s share with the project IDs go24 and eth8. Support from the Google Research Scholar Award in Quantum Computing and the Quantum Center at ETH Zurich is gratefully acknowledged. S.C's contribution to this work is based on work supported by the U.S. Department of Energy, Office of Science — High Energy Physics Contract KA2401032 (Triad National Security, LLC Contract Grant No. 89233218CNA000001) to Los Alamos National Laboratory. S.C is supported by a Duke subcontract based on this grant. S.C's work is also supported in part by the U.S. Department of Energy, Office of Science, Nuclear Physics program under Award No. DE-FG02-05ER41368.
Supplementary Material
§ UNIVERSAL VALUES OF Υ FOR Λ = 0 AND Λ≠ 0
In this section we explain the two different values of the helicity modulus Υ for our model when λ=0 and λ→ 0. When λ=0 our model maps into two identical but decoupled layers of closed packed classical dimer models. As has already been explained in the literature (see for example <cit.>), each layer can be mapped to the theory of a free compact scalar field with the action
S = 1/2 t∫ d^2 x (∂_μθ(x))^2.
with t=4π. One can compute Υ starting with <ref>, by noting that the scalar fields have winding number configurations labeled by n_x:
θ(x) = 2 π x n_x/L_x + φ(x),
where φ(x) is a smooth fluctuation that is independent of winding number n_x. The value of the action in each winding sector in a finite space-time volume is then given by
S(n_x) = 2π^2 n_x^2/tL_y/L_x + S_0,
where S_0 is the action from the usual fluctuations in the zero winding number sector. Using L_x = L_y, we can compute Υ using its connection to the average of the square of the winding numbers,
Υ = ⟨ (Q_x)^2 |=⟩∑_n_x n_x^2 · e^- 2 π^2 n_x^2/t/∑_n_x e^-2π^2 n_x^2/t
Numerically evaluating this expression for t=4π we obtain Υ = 0.303426... for a each layer of our dimer model. Our value of 0.606852... is due to the presence of two decoupled layers.
In contrast, in the limit λ→ 0, we need to consider the physics at the BKT transition and so we begin with the action
S = ∫ d^2x [ 1/2t̃ (∂_μθ_1)^2 + t̃/8π^2 (∂_μθ_2)^2 -
A t̃/4π^2cosθ_2 ]
and focus at t̃=π/2. At this coupling the last term is irrelevant and Υ gets dominant contribution from the θ_2 field. In this we can still use <ref> but need to substitute t = 4π^2/t̃ = 8π. Substituting we get Υ = 0.636508... which is approximately 2/π.
§ WORM ALGORITHM
In this section, we discuss the worm algorithm we use to simulate the model with the partition function,
Z = ∫ [d d] [d d] exp(λ ∑_i _i _i _i_i)
× exp( ∑_⟨ ij⟩( _i _i _j_j + _i _i _j _j))
as introduced in the main paper. These algorithms are well known <cit.>, and can be divided into three parts: Begin, Move, and End.
* Begin: pick a site at random and denote it as tail, and there are the following
two possibilities: (A) either it has a bond connected to it on the other layer (which we call
an instanton, or an interlayer dimer), or, (B) it has a bond connected to it on the same layer
(which we call a dimer).
* For the case (A), propose to remove the instanton, and put the worm head on the
same site at the different layer, with a probability 1/λ. If accepted, then begin the
worm update, otherwise go to (1).
* For the case (B), pick the other site to which the dimer is connected as the head,
and begin the worm update.
* Move: Propose to move the worm head to one of the (2D+1) neighbor sites of head
with an equal probability, which can either be on the same layer (2D choices), or on the different
layer (one choice). Denote the proposed new site as site0, and the following possibilities can
occur, provided that site0 is not the tail:
* site0 is on the same layer, and has an instanton connected to it. Propose to
remove the instanton with a probability 1/λ. If accepted, place the head
at site0, but on the different layer.
* site0 is on the same layer, and has a dimer connected to it (joining site0
and y). Move the head to the site y with a probability 1, and simultaneously insert
a dimer between head and site0.
* site0 is on the different layer, then propose if an instanton can be created. If
yes, then move the position of the head to y in the other layer, where y is the other
end of the dimer connecting site0 and y.
* End: If at any stage in the algorithm, the site0 is the tail, then propose to end
the worm update. If the site0 = tail is on the same layer, then end the update by putting
a dimer between the head and tail with a probability 1. If, on the other hand, they are
on different layers, the worm update ends with a probability λ, leading to the addition of an
extra instanton.
§ EXACT VS MONTE CARLO RESULTS ON A 2 × 2 LATTICE
In this work, we compute two independent fermion bilinear susceptibilities defined as
χ_1 = 1/2V∑_i,j
i≠ j⟨ψ̅_i ψ_i ψ̅_j ψ_j ⟩,
χ_2 = 1/2V∑_i,j
i≠ j⟨ψ̅_i ψ_i χ̅_j χ_j ⟩,
where χ_1 is an observable that can be defined even on a single layer, while χ_2 is involves both the layers. When the coupling λ = 0, the two layers are completely decoupled from each other and we get χ_2 = 0. Another quantity we compute is the average density of Fock vacuum sites or inter-layer dimers (which we also view as instantons), defined as
ρ = 1/V∑_i ⟨ψ̅_i ψ_i χ̅_i χ_i ⟩,
where the expectation value is defined as
⟨ O⟩ = 1/Z∫ [𝒟ψ̅𝒟ψ]
[𝒟χ̅𝒟χ] O
e^-S[ψ̅,ψ, χ̅,χ].
Since every site is populated by either a Fock-vacuum site or an intra-layer dimer, the average intra-layer dimer density is not an independent observable. We can always compute it from the Fock vacuum sites (instanton) density ρ.
In order to test out algorithm, we focus on exact results on a 2× 2 lattice. The partition function in this simple case is given by
Z = 64 + 16 λ^2 + λ^4,
while the instanton density and the two independent susceptibilities are given by
ρ = 1/4Z (32 λ^2 + 4 λ^4),
χ_1 = 1/2Z (32 + 4 λ^2),
χ_2 = 1/2Z (8 λ).
Note that ρ is zero when λ = 0 and approaches one for large couplings. Also, as expected χ_2=0 when λ=0. In <ref> we compare results for three different observables, instanton density (ρ), fermion bilinear susceptibility (χ_1), and helicity modulus (Υ) on a 2 × 2 lattice obtained from an exact calculation against the results obtained using the worm algorithm.
Interestingly, when λ≠ 0 we find that both χ_1 and χ_2 become similar as L increases. The difference also becomes smaller as λ increases. We show this behavior in the <ref>.
Due to this similarity we only focus on χ_1 in our work.
§ PLOTS OF Ρ AND Χ_1
We have computed the fermionic XY model at various values of λ on square lattices up to L = 4000 using the
worm algorithm described above. For our simulations, after allowing for appropriate thermalization, we have recorded between 8 × 10^3 and 48 × 10^3 measurements, each averaged over 2000 worm updates. A comparable number of measurements were also made for the bosonic model.
In <ref>, we plot ρ for various lattice sizes at different values of λ on the left. We note that ρ increases monotonically and approaches the thermodynamic limit by L=160 which is shown on the right.
In <ref>, we plot χ_1 as a function of system size, L for different values of λ. When λ is small, we find that our data is consistent with the behavior χ_1 ∼ AL^2-η expected in a critical phase. However, for larger values of λ, the susceptibility begins to saturate as χ_1 ∼ A which means η≈ 2. For λ=0, since the model describes two decoupled layers of closed packed dimer models we expect η=0.5 <cit.>. However, when λ is small, since we expect our model to describe the physics at the BKT transition, we expect η∼ 0.25. This is consistent with our findings.
The values of constant A and η for various values of λ obtained from a fit are given in <ref>.
§ STEP SCALING FUNCTION
In order to argue that the traditional XY model at the BKT transition and the two layer interacting dimer model are equivalent we compute the step scaling function (SSF) in both of them. We refer to the traditional XY model defined through the lattice action
S = -β∑_⟨ ij ⟩cos(θ_i-θ_j),
as the bosonic XY model and dimer model defined in <ref> as the fermionic XY model. In order to compute the step-scaling function we first compute the second moment correlation length defined in a finite box of size L using the expression
ξ(L) = 1/2sin(π/L)√(χ/F - 1),
where
χ = ∑_i ⟨ O^+_i O^-_0⟩,
F = ∑_i ⟨ O^+_i O^-_0⟩cos(2π x /L),
where i=(x,t) is the space-time lattice site and O^+_i, O^-_i are lattice fields in the two lattice models. In the bosonic XY model, O^+_i = e^iθ_i and O^-_i = e^-iθ_i, while in the fermionic model O^+_i = O^-_i = ψ_iψ_i.
The SSF for the bosonic XY model is computed in the massive phase close to the critical point, for β < β_c = 1.1199 <cit.>. To study the step scaling function, we prepare several pairs of data at (β, L) and (β,2L), and compute both ξ(2L)/ξ(L) and ξ(L)/L using the data presented in <ref>. We follow certain criteria as explained in <cit.>, to ensure the minimization of finite volume and finite lattice spacing errors. In particular, we only choose lattices of sizes L ≥ L_min, where L_min = 80 for couplings β≥0.92. Since the correlation length increases for β close to the β_c, larger lattice sizes are essential. The similar criteria for choosing the lattices sizes and couplings in the fermionic model is L ≥ L_min, where L_min = 80 for 0.62≤λ≤0.9, and L_min = 640 for λ < 0.6.
In order to compute the expectation value and error of ξ(L)/L, we use the jackknife analysis. We report the results here for the analysis with 40 jackknife blocks. The effect of variation of the jackknife blocks did not change the errors significantly, and were consistent with the errors obtained using a bootstrap analysis. In <ref>, we show an example of the variation of the average and error of ξ(L)/L at λ=0.353 and L=320 for the fermionic model using both the jackknife and the bootstrap analysis as a function of block size. For both methods, we use the same number of block sizes, but in order to show the distinction between them, we have displaced the data on the x-axis by multiplying nBlock by a factor of 1.1 for the bootstrap analysis.
In order to compare the SSF between the bosonic and the fermionic models we tried to parameterize the function in two different ways. In the first approach, we follow the idea discussed in <cit.> where it was proposed that
Σ(x) = 1 + a_1 e^-1/x + a_2 e^-2/x + a_3 e^-3/x + a_4 e^-4/x,
where x = ξ(L)/L and Σ = ξ(2L)/ξ(L). The behavior of this function is such that, as x → 0, the function Σ(x) approaches 1. While this function is strictly valid only for small x we find that this form fits our data well. The fit results are given in <ref>. We see that while we get good fits by including all four fit parameters, we can also fix a_2=0 and still get a good fits.
In the second approach, to parameterize our SSF we used a cubical spline to interpolate the data. In <ref>, we provide a tabulation of the spline function that helps parameterize the SSF for both the bosonic and the fermionic models. The errors are obtained using a jackknife analysis.
In order to show how these two different parameterizations help capture our data we show the corresponding curves for the bosonic model in <ref> and for the fermionic model in <ref>. We believe that a combined parameterization would best capture the true function. Hence, we use <ref> for ξ(L)/L ≤ 0.572 and the cubical spline interpolation for ξ(L)/L ≥ 0.572. This combined form in the bosonic model is shown in <ref>, along with the bosonic model data. The dark line of this plot is used in the main paper to compare with the fermionic model.
§ INFINITE VOLUME CORRELATION LENGTH
We can compute the infinite volume correlation length ξ_∞ using the SSF. Here we try to understand how ξ_∞ depends on λ in the fermionic XY model. In order to reliably estimate the errors in ξ_∞ we again use the jackknife analysis. We start with 40 jackknife blocks, where each block contains a pair (ξ(L)/L, ξ(2L)/ξ(L)) for different coupling values (0.01 ≤λ≤ 0.8). We obtain 40 different cubical splines using each jackknife block. We then start with the initial ξ(L)/L at L=640 in each block and evaluate ξ(2^n L) using the spline function for arbitrary values of n, until the correlation length ξ(2^n L) becomes insensitive to L. Finally, the jackknife mean and error is then computed from the 40 values. These results for ξ_∞ and their errors are quoted in <ref>.
Since the correlation lengths increase exponentially as λ becomes small, we were able to extract the infinite volume correlation length only in the range 0.3≤λ≤0.8. Below λ < 0.3, our extrapolation methods fail.
Using the data in <ref> we study the λ dependence of ξ_∞. For the bosonic XY model, it is well known that as one approaches the BKT phase transition, the leading divergence of the infinite volume correlation length is captured by
ξ = C exp( b/√(β_c - β)),
where β_c is the critical coupling, and b and C are non-universal constants. For the fermionic XY model since the partition function is an even function of λ we expect ξ_∞ to be a function of λ^2. Since the BKT critical point appears when λ→ 0, we conjecture that
ξ^(1)_∞ = a_1 exp( b_1/√(λ^2)).
We test this conjecture numerically by fitting the data in <ref> to it. We also compare this to other fit forms including ξ^(2)_∞ = a_2 exp(b_2/(λ^2)^1/4) and ξ^(3)_∞= a_3 exp(b_3/√(λ^2) + c_3 log(λ^2)/2). The results are shown in <ref>. We observe that <ref> is clearly quite good if we expect the constants a and b to be numbers which are not unnatural. We cannot rule out the presence of a power law correction to the expected form.
In <ref>, we show the data in <ref> and the various fits. The first form is the expected behaviour from <ref>. The second form explores a possible dependence on square-root of λ which is clearly unnatural. Finally the third form allows for a logarithmic correction in the exponential (which is equivalent to including a 1/λ dependence outside the exponential). We note that in this extended form the data in the larger range of 0.3 ≤λ≤ 0.8 can be fit.
§ MONTE CARLO RESULTS
We tabulate all of our Monte Carlo data in <ref> for both the bosonic XY and the fermionic XY models, for various values of L and couplings. The errors in these primary quantities have been obtained with 20 jackknife blocks.
|
http://arxiv.org/abs/2307.04961v1 | 20230711015143 | Still Waters Run Deep: Extend THz Coverage with Non-Intelligent Reflecting Surface | [
"Chong Han",
"Yuanbo Li",
"Yinqin Wang"
] | cs.IT | [
"cs.IT",
"eess.SP",
"math.IT"
] |
Still Waters Run Deep: Extend THz Coverage with Non-Intelligent Reflecting Surface
Chong Han, Member, IEEE,, Yuanbo Li, Yiqin Wang
Chong Han, Yuanbo Li, and Yiqin Wang are with the Terahertz Wireless Communications (TWC) Laboratory, Shanghai Jiao Tong University, Shanghai, China (e-mail: {chong.han, yuanbo.li, wangyiqin}@sjtu.edu.cn).
August 12, 2023
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empty
Large reflection and diffraction losses in the Terahertz (THz) band give rise to degraded coverage abilities in non-line-of-sight (NLoS) areas. To overcome this, a non-intelligent reflecting surface (NIRS) can be used, which is essentially a rough surface made by metal materials. NIRS is not only able to enhance received power in large NLoS areas through rich reflections and scattering, but also costless and super-easy to fabricate and implement. In this article, we first thoroughly compare NIRS with the lively discussed intelligent reflecting surface (IRS) and point out the unique advantages of NIRS over IRS. Furthermore, experimental results are elaborated to show the effectiveness of NIRS in improving coverage. Last but not least, open problems and future directions are highlighted to inspire future research efforts on NIRS.
Non-intelligent reflecting surface, Terahertz communications, Coverage extension.
§ INTRODUCTION
Over the past few decades, wireless communication networks have experienced revolutionary developments, from the first generation (1G) to the most recent fifth generation (5G).
Nonetheless, looking towards 2030, the mobile communication network will further evolve to the sixth generation (6G), where internet-of-everything (IoE) is expected to be achieved with ubiquitous network coverage and massive connectivity <cit.>. Various and abundant intelligent devices, such as smartphones, mixed reality (MR) headsets, as well as sensors and machines, will generate a large amount of message and data for wireless communications. As electromagnetic infrastructure to support them, ultra high data rates (e.g., up to 1 Terabits per second) are needed, which can not be fullfilled by current spectrum resource and thus motivate the exploration of the Terahertz (THz) band. Spanning the frequency between 0.1THz to 10THz, the THz band is envisioned as a key technology to address the spectrum scarcity and capacity limitations of current wireless systems <cit.>, thanks to its broad contiguous bandwidth (from tens up to hundreds of GHz).
Wonderful as THz communication is, however, it has its own drawbacks. Among others, one key problem is the weak coverage ability of THz communications in non-line-of-sight (NLoS) areas. At high frequencies, reflection, diffraction, and penetration losses worsen <cit.>. As a result, when line-of-sight (LoS) transmission is blocked, sometimes drastic degradation of link quality may occur. To address LoS blockage problem, one natural solution is to add more active nodes in the network, such as base stations (BSs), access points (APs), and active relays, which however, associate with extra hardware and energy costs. By contrast, energy efficient solutions are preferred, such as intelligent reflecting surface (IRS), which is a passive tunable metasurface and able to redirect propagating THz waves <cit.>.
However, even though the theoretical performance of IRS is extraordinary, realization of IRS in the THz band might be difficult and far from practice, for the following reasons. First, due to high frequencies of THz waves, thousands of elements are required to compensate for the large path loss from IRS to receiver. The fabrication of such large number of IRS elements and corresponding control circuits might be very difficult and costly. Second, attributed to the small wavelength of THz waves, tiny antenna elements in the order of sub-millimeter could be fabricated and densely placed to form ultra massive multiple-input-multiple-output (UM-MIMO) system, resulting in improved spectral efficiency and coverage capability. Integrating UM-MIMO and IRS, the concatenated channel from transmitter (Tx) to IRS to receiver (Rx) is expressed with a channel tensor with dimensions of N_t× N_IRS× N_r, with N_t, N_IRS, and N_r denoting the numbers of elements of transmitter array, IRS, and receiver array, respectively. The massive IRS elements would make the accurate channel estimation of the large-scale channel tensor computationally complex, for which the joint optimization of the UM-MIMO and IRS is hard to achieve. Therefore, still a long and spiny path is in front to practically implement IRS in the THz band.
By contrast, a more realistic and easier way is to use non-intelligent reflecting surface (NIRS), which is essentially a rough surface simply made of metal materials. Compared to IRS, NIRS loses the ability to adapt to mobile users or suppress interference from neighbouring BS, while gaining advantages such as nearly no cost, no fabrication, and super-easy deployment. We hereby note that NIRS is different from the frequently mentioned reflectors in cmWave and mmWave bands <cit.> as follows. NIRS is rough and require no specific design, while reflector is a smooth surface acting as electromagnetic mirrors. The reason that NIRS is preferred than reflectors is two-fold. On one hand, due to the small wavelength, THz waves are more sensitive to surface roughness, resulting in a stricter requirement for a reflector to be smooth considering the sub-millimeter wavelength. Lower fabrication difficulty is the key advantage of NIRS, compared to reflectors. On the other hand, the high sensitivity of THz waves lead to strong scattering, i.e., non-specular reflections, especially when interacting with rough metal surfaces. Even though NIRS performs worse than reflectors in specular directions, it can simultaneously enhance signal strength in non-specular directions, thus covering wider NLoS areas. In summary, even though NIRS appears more clumsy compared to IRS or reflectors, the outstanding low cost, low utilization difficulty, and wide coverage of NIRS promote it to be a good technique for coverage extension in THz networks.
Fig. <ref> illustrates several typical use cases of NIRS, in both indoor and outdoor scenarios. For instance, in L-shaped corridors, the LoS path is blocked by walls, which results in significant link performance degradation. In light of this, NIRS can be deployed on the walls near the turning corner, to provide once-scattering or high-order reflections path to enhance the coverage in the NLoS region. Moreover, objects in indoor rooms, such as bookshelf in library, server rack in data center, etc., can also shelter the receivers from the access points. In this regard, the NIRS can be deployed on walls or ceilings, to bypass the blocking objects. Similarly, the blockage of high-rise buildings in urban areas can also be address by deploying NIRS on building surfaces. Last but not least, for pedestrian communicating with lamppost base stations, the blockage of human body and foliage can be severe, due to the weak penetration ability of THz waves. Therefore, the NIRS deployed on nearby walls or grounds can help redirect a reliable link.
To this end, NIRS is a promising technique for THz communications to address LoS blockage problem, by exploiting the benefits from rough surface scattering. Motivated by this, we provide an overview of NIRS, including the main advantages and disadvantages of NIRS compared to IRS, in terms of flexibility, fabrication and design difficulty, and compatibility to UM-MIMO systems. Moreover, to show the efficacy of NIRS, a preliminary experiment in an indoor corridor scenario is elaborated. Coverage and capacity are improved with deployment of NIRS.
Furthermore, open problems and future challenges are highlighted to inspire future research, including NIRS channel modeling, reliable design, deployment and coordination optimization, joint communication and sensing enhancement.
§ IRS V.S. NIRS: TRADE-OFF OF PERFORMANCE AND IMPLEMENTATION DIFFICULTY
As an overview, the comparison of IRS and NIRS is shown in Table <ref>. In short, the attractive excellent flexibility and intelligence of beam control via IRS comes at prices of high hardware and computation costs, which may prevent its realization in the THz band. On the contrary, the design and usage of NIRS are much easier in practice, with noticeable coverage enhancement.
§.§ Beam Control Ability
High flexibility and adaptability are the key advantage of IRS compared to NIRS, which also distinguish their intelligence. IRS is composed of a large number of reflecting elements, which can manipulate the reflecting amplitude and phase shift of the impinging THz waves. Enabled by them, IRS has the so-called passive beamforming ability, i.e., the scattering pattern can be electrically controlled to realize certain purposes. For instance, when used for coverage extension, IRS can intelligently concentrate the signal power towards the directions of users. As the user moves, the IRS beam constantly steers to follow, achieving reliable communication links. Moreover, just like active beamforming of antenna arrays, IRS can also create zero-directions to suppress interference from neighbouring BS in down-link or to them in up-link.
In this regard, NIRS is much more clumsy and thus non-intelligent. It can neither simultaneously change the scattering pattern to track mobile users, nor cancel interference in multi-user or multi-BS scenarios. In fact, NIRS may cause more interference since the NLoS signals are all enhanced, no matter from the targeted BS/AP or other BSs/APs. Therefore, before deployment in realistic communication networks, it is of importance to appropriately design and deploy the NIRS in positions maximizing the signal-to-interference and noise ratio (SNR).
§.§ Link Path Loss
A key factor to evaluate link performance is the path loss, which determines the SNR at the receiver side. Due to the different design considerations, the path losses of NIRS and IRS-aided communication links are fundamentally different. Particularly, for IRS, the communication channel is concatenated by two segments, namely the Tx-IRS channel and the IRS-Rx channel. Moreover, to steer the IRS beams toward any user location, IRS elements are designed to scatter the incident signal omnidirectionally. In other words, the reflected signal from a single IRS element is just like being radiated by an omnidirectional antenna. As a result, the path loss of IRS-aided communication link is inversely proportional to the product of distance from Tx to IRS and distance from IRS to Rx, as clearly explained as the product-distance path loss model in <cit.>. Considering only the LoS path, by using the Friis' formula, the free space path loss at 300GHz could exceed 80dB for a distance of only 1m. As a result, an overall path loss of Tx-IRS-Rx link would surpass 160dB if Tx-IRS and IRS-Rx distances are both 1m. To overcome the severe path loss, high beamforming gains and thus very large amount of IRS elements are needed. For example, as analyzed in <cit.>, to outperform a direct link, the number of IRS elements needs to exceed 4096 in the THz band.
Unlike the omnidirectional IRS elements that spread signal energy uniformly to all directions, the reflected/scattered signals from NIRS is concentrated on several directions, such as the specular direction. Therefore, the path loss of Tx-NIRS-Rx link consists of two parts, namely the spreading loss part and additional reflection loss part. The spreading loss part is dependent on the overall link distance, i.e., reversely proportional to the summation of Tx-NIRS and NIRS-Rx distances, following the sum-distance path loss model <cit.>. Moreover, the additional reflection loss part is dependent on reflection angles of NIRS-Rx direction. In strong reflection/scattering directions, such as the specular direction, the additional reflection loss could be as low as several dB, while in other directions, the additional reflection loss could increase up to tens of dB. Generally speaking, an overall path loss of 100140dB could occur for NIRS-aided links, depending on propagation distance and receiver locations <cit.>. Nonetheless, compared to the situations in NLoS areas without NIRS, the path loss values are mitigated by 317dB.
§.§ Fabrication and Control Difficulty
The super-easy fabrication and control is the key strength of NIRS over IRS. As mentioned above, to explore the potential of IRS in the THz band, thousands of elements are needed to compensate for the significant path loss, each of which is electrically controlled. Moreover, due to the small wavelength of THz waves, IRS elements need to be rather small to omnidirectionally scatter the signals, e.g., a fifth of the wavelength (tends of micrometer). Such ultra-massive yet ultra-tiny IRS elements, and more importantly, their corresponding control circuits, would make it extremely costly and difficult to fabricate. On the contrary, NIRS require no specific design, whose fabrication is much easier and costless. As reported in <cit.>, NIRS can be simply made with super cheap aluminium foils, which yet realize considerable coverage extension for THz communications.
§.§ Jointly Usage With UM-MIMO
To exploit high spectral efficiency with spatial multiplexing, UM-MIMO is a critical mass in the THz band <cit.>. When embedding IRS in UM-MIMO systems, the joint optimization requires knowledge of the concatenated Tx array-IRS-Rx array channel, which however, is computationally complex due to the high dimensions. By contrast, the joint usage of NIRS and UM-MIMO system is natural since the NIRS needs no real-time control and thus does not add additional signal processing burden. Moreover, since NIRS creates rich reflection and scattering, the spatial degree of freedom improves, which is originally limited in the THz UM-MIMO system due to the sparse THz channel. Consequently, the rank of the UM-MIMO channel matrix and spatial multiplexing gain increase, which further improves the channel capacity.
§ EXPERIMENTAL RESULTS FOR NIRS-AIDED THZ COVERAGE EXTENSION
In this section, we elaborate an experiment of using NIRS to extend THz coverage in a corridor scenario. The experiment set-up, deployment, and results are explained in the following part.
§.§ Experiment Set-up and Deployment
Experiments are conducted with a vector network analyzer (VNA)-based channel sounder, which is introduced in detail in <cit.>. Specifically, the measured frequency bands are 306321GHz and 356371GHz. During the channel measurement, the transmitter is only equipped with a standard waveguide WR2.8 for large coverage with wide beam, which has 7dBi antenna gain and a 30^∘ half-power beamwidth (HPBW). For the Rx side, to obtain omnidirectional channel observations, direction-scan sounding (DSS) scheme is utilized. Particularly, equipped with a directional horn antenna with a 25dBi antenna gain and a 8^∘ HPBW, the Rx scans the spatial domain with 10^∘ angle steps, from 0^∘ to 360^∘ in the azimuth plane and -20^∘ to 20^∘ in the elevation plane.
The measurement scenario is an indoor corridor on the second floor of the Longbin Building in Shanghai Jiao Tong University. Particularly, the transmitter is placed in the middle of the corridor near room a and remains static, while 9 Rx positions in NLoS areas in room d are selected, as shown in Fig, <ref>. To test the effectiveness of NIRS, a homemade NIRS with a size of 1.2m×1.2m is glued near the turning corner. As shown in Fig. <ref>, the NIRS is a foam board overlaid by aluminium foils that are manually pasted, resulting in a rough and irregular metal surface.
§.§ Power Enhancement By Including NIRS
To observe the performance of NIRS, we compare the measured path loss with/without NIRS. Since only limited Rx positions are measured due to high time consumption of channel measurements, to analyze the coverage situations in the whole area, the path loss in positions between adjacent Rx locations are obtained through linear interpolation. As a result, the power enhancement is calculated as the difference of path loss before/after adding the NIRS, where the results are shown in Fig. <ref> and several observations are made as follows.
First, for both 306321GHz and 356371GHz bands, received power is enhanced in most areas. Specifically, 63.6% area at 306321GHz and 51.6% area at 356371GHz obtains power enhancement of more than 3dB. Moreover, the maximum power enhancement is 12.56dB and 9.56dB at 306321GHz and 356371GHz, respectively. This proves the effectiveness of the NIRS. Second, the power enhancement by adding the NIRS is not uniform in the NLoS areas. At certain Rx positions, such as the top-middle receiver location, the path loss is decreased by nearly 10dB, while in other Rx positions, especially those receiver positions that are far away from the NIRS, the received power barely changes. Third, comparing the two frequency bands, the power enhancement shows different patterns. Therefore, it might be hard to control the NIRS to enhance the received power at certain Rx locations.
§.§ Channel Capacity With/Without NIRS
To clearly show the effectiveness of NIRS, SNR is calculated based on the measured path loss results, assuming a realistic THz communication link with reference parameters in <cit.>. Key parameters include a bandwidth of 15GHz, transmitter power of 13dBm, Tx and Rx antenna gains of 25dB. Furthermore, based on the SNR results, the channel capacity can be evaluated, as shown in Fig. <ref>. The results show that by adding the NIRS, the channel capacity in NLoS areas are greatly increased, especially in the 306321GHz band. Specifically, the average channel capacity increases from 5.42Gbps to 13.55Gbps at 306321GHz, and from 3.46Gbps to 7.97Gbps at 356371GHz, respectively. Moreover, with NIRS, in the best ten percent areas, the channel capacity exceeds 27.08Gbps and 15.85Gbps at 306321GHz and 356371GHz, respectively, while the values are only 8.96Gbps and 4.73Gbps at these two frequency bands without NIRS, respectively. Therefore, by including the NIRS, the channel capacity doubles or even triples than without using NIRS, proving its effectiveness.
§ OPEN PROBLEMS AND FUTURE DIRECTIONS
To effectively make use of NIRS, several open problems need to be addressed, including the channel modeling of the NIRS-aided communications, reliable design for site-specific coverage extension, optimal deployment and coordination of multiple NIRS, and possible joint communication and sensing enhancement.
§.§ NIRS Channel Modeling
Unlike IRS and reflectors, which involve either diffusely scattering or specular reflection, the scattering phenomenon on rough NIRS depends on multiple factors, including surface roughness, material, surface size, etc. To characterize NIRS channels and further evaluate the link performance of NIRS-aided communications, an accurate yet efficient channel model is necessary. Since the problem of wave scattering from rough surfaces has no closed-form solutions, existing studies usually use approximate solutions. For instance, the Kirchhoff scattering theory might be used to calculate the scattering efficient, which further depends on rough surface height standard deviation <cit.>. However, since the fabrication and design of NIRS is casual, such quantitative characteristics might not be available. Therefore, other models, such as statistical models or empirical fitting results based on real measurements, may be preferred in practice.
Another key problem related to NIRS channel modeling is the assessment of multipath richness and near-field effect resulted by NIRS. As mentioned above in Sec.II-D, NIRS can be easily embedded into UM-MIMO systems. The spatial multiplexing gain and channel capacity of UM-MIMO links highly rely on the number of significant paths in the communication channels. Therefore, by involving NIRS, the surrounding environment become more sensitive to THz waves, for which the originally weak high-order reflection/scattering paths can become more significant. Meanwhile, far-field propagation might convert to near-field, or cross near-and-far-field after experiencing NIRS scattering. Thus, the spatial multiplexing gain and channel capacity may increase. Extensive channel measurements are needed to analyze the multipath channel and analyze channel capacity of NIRS-aided THz communications.
§.§ Reliable NIRS Design for Site-Specific Coverage Extension
With the low fabrication cost, NIRS can enhance the THz coverage in the NLoS areas, as discussed in Sec. III. However, casual design of NIRS brings drawbacks such as random scattering pattern. Without careful design, it is hard to control which NLoS area to be enhanced, except for the specular directions that always receives better coverage due to smaller reflection loss. However, usage of NIRS in reality may be very site-specific, i.e., one usually has a target area or direction whose coverage needs to improved. In such cases, the random scattering pattern of NIRS prevents its effective usage.
There are several possible research directions to address this issue. First, accurate modeling of the scattering pattern could enable reliable designs for practical use, which however, might be difficult due to the reasons aforementioned. Second, one way to obtain desired scattering pattern is to control the roughness of NIRS. For this purpose, special structure might be explored, such as placing polished metal cubes of different heights in a grid pattern. This scheme is potential, yet causing design difficulty. Third, a possible cost-effective and simple solution is to add more NIRS in the NLoS area, similar to the usage of double IRS for improved performance <cit.>. However, this may incur interference problems, for which the joint deployment and coordination of multiple NIRS need to be considered.
§.§ NIRS Deployment and Coordination Optimization
Since NIRS is unable to change the beam steering direction after placement, where to deploy the NIRS is a key question to investigate. Generally speaking, since the specular reflection produces the strongest reflection, it is beneficial to place the NIRS in the specular reflection points between transmitter and receiver. Nonetheless, since NIRS can also enhance high-order reflections and scattering, practical deployment is rather more complicated. Moreover, considering the mobility of users, it is usually preferred to obtain a good coverage enhancement in most of the NLoS area, rather than great improvement at several locations while neglecting others. Therefore, the NIRS deployment optimization is a question that needs to be answered.
Furthermore, as the NIRS scattering pattern is hard to control, it is intuitive to place more NIRS to fully cover the NLoS areas. Ideally speaking, by placing multiple NIRS in the appropriate positions, the coverage ability of THz communications can be greatly extended. The first NIRS closest to Tx can cover part of the NLoS area with first-order reflection/scattering, while the second and later NIRS can further extend the coverage in deep NLoS areas, i.e., those areas that are far from the LoS region and barely receive enough signal strength. To achieve this, coordination of multiple NIRS is a key problem to be solved. With more NIRS, the dimension of the optimization problem grows, for which an computationally efficient and effective method to find the global maximum is needed.
§.§ Joint Communication and Sensing Enhancement with NIRS
In 6G and beyond wireless systems, it is expected that high-level integration of sensing and communication (ISAC) will play an important role. This is even enticing in the THz band, which promises unprecedented millimeter-level sensing accuracy. Even though NIRS is proposed to extend the coverage ability of THz communications, it is also potential to improve the sensing ability. By including NIRS in surrounding environment, the reflection and scattering loss are reduced. This leads that the back-scattered echo signal for sensing amplifies and therefore, the sensing SNR is increased and higher sensing accuracy can be achieved. However, it is also possible that if being placed in inappropriate positions, scattering from NIRS can cause stronger interference to the sensing echo signals. Experiments are needed to verify whether gain or loss NIRS may bring to THz sensing systems. Furthermore, due to the different metrics of communication and sensing systems, effective algorithms and methods are needed to joint optimize communication and sensing performance, or putting forward a good balance between them.
§ CONCLUSION
In this article, we provided an overview of the non-intelligent reflection surface (NIRS), which is a rough surface simply made of metal materials. The advantages and disadvantages of NIRS compared to IRS are presented. Still waters run deep - with almost nil-cost and extremely low fabrication difficulty, NIRS can effectively solve the LoS blockage problem, as well as enhance coverage, channel capacity and even sensing capabilities. Experimental results show that by using the NIRS, the channel capacity in the NLoS area could double on average. To shed light on studying THz NIRS, open problems and future directions are elaborated, including the NIRS channel modeling, reliable design of site-specific usage, deployment and coordination optimization, and joint communication and sensing enhancement.
IEEEtran
|
http://arxiv.org/abs/2307.05670v1 | 20230711180001 | Mid-Infrared Outbursts in Nearby Galaxies: Nuclear Obscuration and Connections to Hidden Tidal Disruption Events and Changing-Look Active Galactic Nuclei | [
"Sierra A. Dodd",
"Arya Nukala",
"Izzy Connor",
"Katie Auchettl",
"K. D. French",
"Jamie A. P. Law-Smith",
"Enrico Ramirez-Ruiz"
] | astro-ph.GA | [
"astro-ph.GA"
] |
0000-0002-3696-8035]Sierra A. Dodd
Department of Astronomy and Astrophysics,
University of California,
Santa Cruz, CA, 95064, USA
Castilleja School, Palo Alto, CA, 94301, USA
Department of Astronomy and Astrophysics,
University of California,
Santa Cruz, CA, 95064, USA
Department of Astronomy and Astrophysics, University of California,
Santa Cruz, CA, 95064, USA
0000-0002-4449-9152]Katie Auchettl
School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia
ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA, 95064, USA
0000-0002-4235-7337]K.D. French
Department of Astronomy, University of Illinois, 1002 W. Green St., Urbana, IL, 61801, USA
0000-0001-8825-4790]Jamie A.P. Law-Smith
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, 60637, USA
Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, 02138, USA
0000-0003-2558-3102]Enrico Ramirez-Ruiz
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA, 95064, USA
We study the properties of galaxies hosting mid-infrared outbursts in the context of a catalog of five hundred thousand galaxies from the Sloan Digital Sky Survey. We find that nuclear obscuration, as inferred by the surrounding dust mass, does not correlate with host galaxy type, stellar properties (e.g. total mass and mean age), or with the extinction of the host galaxy as estimated by the Balmer decrement. This implies that nuclear obscuration cannot explain any over-representation of tidal disruption events in particular host galaxies. We identify a region in the galaxy catalog parameter space that contains all unobscured tidal disruption events but only harbors ≲ 9% of the mid-infrared outburst hosts. We find that mid-infrared outburst hosts appear more centrally concentrated and have higher galaxy Sérsic indices than galaxies hosting active galactic nuclei (AGN) selected using the BPT classification. We thus conclude that the majority of mid-infrared outbursts are not hidden tidal disruption events but are instead consistent with being obscured AGN that are highly variable, such as changing-look AGN.
§ INTRODUCTION
The luminosity of supermassive black holes residing in the nucleus of most if not all galaxies is directly related to the rate at which they are supplied with matter. The fraction of supermassive black holes (SMBHs) that are in a highly luminous state has been observed to peak a few billion years after the big bang <cit.>
and to gradually decline to the present day, where only one black hole out of every one hundred radiates close to its maximum allowed luminosity <cit.>.
However, most SMBHs in the local universe still show some degree of activity <cit.>, ranging from outbursts with modest luminosities to highly luminous flares.
Black hole activity in the local universe has previously been associated with gas-rich mergers <cit.>, which is thought to power SMBH growth in the early universe. However, AGN activity has been shown not to correlate with merger activity
<cit.>, and many SMBHs live in gas-poor environments that may be incapable of powering highly variable outbursts <cit.>.
This is one of the main reasons why tidal disruption events (where a star approaches a SMBH close enough to be torn apart by tidal forces) are commonly invoked to explain some highly variable SMBH activity in the local universe <cit.>. It has been recently argued that SMBHs in the local universe accrete depending on how recent their last episode of star formation occurred <cit.>, which also seems to have profound consequences for moderating the rate of tidal disruption events <cit.>.
At high redshift, black hole activity is primarily driven by the accretion of gas, which is plentiful in the early universe as compared to today <cit.>. Thus, there clearly exists a transition in how black holes are fed when the gas content of galaxies is drastically reduced <cit.>. Yet, the material surrounding accreting supermassive black holes is thought to be related to the active galactic nucleus with its host galaxy <cit.>. For this reason, to probe the AGN–host galaxy connection directly in the local Universe, one needs to understand the structure and kinematics of the parsec-scale dust and gas that surrounds accreting SMBHs <cit.>.
During the last decade, mid-infrared interferometry has represented a major step forward in the characterization of nuclear dust in nearby AGNs <cit.>.
Our current understanding of the close environment of accreting supermassive black holes obtained from infrared and X-ray studies of local active galactic nuclei suggests that the structure of the surrounding gas is complex, clumpy, and highly variable <cit.>.
Although scarcely explored, mid-infrared outbursts hold the potential of revealing some of the most dramatic obscured AGN activity, such as the disruptions of stars by SMBHs. What is more, they can be used to directly probe the parsec-scale dust and gas content in nearby AGN and explore the AGN–host galaxy connection for highly variable accretion episodes in nearby galaxies. In this Letter, we study the properties of galaxies hosting mid-infrared outbursts in the local Universe, presented in <ref>, in the context of a catalog of five hundred thousand galaxies from the Sloan Digital Sky Survey, described in <ref>, with the goal of constraining their origin.
§ METHODS
§.§ Reference Catalog
We use the galaxy catalog from <cit.> and <cit.>. It consists of ≈ 5 × 10^5 galaxies from the Sloan Digital Sky Survey Data Release 7 MPA-JHU catalog[<http://wwwmpa.mpa-garching.mpg.de/SDSS/DR7>] <cit.> with additional derived properties from <cit.> and <cit.>, yielding a wide range of host galaxy properties. These include velocity dispersion, emission line fluxes, Lick Hδ_A, and star formation rate (SFR) from the MPA-JHU catalog; redshift, bulge g-r, bulge and galaxy magnitudes, galaxy half-light radius, galaxy Sérsic index, bulge to total light fraction, galaxy asymmetry indicator and galaxy inclination from <cit.>; and bulge and total stellar masses from <cit.>. Line fluxes in the catalog are calculated using the methodology described in Section 2.1 of <cit.> and are corrected for stellar absorption features. We derive SMBH masses using the M_ bh–σ_e scaling relation from <cit.>. We refer the reader to <cit.> for additional discussion.
A sample of AGN is selected from the host galaxy catalog using the BPT classification <cit.>, which considers the relative strength of OIII (5007Å), Hβ, NII (6584Å), and Hα emission lines to infer AGN activity. We require the signal-to-noise for each of these lines to be ≥ 3, yielding a final sample of ≈ 5 × 10^4 AGN.
§.§ MIRONG
<cit.> conducted a systematic search of low-redshift (z<0.35) galaxies that sustained mid-infrared outbursts based on Wide-field Infrared Survey Explorer (WISE) light curves, yielding a sample of 137 mid-infrared outbursts in nearby galaxies (MIRONG). 103 of the 137 MIRONG from <cit.> are contained in our reference catalog and constitute our MIRONG sample. They are listed in Table <ref>. Because the <cit.> MIRONG sample is constructed from galaxies in the SDSS spectroscopic catalog with z < 0.35, we recover a high fraction of matches with the original sample.
Of our 103 MIRONG sample, 3 were previously reported turn-on changing-look AGN (CL AGN): J0915+4814, J1133+6701, and J1115+0544 <cit.>. We also recover 4 unclassified optical transients (J0045-0047, J0841+0526, J1533+2729, J1647+3843) and 1 spectroscopically-confirmed supernova (J1540+0054/ ASASSN-16eh).
<cit.> obtain multi-epoch follow-up spectra of 54 of the 137 originally reported MIRONG, of which 43 are in our host galaxy sample. They propose tentative classifications of either turn-on CL AGN, AGN flare, or TDE for the 22 MIRONG exhibiting emission line variability, 16 of which are in our sample. Based on their analysis, 9 of the 16 in our sample are tentatively classified as TDEs, 5 as non-specified AGN flares, and 2 as turn-on CL AGN.
Studying the host galaxy properties of MIRONG provides us with an alternative classification scheme to constrain their possible identities, for those with and without spectroscopic follow-up. This can be effectively done if nuclear obscuration is fairly independent of galaxy type. We turn our attention to this critical issue in Section <ref>. In summary, we set out to analyze the host galaxy properties of 103 MIRONG, 43 of which have additional follow-up spectroscopy as described in <cit.>, with the goal of further understanding their hidden origin.
§ HOST GALAXY PROPERTIES
We begin by considering the star formation rate (SFR) and total stellar mass, M_∗, of the host galaxies of MIRONG in relation to our galaxy catalog (see the top panels of Figure <ref>). We also include in our comparison a sample of 8 TDEs from <cit.>, 15 CL AGN from <cit.>, as well as the 4 TDEs from <cit.> that are contained in our reference catalog. We define the star-forming main sequence (SFMS) of galaxies in this plane to fall along the solid blue line <cit.>. Dashed lines spaced by 1σ (the median scatter in the SFR measurements) are added to indicate degrees of quiescence. Contours of galaxies and AGN plotted in Figure <ref> and throughout this analysis are spaced by 0.5σ (11.8%; note that percentages associated with σ in 2D histograms differ from 1D).
Two distinct groupings of galaxies are observed in the SFR - M_∗ plane for galaxies (upper left panel of Figure <ref>). The first, located in the top left and along the SFMS, consists of spiral, late-type galaxies. The SFMS provides a critical tool for studying the quenching of star formation <cit.> and the possible emergence of quiescent galaxies (as indicated by the second grouping in the lower right). Although the exact process by which this transformation takes place is debated, galaxies falling between late- and early-type galaxies are understood to be in a state of transition <cit.>, and are often referred to as green valley galaxies. Following <cit.> and <cit.>, we designate the green valley region as falling between the lower blue dashed line and the orange dashed line in the SFR - M_∗ plane.
AGN are generally seen to activate preferentially in SF galaxies <cit.>, and persist through (and possibly drive) the eventual quenching phase (see the upper right panel of Figure <ref>). As expected, the distribution of MIRONG appears to be more closely aligned in the SFR - M_∗ plane with AGN than galaxies. The MIRONG hosts also do not exhibit any large degrees of clumping or grouping like that seen in CL AGN hosts (although the larger sample size is certainly relevant), but the populations do appear to slightly overlap. What is more, the general MIRONG population does not closely follow the region occupied by TDE hosts. We investigate this further in subsequent sections.
As clearly seen in Figure <ref>, TDEs are observed to preferentially take place in post-starburst host galaxies, also known as E+A galaxies <cit.>. These unique hosts can be identified through the Hα equivalent width vs. Lick Hδ_A absorption plane. Hα equivalent widths are associated with current star formation, while Lick Hδ_A absorption results primarily from A-type stars. E+A galaxies lie in the bottom right of this plane, as seen in the lower panels of Figure <ref>, where low values of Hα equivalent width indicate little to no ongoing star formation, and high values of Lick Hδ_A absorption indicate a starburst in the last ≈ 1 Gyr. As in <cit.> and <cit.>, we define E+A galaxies as residing in the rectangle created by the solid black lines in the lower panels of Figure <ref> and Balmer-strong quiescent galaxies as those falling in the larger rectangle created by the dashed black lines.
Unlike TDEs, of which 6/12 (50%) fall within the quiescent Balmer-strong region, only 2/103 (≈1.9%) of MIRONG occupy the same region. As in the SFR - M_∗ plane, MIRONG hosts seem to broadly follow the distribution of AGN. <cit.> demonstrated that the overrepresentation of TDEs in post-starburst galaxies persists even after controlling for possible selection effects such as black hole mass, redshift, presence of a strong AGN, bulge colors, and surface brightness of host galaxies. <cit.> and <cit.> also found rate enhancements of TDEs in quiescent Balmer-strong galaxies, the rates of which were demonstrated by <cit.> to all be consistent with one another.
Whether the over-representation of TDEs in post-starburst galaxies is physically driven or the result of observational biases is the subject of ongoing study. <cit.> suggest that the intrinsic rate of TDEs in highly SF galaxies can dominate over that in post-starburst galaxies, but that high nuclear dust content in such galaxies could make detection extremely difficult. Are TDEs in SF galaxies preferentially obscured? It is to this question that we next turn our attention to by studying nuclear obscuration as a function of galaxy type and interstellar extinction.
§ NUCLEAR OBSCURATION, GALAXY TYPE AND GALACTIC-SCALE EXTINCTION
As demonstrated in the previous section, MIRONG hosts appear to more closely resemble AGN than typical TDE host galaxies. The derived dust properties in the nuclear region of MIRONG hosts <cit.> allow us to investigate the possibility that TDEs in high dust-content galaxies might be preferentially hidden. Dust mass is derived by assuming that thermal emission from dust is powering the MIR emission. The assumed dust grain size distribution comes from Mathis, Rumpl, and Nord <cit.> and is roughly power-law in nature. W1 and W2 fluxes are then fit to a modified blackbody to obtain dust temperature and corresponding masses under the assumption of a high dust covering fraction <cit.>. This assumption seems to be additionally supported by our findings that dust mass is not correlated with galaxy orientation in our galaxy host sample.
To investigate how nuclear dust mass varies with galaxy type, we revisit the SFR - M_∗ plane, now with each MIRONG shaded according to their dust mass content (in units of log M_⊙; see the left and middle panels of Figure <ref>). Contours of AGN are shown in the background. No obvious trend is visible between the location on the evolutionary sequence of galaxies and nuclear dust mass. As discussed in the previous section, it has been suggested that TDEs could occur at elevated rates in highly SF galaxies but would be obscured by higher levels of nuclear dust
<cit.>. Since TDEs are preferentially observed in green valley host galaxies, we compare the nuclear dust content of SFMS and green valley MIRONG galaxies in the middle panel of Figure <ref> to see whether nuclear dust levels are elevated in highly SF galaxies. The two distributions are similar, as also seen by their median values (solid black lines) and ± 1σ (dashed black lines). We perform a Kolmogorov-Smirnov test on the two samples. The resultant p-value of 0.37 implies that we are unable to reject the null hypothesis that the two populations are drawn from the same underlying distribution, suggesting they are consistent with one another. This is in stark contrast to predictions that advocate for an elevated rate of obscured TDEs in SF galaxies. We also find no trend between D_n(4000), an indicator of the age of the galaxy stellar population <cit.>, and nuclear dust mass. We can thus conclude that nuclear obscuration does not correlate with stellar properties such as stellar mass and age. Combined with the finding that nuclear dust does not depend on SFR, we can robustly conclude that obscured TDEs would not prefer one type of galaxy over another. This implies we are not missing a population of TDEs in more highly-SF galaxies, and that their observed green valley preference might be based on underlying physical mechanisms controlling the rate of TDEs <cit.>.
Having shown that nuclear obscuration does not depend on galaxy type, here we consider how the Balmer decrement, which is thought to correlate with the dust mass content in a galaxy, correlates with the abundance of dust surrounding SMBHs. Balmer decrement, which consists of the ratio between Hα and Hβ emission lines from the Balmer series (n=2), is commonly used as a measure of galaxy interstellar extinction. However, the presence of an AGN can alter the region probed by this measurement. To this end, the Balmer decrement has been used to measure dust extinction in the broad line regions of type I AGN and quasars <cit.>, as well as in the narrow line region of AGN <cit.>. For partially obscured AGN, such as the ones in our MIRONG sample, we can expect the Balmer decrement to reflect extinction on both nuclear and galactic scales. Because most of the MIRONG in our sample (76; 52 of which have strong (≥ 3) signal-to-noise ratio in the BPT line measurements) are classified as AGN in the BPT diagram, we expect a large degree of scatter in their Balmer decrements.
The right-most panel of Figure <ref> shows the Balmer decrement of all MIRONG vs. nuclear log dust mass, separated into galaxy hosts with and without AGN activity. As anticipated, we see a large degree of scatter in the AGN group. The non-AGN show considerably less scatter. Neither group shows a strong correlation between Balmer decrement and nuclear dust mass[This statement is supported by correlation coefficients ≲ 0.5 for each population in both Pearson and Spearman tests (Correlation coefficient values: AGN MIRONG Pearson = 0.39, AGN MIRONG Spearman = 0.41, non-AGN MIRONG Pearson = 0.53, non-AGN MIRONG Spearman = 0.37). We note that the highest correlation coefficient of 0.53 for non-AGN MIRONG could be consistent with a weak correlation with significant scatter.]. As the Balmer decrements of the non-AGN group are more likely to be indicative of galactic-scale dust, this suggests no clear correlation between nuclear- and galactic-scale dust in these systems. This is in agreement with works such as <cit.>, which find that the degree of nuclear obscuration appears uncorrelated with larger-scale galactic properties including SFR and total stellar mass. This provides an additional clue that dust content of galaxies in the local Universe at the host-scale do not determine their innermost dust content.
§ ON THE ORIGIN OF MIRONG SOURCES
Finding any correlation between nuclear- and galactic-scale dust content is key to understand any potential host galaxy preference for the MIRONG population. With this information, we can examine the notion that MIRONG in highly SF regions (where galactic dust is expected to be abundant) could plausibly be obscured TDEs that until now we had no means of uncovering.
Having proven that nuclear obscuration does not depend on host galaxy type, it is clear that the hidden population should mirror the unobscured population. This enables us to use host galaxy properties of unobscured nuclear transients to identify possible MIRONG origins. In this Section, we estimate the fraction of MIRONG that might be TDEs, as well as explore possible alternative explanations for the origin of the bulk of the MIRONG population.
§.§ Host Galaxy Matching in the SFR - M_∗ Plane
We identify MIRONG hosts located within a range of SFR and total stellar mass of TDE hosts using the matching methodology of <cit.> and <cit.> but with a 15% tolerance window. The top left panel of Figure <ref> shows the results of this matching and Table <ref> summarizes our findings. We locate 9 MIRONG with matching properties to TDE hosts, corresponding to 8.7% of MIRONG in our sample. As such, the overall MIRONG population does not appear to mirror the TDE host population. For comparison, AGN and galaxies matched to TDE hosts in this plane represent 13.6% and 8.8%, respectively. We note that one of the MIRONG identified through this analysis, VT 1548 <cit.>, is a previously-observed nuclear radio flare. This object is a promising candidate for further study and has been interpreted as either a TDE or an extreme flare of an AGN, in both cases obscured by a dusty torus.
We perform the same matching for CL AGN hosts and find 41 MIRONG with similar host properties in the SFR - M_∗ plane, corresponding to 39.8% of the total MIRONG population[Although the CL AGN sample consists of 15 objects with SFR and M_∗ measurements, in comparison to 12 TDEs, performing the matching analysis while excluding 3 CL AGN at random does not reduce the number of matches significantly to yield anything similar to the 9% found for TDEs.]. This includes the 3 CL AGN contained in the MIRONG sample. These results can be seen in the right panel of Figure <ref> and in Table <ref>. We note that the matched CL AGN and TDE samples have 4 events in common. This is not surprising given that both populations have a strong preference for green-valley galaxies <cit.>. These 4 events are shown in purple in both panels of Figure <ref>. Of the 41 MIRONG with similar host properties to CL AGN, 21 are classified as AGN in the catalog (44.2%; see Section <ref> for details on AGN classification). For comparison, the relative percentages of AGN and galaxies near CL AGN hosts are 34.0% and 28.1%, respectively. This implies that the MIRONG population more closely resembles the CL AGN host population than a typical AGN. This is
as one might expect, given that MIRONG are selected such that they need to be associated with a luminous and highly variable nuclear source.
As anticipated, most MIRONG hosts broadly resemble the general AGN population in the SFR and M_∗ plane. We confirm this using a 2D Kolmogorov–Smirnov test and are thus unable to reject the null hypothesis that the two samples were drawn from the same distribution (p value: 0.11). This is however not the case when compared to the galaxy population (p value: 6e-8). Despite their apparent similarity to typical AGN, the identification of MIRONG via their extreme outbursts points to a more dramatic type of variability than what is expected for typical AGN in the local Universe. CL AGN represent the extreme end of these highly-variable AGN. A useful phenomenological distinction between AGN and highly variable (HV) AGN is galaxy Sérsic index <cit.>. In the following section we thus analyze MIRONG in this context.
§.§ Sérsic Index Comparison
Sérsic index measures the light concentration of the galaxy surface brightness. Higher values correspond to highly centrally concentrated light profiles, such as those often exhibited by elliptical galaxies, while lower values are more consistent with diffuse profiles, such as those seen in spirals. Sérsic index is generally thought to provide a measurement of the density profile of stars and, to a lesser extent, the kinematic state of the star-forming gas in the nuclear region <cit.>. As mentioned previously, highly variable AGN and CL AGN have been shown to have higher Sérsic indices than AGN <cit.>, even when controlling for black hole mass. TDE hosts also exhibit this trend to a slightly lesser degree <cit.>.
Figure <ref> shows Sérsic index for AGN, TDEs, MIRONG, and CL and HV AGN. The four HV AGN used in our sample are the same from <cit.> and consist of KUG 1624+351 <cit.>, J094608+351222
<cit.>, 2MASS J09392289+3709438 <cit.>, and Swift J1200.8+0650
<cit.>. Intriguingly, the last object listed (Swift J1200.8+0650) is also a MIRONG source, even though it was selected independently. The MIRONG distribution differs markedly from that of AGN. MIRONG Sérsic indices tend to fall between those of TDEs and CL and HV AGN. We also performed this analysis with a sample of AGN matched to MIRONG based on SFR, total stellar mass, and redshift, and found no noticeable difference in the resulting distribution compared to typical AGN. Comparing the distribution of MIRONG Sérsic indices to these other populations suggests that HV AGN are the likeliest source of MIRONG behavior, as opposed to standard AGN-type flares.
§.§ The rate of TDEs in the MIRONG population
We now examine if our derived rate of TDEs in MIRONG based on host galaxy matching of ≲ 10% is consistent with the classification of MIRONG from <cit.>. The authors spectroscopically monitored 53 of the 103 original MIRONG sources over a roughly 4-year period. Of these 53, 22 (41.5%) displayed variability in the broad Hα emission line (EL). <cit.> perform their subsequent classification on this subsample of 22 objects, based on the notion that any light from a transient event located at or near the central, obscured SMBH would have been reprocessed by the dust over this monitoring timescale.
TDEs are selected from this group of 22 EL-variable sources as follows. Sources are split into likely AGN and quiescent groups. Quiescent sources are classified as TDEs with the exception of one turn-on AGN candidate. Any AGN MIRONG with iron coronal lines and He IIλ4686 features are also tentatively classified as TDEs given the association of those features with TDE spectra <cit.>. This results in 14 TDE candidates (≈ 26%).
The authors highlight the challenges associated with classifying obscured nuclear transients. This includes the possibility of a myriad of origins for AGN flares, for which we have very few expected spectral templates for comparison. As such, we hope to offer a complementary view to the work of <cit.> based on host galaxy properties of unobscured TDEs. We have 6 of the 14 <cit.> TDE candidates in our galaxy catalog. Only 1 is TDE-like in the SFR and M_∗ plane, corresponding to ≈ 16%. The Sérsic index of this object is on the lower end of the general MIRONG distribution, at a value of 1.7. We also find that another 1 of these 6 tentative <cit.> TDEs is similar to CL AGN in the SFR - M_∗ plane, this time with a much higher Sérsic index of 7.9. The relatively low fraction of <cit.> tentative TDEs with similar host properties to unobscured TDEs (≈ 16%) suggests that, as the authors posit, there is likely other variability at play that can drive MIRONG flares besides TDEs, including HV AGN-type flares.
Finally, an additional estimate of the rate of TDEs in MIRONG can be made by again utilizing our finding from Section <ref> that the obscured hosts should mirror the unobscured population of TDE hosts. Briefly revisiting the bottom panel of Figure <ref>, we see that very few of the MIRONG occupy the post-starburst region favored by TDEs and shown by the dotted black lines. Using a simple analysis, if 6 out of 12 TDE hosts (50%) lie in this region, and 2 out of 103 MIRONG (1.9%) occupy this region as well, we can estimate the number of obscured TDEs responsible for MIRONG behavior to be somewhere around 4 (3.8%).
§ SUMMARY AND CONCLUSIONS
The arguments leading to the conclusion that the bulk of MIRONG are obscured HV AGN can be broadly summarized as follows:
* MIRONG host galaxies appear more similar to AGN host galaxies in the SFR - M_∗ plane than galaxies or TDE hosts (Figure <ref>).
* Nuclear dust content does not appear to correlate with host galaxy type, stellar properties, or galaxy orientation. We also find no strong correlation between nuclear dust content with galactic-scale dust content as estimated by the Balmer decrement in MIRONG host galaxies showing no AGN activity. This suggests that the obscured host galaxy transient population should trace the unobscured one and that we are not missing a population of TDEs in highly SF galaxies (Figure <ref>). This also implies that the observed post-starburst preference of TDEs is intrinsic, as opposed to an observational bias, raising many questions about the uniqueness of these systems. What is more, we conclude that the intrinsic rate of TDEs in star-forming galaxies is significantly lower than in post-starburst hosts.
* Based on a comparison with the unobscured population, we estimate the relative fraction of TDEs responsible for MIRONG to be ≲ 9% (Figure <ref>).
* We conclude that the majority of MIRONG appear to be driven by HV AGN activity, as suggested by the higher Sérsic index distribution when compared to typical AGN (Figure <ref>).
Having shown that TDEs are likely a smaller fraction of MIRONG than originally anticipated and that HV AGN might represent a higher fraction as shown by host galaxy and Sérsic index analyses, we lastly consider whether or not the observed rates of these transient populations are consistent with one another. Figure <ref> shows an adapted version of the luminosity function (LF) from <cit.>, with the additional inclusion of a population of soft X-ray AGN from <cit.>. The two lines represent fits to the MIRONG population for different assumed dust covering factors: the red line for a factor of 1, the fainter pink line for a factor of 0.3 <cit.>. As illustrated in <cit.>, MIRONG appear similar in rate to optical and X-ray TDE fractions, especially given the uncertainty in the obscuration factor. Soft X-ray AGN exhibit higher values of Φ (i.e. have a higher occurrence rate in the local universe) than MIRONG across luminosity bins. Given that HV AGN are by nature a smaller subset of the local AGN population, the LF of MIRONG would likely be consistent with that of HV AGN. This is consistent with our findings that HV AGN could be responsible for most MIRONG.
§.§ Relationship between nuclear obscuration and host galaxy properties
The MIRONG population offers us an exciting opportunity to study the observational appearance of obscured AGN and TDEs, which is not only determined by their intrinsic emission properties but also by the state of the intervening material along the line of sight. We find that nuclear dust obscuration does not correlate with host galaxy type or with the extinction
of the host galaxy. This is in agreement with <cit.>, which finds no significant difference between the mean stellar masses and star formation rates of obscured and unobscured AGN hosts selected by X-ray flux. In essence, it is commonly understood that the physical state of intervening material is largely insensitive to the wider scale galactic conditions but appears to be mainly determined by radiation properties of the nuclear region <cit.>. For highly luminous AGN, the dust content inferred from the column of material responsible for the X-ray absorption is commonly larger than the one inferred from the reprocessing luminosity. This is consistent with the idea that X-ray absorbing gas is located within the dust sublimation radius, whereas the mid-IR flux arises from an area farther out <cit.>. Given that the MIRONG population, as argued in this Letter, is likely to be associated with highly luminous nuclear activity, we expect the X-ray absorbing region to be located within the dust sublimation radius. Swift J1200.8+0650 provides us with the best example to test this hypothesis, given that it was identified independently by the MIRONG and the high Galactic latitude Swift survey. The X-ray absorbing gas measurement in Swift J1200.8+0650 <cit.> implies a much higher dust mass than the one derived from the mid-IR luminosity under the assumption of a high covering factor <cit.>. As expected, the dust mass inferences in the MIRONG population are thus likely to be highly sensitive to the physics of dust sublimation. Be that as it may, radiative feedback in Swift J1200.8+0650 and other MIRONG sources is still unable to completely clear out the circumnuclear dust environment. In sources for which removal of dusty gas might be ultimately efficient, the nuclear source may decline in luminosity, giving rise to unabsorbed sources at lower luminosities.
In closing, here we confirm the general view that there is little physical connection between the gas accreting onto the SMBH and the material out of which stars form throughout the galaxy by demonstrating the lack of a connection between obscuring dust within the sphere of influence of the SMBH and the galaxy-wide properties (stellar age, stellar mass, SFR and dust content). This is consistent, for example, with the findings derived from large comprehensive studies using Herschel <cit.> and XMM–Newton in the COSMOS field <cit.>.
We would like to express gratitude for insightful discussions with Julianne Dalcanton, Jenny Greene, Ryan Foley, and Morgan MacLeod. S.A.D. and E.R.R. thank the Heising-Simons Foundation, NSF (Graduate Research Fellowship, AST-1615881 and AST-2206243), Swift (80NSSC21K1409, 80NSSC19K1391) and Chandra (22-0142) for support. K.D.F. acknowledges support from NSF grant AST–2206164.
cccccccc
MIRONG Sample
Name R.A. Dec z SFR log M_∗/M_⊙ log M_ dust Sérsic Index
SDSS J110501.98+594103.5*a 166.2583 59.6843 0.0337 0.6938 10.316 -1.58 5.15 ± 0.04
SDSS J163246.84+441618.5*a 248.1952 44.2718 0.0579 -1.3619 10.525 -1.88 4.28 ± 0.17
SDSS J165726.81+234528.1*a 254.3617 23.7578 0.0591 -0.3956 9.971 -0.53 1.74 ± 0.11
SDSS J104306.56+271602.1*a 160.7774 27.2673 0.1281 0.1142 10.654 -1.25 7.96 ± 0.15
SDSS J140221.26+392212.3*a 210.5886 39.3701 0.0638 0.3251 10.547 -1.29 1.23 ± 0.03
SDSS J151345.76+311125.0*a 228.4407 31.1903 0.0718 0.5689 10.829 -1.55 2.50 ± 0.03
SDSS J164754.38+384342.0*a 251.9766 38.7283 0.0855 0.2108 10.214 -1.82 1.85 ± 0.13
SDSS J154955.19+332752.0*a 237.4800 33.4644 0.0856 0.1408 10.017 -1.36 1.27 ± 0.06
SDSS J111536.57+054449.7*a 168.9024 5.7471 0.0900 -0.8071 10.610 -0.92 4.12 ± 0.22
SDSS J131509.34+072737.6*b 198.7889 7.4605 0.0918 0.7665 11.136 -1.34 5.09 ± 0.10
SDSS J153711.29+581420.2*b 234.2971 58.2389 0.0936 0.3130 10.583 -1.08 3.41 ± 0.19
SDSS J113355.93+670107.0*b 173.4831 67.0186 0.0397 -0.4329 10.822 -1.66 3.76 ± 0.02
SDSS J133212.62+203637.9*b 203.0526 20.6105 0.1125 1.0122 10.753 -0.78 5.71 ± 0.18
SDSS J123852.87+081512.0*b 189.7203 8.2533 0.1138 0.4280 10.694 -1.11 3.31 ± 0.14
SDSS J100350.97+020227.6*c 150.9624 2.0410 0.1247 -0.0666 10.799 -1.50 3.91 ± 0.23
SDSS J144227.57+555846.3*c 220.6149 55.9795 0.0769 0.8001 10.939 -0.82 4.11 ± 0.02
SDSS J124521.42 -014735.4* 191.3393 -1.7932 0.2154 0.5375 11.050 -0.58 7.81 ± 0.29
SDSS J110958.34+370809.6* 167.4931 37.1360 0.0260 -1.1838 10.594 -1.72 4.95 ± 0.01
SDSS J155743.52+272753.0* 239.4314 27.4647 0.0316 0.9564 10.558 -0.83 2.44 ± 0.01
SDSS J112916.12+513123.5* 172.3172 51.5232 0.0329 0.3398 10.778 -0.78 1.41 ± 0.01
SDSS J160052.26+461242.9* 240.2178 46.2119 0.1974 0.8476 11.191 -0.88 1.64 ± 0.15
SDSS J010320.42+140149.8* 15.8351 14.0305 0.0418 0.5445 10.929 -0.81 1.01 ± 0.01
SDSS J083536.49+493542.7* 128.9020 49.5952 0.0424 -0.3198 10.749 -1.30 2.73 ± 0.02
SDSS J165922.65+204947.4* 254.8444 20.8298 0.0451 0.0911 11.144 -1.46 2.97 ± 0.01
SDSS J120338.31+585911.8* 180.9097 58.9866 0.0469 -0.5879 9.730 -2.02 1.51 ± 0.13
SDSS J130815.57+042909.6* 197.0649 4.4860 0.0483 -1.2743 10.627 -1.85 4.78 ± 0.06
SDSS J105801.52+544437.0* 164.5064 54.7436 0.1306 0.6060 10.530 -0.86 2.53 ± 0.24
SDSS J113901.27+613408.5* 174.7553 61.5691 0.1346 1.0686 10.605 -0.84 0.95 ± 0.04
SDSS J162810.03+481047.7* 247.0419 48.1799 0.1245 0.7825 10.314 -1.25 1.17 ± 0.06
SDSS J132259.94+330121.9* 200.7498 33.0227 0.1269 0.7953 10.893 -1.10 1.03 ± 0.05
SDSS J112018.31+193345.8* 170.0763 19.5627 0.1278 1.1259 10.759 -1.17 4.35 ± 0.22
SDSS J155437.26+525526.4* 238.6553 52.9240 0.0664 -0.5625 10.502 -1.27 5.97 ± 0.11
SDSS J155539.95+212005.7* 238.9165 21.3349 0.0709 -0.7768 10.642 -0.99 3.26 ± 0.12
SDSS J132902.05+234108.4* 202.2585 23.6857 0.0717 -0.4885 10.565 -0.94 1.97 ± 0.05
SDSS J130532.91+395337.9* 196.3871 39.8939 0.0725 -1.2688 10.653 -1.59 3.83 ± 0.12
SDSS J112446.21+045525.4* 171.1926 4.9237 0.0740 0.9605 10.729 -0.83 2.08 ± 0.03
SDSS J094303.26+595809.3* 145.7636 59.9693 0.0749 -1.2882 10.356 -1.64 4.62 ± 0.28
SDSS J081403.78+261144.3* 123.5158 26.1956 0.0757 0.5499 10.917 -0.86 2.67 ± 0.07
SDSS J150844.22+260249.1* 227.1843 26.0470 0.0826 0.3946 10.404 -1.02 0.76 ± 0.03
SDSS J140648.43+062834.8* 211.7018 6.4763 0.0850 1.1048 10.770 -1.04 1.32 ± 0.03
SDSS J141235.89+411458.5* 213.1496 41.2496 0.1025 0.0925 11.012 -1.83 2.98 ± 0.12
SDSS J103753.68+391249.6* 159.4737 39.2138 0.1068 0.5869 10.446 -0.98 1.10 ± 0.06
SDSS J090924.55+192004.8* 137.3523 19.3347 0.1072 -0.2895 10.532 -0.82 1.78 ± 0.10
SDSS J075709.69+190842.8 119.2904 19.1452 0.1050 0.6167 10.767 -1.37 3.09 ± 0.12
SDSS J100120.37+182926.6 150.3349 18.4907 0.1060 0.0215 10.498 -1.56 1.45 ± 0.06
SDSS J101708.94+122412.0 154.2873 12.4034 0.1076 0.0808 10.567 -1.58 6.89 ± 0.33
SDSS J074547.87+265537.9 116.4495 26.9272 0.1148 0.9047 10.312 -0.78 7.98 ± 0.11
SDSS J144024.32+175852.7 220.1013 17.9813 0.1157 -0.1135 10.502 -1.73 5.68 ± 0.39
SDSS J151257.19+280937.5 228.2383 28.1604 0.1155 -0.8570 10.941 -1.10 5.40 ± 0.27
SDSS J100809.02+154951.3 152.0376 15.8309 0.1176 0.3173 10.982 -1.10 1.74 ± 0.06
SDSS J084752.78+514236.2 131.9699 51.7101 0.1200 0.8325 10.756 -1.11 1.34 ± 0.10
SDSS J151117.94+221428.2 227.8248 22.2412 0.1205 -0.8159 11.116 -0.98 4.29 ± 0.30
SDSS J012100.67+140517.3 20.2528 14.0881 0.1294 0.8598 10.930 -0.90 3.23 ± 0.18
SDSS J000046.46+143813.0 0.1936 14.6370 0.1366 0.5935 11.053 -1.07 3.38 ± 0.18
SDSS J081451.87+533732.5 123.7161 53.6257 0.1390 0.8186 10.857 -1.18 1.72 ± 0.10
SDSS J104138.79+341253.5 160.4117 34.2149 0.1403 0.7567 10.902 -1.51 5.12 ± 0.28
SDSS J100955.70+220949.3 152.4821 22.1637 0.1415 -0.5718 11.131 -1.19 4.67 ± 0.19
SDSS J115326.76+403719.1 178.3615 40.6220 0.1451 0.8811 10.406 -1.13 0.97 ± 0.04
SDSS J232452.26+154251.0 351.2178 15.7142 0.1511 0.6084 10.973 -1.58 4.85 ± 0.38
SDSS J105344.12+552405.7 163.4339 55.4016 0.1517 1.1808 10.711 -1.07 1.03 ± 0.05
SDSS J085959.47+092225.7 134.9978 9.3738 0.1519 1.1258 10.773 -0.74 2.80 ± 0.10
SDSS J111431.83+405613.8 168.6327 40.9372 0.1525 -0.7727 11.104 -1.06 4.68 ± 0.43
SDSS J100256.90+442457.8 150.7371 44.4161 0.1545 -0.5432 11.393 -0.31 3.12 ± 0.11
SDSS J084157.98+052605.8 130.4916 5.4349 0.1563 0.3459 11.221 -1.04 4.45 ± 0.24
SDSS J131022.77+251809.3 197.5949 25.3026 0.1604 1.1384 10.806 -1.20 4.02 ± 0.35
SDSS J135241.36+000925.8 208.1724 0.1572 0.1660 0.9115 11.165 -1.12 2.07 ± 0.09
SDSS J085434.65+111334.7 133.6444 11.2263 0.1672 0.8626 11.596 -0.87 7.93 ± 0.07
SDSS J134105.98 -004902.5 205.2749 -0.8174 0.1754 1.0146 11.074 -1.11 6.15 ± 0.33
SDSS J120145.97+352522.5 180.4416 35.4229 0.1903 0.5322 11.192 -0.68 5.32 ± 0.30
SDSS J112238.84+143348.4 170.6619 14.5634 0.1942 0.9404 11.252 -1.06 7.96 ± 0.12
SDSS J104609.61+165511.4 161.5401 16.9199 0.2069 0.8473 11.346 -0.81 8.00 ± 0.13
SDSS J102959.95+482937.9 157.4998 48.4939 0.2324 1.5759 10.655 -0.75 3.90 ± 0.39
SDSS J154029.29+005437.2 235.1221 0.9104 0.0117 -2.4216 8.322 -3.71 1.09 ± 0.02
SDSS J154843.06+220812.6 237.1794 22.1368 0.0313 -0.6606 9.945 -1.42 3.80 ± 0.11
SDSS J120057.93+064823.1 180.2414 6.8064 0.0360 -0.2476 11.058 -0.70 3.38 ± 0.02
SDSS J142808.89 -023124.8 217.0371 -2.5236 0.0521 -0.7258 9.923 -2.22 7.89 ± 0.38
SDSS J215648.45+004110.6 329.2019 0.6863 0.0539 -0.3556 10.549 -1.80 5.83 ± 0.20
SDSS J004500.47 -004723.1 11.2520 -0.7897 0.0568 -0.0044 9.426 -2.20 1.19 ± 0.08
SDSS J150440.39+010735.0 226.1683 1.1264 0.1283 0.6755 11.187 -0.64 3.79 ± 0.15
SDSS J095754.76+020711.2 149.4782 2.1198 0.1253 0.5030 10.736 -1.42 3.62 ± 0.23
SDSS J134123.20+151650.4 205.3467 15.2807 0.1255 1.0196 10.906 -1.03 1.99 ± 0.07
SDSS J120942.22+320258.8 182.4259 32.0497 0.0590 0.2593 10.784 -1.36 4.44 ± 0.05
SDSS J140950.27+105740.2 212.4595 10.9612 0.0597 0.3989 11.246 -0.83 1.82 ± 0.01
SDSS J114922.02+544151.4 177.3418 54.6976 0.0619 0.4912 10.566 -1.72 1.17 ± 0.02
SDSS J084232.87+235719.6 130.6370 23.9555 0.0635 0.4795 10.464 -2.08 4.99 ± 0.08
SDSS J105145.47+210132.1 162.9395 21.0256 0.0659 -0.6158 10.304 -2.02 4.54 ± 0.25
SDSS J144829.01+113732.1 222.1209 11.6256 0.0666 0.3213 10.711 -1.36 2.28 ± 0.04
SDSS J142420.78+624916.5 216.0866 62.8213 0.1091 0.4550 10.960 -1.56 3.58 ± 0.13
SDSS J081121.40+405451.8 122.8392 40.9144 0.0670 0.8102 10.217 -1.17 5.98 ± 0.20
SDSS J153310.02+272920.2 233.2918 27.4890 0.0719 -0.4336 10.972 -1.11 4.96 ± 0.05
SDSS J161258.17+141617.5 243.2424 14.2715 0.0720 -0.0619 10.510 -1.39 6.26 ± 0.28
SDSS J020552.15+000411.7 31.4673 0.0699 0.0765 0.8150 10.345 -0.94 1.42 ± 0.06
SDSS J143016.05+230344.4 217.5669 23.0623 0.0810 -0.2164 11.228 -1.33 3.88 ± 0.06
SDSS J121907.89+051645.7 184.7829 5.2794 0.0825 0.0719 10.226 -1.31 2.19 ± 0.11
SDSS J152438.13+531458.7 231.1589 53.2496 0.0851 0.1311 10.011 -2.02 1.36 ± 0.07
SDSS J214142.90 -085702.3 325.4288 -8.9507 0.0873 -0.3762 10.499 -1.56 2.65 ± 0.15
SDSS J085835.90+412113.8 134.6496 41.3539 0.0870 -1.0266 10.498 -1.06 3.81 ± 0.26
SDSS J093135.46+662652.2 142.8978 66.4478 0.0873 -1.0901 10.784 -1.90 6.06 ± 0.28
SDSS J124255.36+253727.9 190.7307 25.6244 0.0879 -1.3684 10.426 -1.57 4.51 ± 0.25
SDSS J134032.49+184218.6 205.1354 18.7052 0.0902 -0.5332 10.609 -1.77 5.79 ± 0.24
SDSS J132848.45+275227.8 202.2019 27.8744 0.0911 0.1618 10.828 -1.35 5.86 ± 0.21
SDSS J083721.86+414342.0 129.3411 41.7283 0.0981 0.6995 11.010 -1.10 3.51 ± 0.12
SDSS J091531.04+481407.7 138.8794 48.2355 0.1005 -0.7665 10.874 -1.54 6.61 ± 0.12
*Have multi-epoch spectroscopy from <cit.>.
aTentative classification from <cit.> as TDE.
bTentative classification from <cit.> as AGN flare.
cTentative classification from <cit.> as turn-on CL AGN.
c c c c
4
TDE and CL AGN host galaxy matching overview. Sérsic index errors are the median of individual Sérsic errors.
Category Fraction Percent Median Sérsic Index
Matching to TDEs
All MIRONG 9/103 8.7 3.80 ± 0.13
Decl. MIRONG 1/6 16.6 1.74 ± 0.11
AGN 7,160/52,613 13.6 2.29 ± 0.07
Galaxies 44,401/500,707 8.8 1.80 ± 0.06
Matching to CL AGN
All MIRONG 41/103 39.8 3.91 ± 0.15
Decl. MIRONG 1/6 16.6 7.96 ± 0.15
AGN MIRONG 23/52 44.2 4.11 ± 0.12
AGN 17,909,52,613 34.0 3.02 ± 0.08
Galaxies 140,831/500,707 28.1 4.03 ± 0.15
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|
http://arxiv.org/abs/2307.03901v2 | 20230708044917 | One-Loop Quantum Effects in Carroll Scalars | [
"Kinjal Banerjee",
"Rudranil Basu",
"Bhagya Krishnan",
"Sabyasachi Maulik",
"Aditya Mehra",
"Augniva Ray"
] | hep-th | [
"hep-th"
] |
=1
|
http://arxiv.org/abs/2307.07341v1 | 20230714134304 | PiTL: Cross-modal Retrieval with Weakly-supervised Vision-language Pre-training via Prompting | [
"Zixin Guo",
"Tzu-Jui Julius Wang",
"Selen Pehlivan",
"Abduljalil Radman",
"Jorma Laaksonen"
] | cs.IR | [
"cs.IR",
"cs.CV"
] |
Aalto University
Espoo
Finland
[email protected]
Aalto University
Espoo
Finland
[email protected]
Aalto University
Espoo
Finland
[email protected]
Aalto University
Espoo
Finland
[email protected]
Aalto University
Espoo
Finland
[email protected]
Vision-language (VL) Pre-training (VLP) has shown to well generalize VL models over a wide range of VL downstream tasks, especially for cross-modal retrieval. However, it hinges on a huge amount of image-text pairs, which requires tedious and costly curation. On the contrary, weakly-supervised VLP (W-VLP) <cit.> explores means with object tags generated by a pre-trained object detector (OD) from images. Yet, they still require paired information, i.e. images and object-level annotations, as supervision to train an OD.
To further reduce the amount of supervision, we propose Prompts-in-The-Loop (PiTL) that prompts knowledge from large language models (LLMs) to describe images. Concretely, given a category label of an image, e.g. refinery, the knowledge, e.g. a refinery could be seen with large storage tanks, pipework, and ..., extracted by LLMs is used as the language counterpart. The knowledge supplements, e.g. the common relations among entities most likely appearing in a scene.
We create IN14K, a new VL dataset of 9M images and 1M descriptions of 14K categories from ImageNet21K <cit.> with PiTL.
Empirically, the VL models pre-trained with PiTL-generated pairs are strongly favored over other W-VLP works on image-to-text (I2T) and text-to-image (T2I) retrieval tasks, with less supervision.
The results reveal the effectiveness of PiTL-generated pairs for VLP.
<ccs2012>
<concept>
<concept_id>10002951.10003317.10003338</concept_id>
<concept_desc>Information systems Retrieval models and ranking</concept_desc>
<concept_significance>500</concept_significance>
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[500]Information systems Retrieval models and ranking
PiTL: Cross-modal Retrieval with Weakly-supervised Vision-language Pre-training via Prompting
Jorma Laaksonen
=============================================================================================
§ INTRODUCTION
Vision-language (VL) models have been advancing rapidly with the introduction of various Vision-language Pre-training (VLP) methods. The models for VLP can adapt to various downstream tasks involving VL modalities, such as cross-modal retrieval <cit.>, visual question answering <cit.>, visual captioning <cit.>, etc.
The secret recipe of a VLP model comes with (1) a decent amount of webly-supervised[Some of the gathered image-text pairs may not be highly relevant as they are not validated by a human.] image-text pairs, (2) effective pre-training objectives which learn cross-modal interaction, and (3) sufficient resources, e.g. GPUs/TPUs, which enable large-scale training. Those models can also scale well with more image-text data, i.e. the models excel better in downstream tasks with more pre-training data <cit.>.
§.§ Weakly-supervised VLP
While the success of VLP methods relying on huge amounts of image-text annotations has been proven,
a less visited research path is emerging to pursue more data-efficient VLP. The data-efficiency of a VL model can be viewed from the amount of supervision. That is, would a VL model pre-trained with less image-text data remain as performant in downstream tasks? This question leads to works <cit.> in weakly-supervised VLP (W-VLP)
that aims at not relying on image-text pairs from, e.g. SBU Captions <cit.> and Conceptual Captions (CC) <cit.>.
Without the aligned images and texts, these works instead resort to a pre-trained object detector (OD) that generates object tags, i.e. the visual entities detected in the given image. The paired images and the object tags offer weaker supervision than those from the image-sentence pairs, but are still effective as the cross-domain bridge. However, training an OD relies on object-level annotations, which is still a form of supervision <cit.>. This seems to deviate these VLP works from the fully unsupervised path, which aims to remove any kind of cross-modal supervision.
§.§ Towards Unsupervised VLP
The unsupervised VLP (U-VLP), which aims at learning a VL model without any supervision across modalities, remains a daunting challenge. As a step towards U-VLP, we introduce Prompts-in-The-Loop (PiTL) that generates highly effective image-text pairs for W-VLP without an OD. PiTL capitalizes on image-level, i.e. a category label per image, instead of object-level supervision from the object bounding boxes and the corresponding object categories. This leads to a much harder W-VLP setting since much underlying information about an entity could no longer be inferred, such as the common co-occurrence of the visual entities, e.g. a chair and a desk, in a scene, and the entity relations. e.g. a person usually sits on a bench.
Specifically, given images with category labels, e.g. a duck as shown in Fig. <ref>, we prompt large language models (LLMs), e.g. GPT-3 <cit.>, to generate descriptions as the external knowledge of the category labels of the images. In fact, different prompts provide different focuses on each target category, e.g. one that emphasizes colors: "Describe the colors seen from a/an <category>?", and another that emphasizes relations with other entities: "Describe what could a/an <category> be seen with?". This encourages a VL model to associate all plausible visual traits, entities, actions, and scenes pertaining to the target categories.
The prompting paradigm is becoming trendy. Recent works <cit.> reveal that textual prompts generated by LLMs lead to significant improvement in zero-shot image classification with VLP models like CLIP <cit.>. For instance, prompting a VL model with the LLM-generated description, "Goldfish are small, orange fish with shiny scales.", is more likely to find matches in the visual domain than the generic "A photo of a goldfish.". Likewise, our work explores if the LLM-generated descriptions could be proved useful as well in the W-VLP setting.
Our contributions are summarized as follows. Firstly, we propose PiTL that generates an image-text dataset IN14K containing 9M images with 1M descriptions of 14K categories from the "Winter21" release of ImageNet-21K <cit.>. Second, trained with half of the samples in IN14K, our models are shown competitive to the state of the arts in image-to-text (I2T) and text-to-image (T2I) retrieval tasks. With full IN14K, our models significantly outperform them, e.g. on MSCOCO-5K <cit.> by 11% and 10%, respectively, on I2T and T2I. Moreover, our models are comparable with VLP models trained with 4M aligned image-text pairs from, e.g. CC3M and SBU Captions, etc. Lastly, PiTL does not only come with the least cross-modal supervision among W-VLP works, but also leads to a small gap between the W-VLP and VLP performances.
§ W-VLP WITH IMAGE-LEVEL SUPERVISION
VLP aims at learning VL alignments given a large number of image-sentence pairs. The methodology is concluded as (1) learning shared semantics around VL modalities, (2) learning cross-modal context, e.g. masked modeling <cit.>, and (3) learning to explicitly match images and texts.
With the same aim, the existing W-VLP methods leverage OD-generated tags to form VL pairs. What is usually neglected is the cost of pre-training such an OD, which usually requires 10+ object-level annotations to be effective. The proposed PiTL aims at relaxing the requirement of having an OD via prompting LLMs <cit.> to generate descriptions for the object categories.
§.§ Forming Image-text Pairs via Prompting
PiTL elicits knowledge about an object category from nine prompts of different perspectives with an LLM. Five descriptions are collected for each prompt. Table <ref> summarizes the nine prompts and their focuses. Some of them are more visually-relevant (P1-6), some focus more on knowledge around the target category (P7-8), and some are more open-ended (P9). We study the effectiveness of the descriptions generated by each prompt later in Sec. <ref>.
Among PiTL-generated pairs, an image can be paired with different descriptions as long as they are of the same category. In pre-training, the positive pairs for the Image-Text Contrastive and Image-Text Matching losses (introduced later in Sec. <ref>) are drawn from the images and descriptions of the same categories, and the negative pairs from those of the different categories. In this setup, pre-training with PiTL-generated pairs encourages the VL models to learn cross-modal alignment at the category level, i.e. images of a target category aligned to a group of descriptions, instead of instance level, i.e. an image aligned with a description as in other VLP works. As such, a given image would be associated with the plausible categorical visual traits, entities, actions, and scenes through the VL models.
§.§ VL Model Architecture
Our model architecture follows a state-of-the-art VL model, ALBEF <cit.>, which has a multi-modal encoder fusing the representations generated by visual and textual encoders. Indeed, any VL model with image-text inputs could also be used instead, as proposing a new VL architecture is not the focus of this work.
Specifically, given an image V and its paired text description T, the vision encoder g_v(·) follows ViT <cit.> consisting of a 12-layer Transformer that generates the image embedding as g_v(V)=(x_v^CLS, x_v^1, …, x_v^n_v). The text encoder g_t(·) is a 6-layer Transformer encoder that embeds the input text as g_t(T)=(x_t^CLS, x_t^1, …, x_t^n_t), where x^CLS_v and x^CLS_t are the representations of the [CLS] tokens summarizing the image and the text, respectively. n_v and n_t are the numbers of image patches and textual tokens, respectively. A fusion encoder g_f(·) consisting of a 6-layer Transformer learns the interaction across the VL modalities encoded as g_v(V) and g_t(T), and generates g_f(g_v(V), g_t(T))=(x_f^CLS, x_f^1, …, x_f^n_t).
§.§ Pre-training Losses
Our PiTL VL-models are pre-trained with four losses <cit.> that all contribute equally to the total loss :
=_ITC+_ITM+_MLM+_IMC,
where each objective is described as follows.
Image-Text Contrastive (ITC) aims to retain high and low similarities between the positive and negative image-text pairs, respectively.
To obtain the ITC loss, one first calculates
p_m^IT(V)=e^s(V,T_m)/τ/∑_i=1^M e^s(V, T_i)/τ,
p_m^TI(T)=e^s(T,V_m)/τ/∑_i=1^M e^s(T, V_i)/τ,
where s(V, T)=(x_v^CLS)^Tx_t^CLS measures the dot-product similarity of an image-text pair (V, T). p_m^IT(V) and p_m^TI(T) are image-to-text and text-to-image similarities, respectively. τ is a learnable temperature parameter and M is the size of the queues storing the image and textual class embeddings. The ITC loss is then defined as
_ITC=1/2 𝔼_V, T ∼D[H(y^IT_V, p^IT(V)) + H(y^TI_T, p^TI(T))],
where D denotes the pool of image-text pairs, y^IT_V and y^TI_T are M-dimensional binary vectors encoding ground-truth similarity, and H(·, ·) refers to the cross-entropy function.
Image-Text Matching (ITM) aims to predict whether an image-text pair is matched.
The token embedding x_f^CLS of the fusion encoder predicts the binary classification probability p^ITM. The ITM loss is defined as
_ITM=𝔼_V,T[H(y^ITM, p^ITM(V,T))],
where y^ITM is a binary vector indicating the matching pairs.
Masked Language Modeling (MLM) predicts the masked tokens in a sentence given an image and the unmasked textual tokens in the same sentence. 15% masking probability is set. The MLM loss <cit.> is denoted as _MLM.
Intra-Modal Contrastive (IMC) aims to differentiate the semantics between the positive and negative pairs within the same modality <cit.>, i.e. image-image and text-text pairs with similarities:
p_m^II(V)=e^s(V,V_m)/τ/∑_i=1^M e^s(V, V_i)/τ,
p_m^TT(T)=e^s(T,T_m)/τ/∑_i=1^M e^s(T, T_i)/τ.
The IMC objective is defined as
_IMC=1/2 𝔼_V, T ∼D[H(y^II_V, p^II(V)) + H(y^TT_T, p^TT(T))],
where y^II_V and y^TT_T indicate whether the pair is matched or not. It is worth noting that _IMC encourages the model to retain the uni-modal semantics provided by the pre-trained weights of the vision and textual encoders, complementing _ITC, _ITM, and _MLM, all of which promote multi-modal alignments.
§ EXPERIMENTS
§.§ Settings
Our vision encoder is instantiated by ViT <cit.> and initialized with DEiT <cit.> or BEiT-B/16 <cit.> pre-trained weights. The textual encoder is initialized with BERT-Base <cit.>. The proposed PiTL W-VLP model is pre-trained on three subsets IN1K, IN6K, and IN14K created from ImageNet21K.
Note that IN6K and IN14K are created in this work.
Specifically, IN14K contains IN6K which also contains IN1K samples. Each prompt, out of the nine shown in Table <ref>, generates five responses for a category. Individual prompts are created for multiple synonyms, e.g. snorkeling and snorkel_diving under the same category. Statistics of IN1K, IN6K, IN14K along with other datasets, e.g. CC3M, BookCorpus (BC) <cit.>, and VL-Full <cit.> used by other VLP methods, are shown in Table <ref> under the Pre-training Corpus column. We assess the pre-training quality on I2T and T2I with MSCOCO-5K and Flickr30K <cit.>. The retrieval models are evaluated on recall at rank K (R@K).
§.§ Quantitative Results
Table <ref> shows the main results of PiTL and the comparisons against the state-of-the-art VLP and W-VLP on the I2T and T2I tasks.
Effects of Initialization and Dataset Sizes.
We initialize the image encoder with weights pre-trained with no supervision (i.e. the self-supervised BEiT†) and with image-level supervision (i.e. ViT, BEiT, and BEiT⋆). The best performances are obtained with BEiT⋆, whose weights contain the strongest visual semantics, on IN1K to IN14K. PiTL's results steadily improve with more images and descriptions.
Compared to models without pre-training, BEiT† pre-trained on IN1K has larger improvements in R@1 than the other initializations, i.e. 8.6% for I2T and 10.8% for T2I R@1 on MSCOCO-5K.
PiTL with BEiT-B/16† vs. W-VLP Works.
To purely assess the generated image-text pairs, models initialized with self-supervised BEiT-B/16† weights are mainly benchmarked. VLMixer starts out as a better model than PiTL when both are not pre-trained on any image-text pairs. However, once pre-trained, PiTL appears to be strongly competitive. For instance, PiTL pre-trained on IN1K outperforms VLMixer pre-trained on CC3M, across all the I2T metrics, with fewer images and texts. Pre-trained on IN6K and IN14K, PiTL is strongly competitive and better than VLMixer pre-trained on VL-Full. On Flickr30K, PiTL is also comparable to E2E-UVLP with more images and fewer texts.
PiTL Models Pre-trained on IN14K vs. W-VLP Works.
While empowered by the pre-trained weights, i.e. ViT, BEiT-B/16, and BEiT-B/16⋆, PiTL is strongly favored over other W-VLP works across all the metrics. We stress that obtaining pre-trained weights requires far less amount of supervision, i.e. 14M labels for 14M images versus 10+ object annotations per image, compared to an effectively pre-trained OD.
§.§ Ablation Studies
We study the retrieval performances on Flickr30K on the types of prompts (P1-P9 in Table <ref>) and the effect of the paired images and descriptions generated by PiTL.
On Effects of Prompts P1-P9.
We dissect the effect contributed by prompts P1-9 to the retrieval performance. We pre-train models initialized with BEiT-B/16⋆ on descriptions from each prompt. Table <ref> shows that, with all the descriptions available, the models achieve the best R@1 scores for both retrieval tasks. On I2T,
P9 results in the best average recall over R@1,5,10. We reckon that P9 is a more open-ended question than the others, which focus specifically on appearances, relations, etc. The open-endedness could result in more diverse descriptions and hence better I2T result. On T2I, P1-4 usually yield better T2I results than the other prompts. This seems to be expected since P1-4 are more visually relevant.
On the Effect of Increased Uni-modal Shuffled Sources. To study whether the improvements actually come from the proposed prompt-based weak supervision, rather than from the pure increase in the number of images and texts, we pre-train the models (initialized with BEiT-B/16⋆) on IN1K and IN6K with shuffled images and descriptions. Indeed, on Flickr30K, the models degrade in R@1 from 60.7 to 56.1 for I2T, and from 48.2 to 42.7 for T2I, when pre-trained on shuffled IN1K and IN6K, respectively. This concludes the effectiveness of PiTL-generated pairs for VLP.
§ CONCLUSION
In this work, we proposed Prompts-in-The-Loop (PiTL), a weakly-supervised method to pre-train VL-models for cross-modal retrieval tasks. Without object-level supervision based on OD, our PiTL comes with image-level supervision from images and their categories described by large language models.
Retrieval results on the cross-modal datasets demonstrated the effectiveness of PiTL.
This work is supported by the Academy of Finland in project 345791.
We acknowledge the LUMI supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CSC and the LUMI consortium.
ACM-Reference-Format
|
http://arxiv.org/abs/2307.04176v1 | 20230709135730 | Electron-phonon driven charge density wave in CuTe | [
"Marco Campetella",
"Giovanni Marini",
"Jianqiang Sky Zhou",
"Matteo Calandra"
] | cond-mat.mtrl-sci | [
"cond-mat.mtrl-sci",
"cond-mat.str-el"
] |
[][email protected]
Consiglio Nazionale Delle Ricerche (CNR) SPIN, 00133 Rome, Italy
Dipartimento di Biotecnologie, Chimica e Farmacia, Università di Siena, Via Aldo Moro 2, Siena, I-53100, Italy
Graphene Labs, Fondazione Istituto Italiano di Tecnologia, Via Morego, I-16163 Genova, Italy
Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252, Paris, France
[][email protected]
Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo, Italy
Graphene Labs, Fondazione Istituto Italiano di Tecnologia, Via Morego, I-16163 Genova, Italy
Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252, Paris, France
The compound CuTe (vulcanite) undergoes a quasi one dimensional charge density wave (CDW) at T< T_CDW=335 K with a 5×1×2 periodicity. The mechanism at its origin is debated. Several theoretical works claimed that semilocal functionals are unable to describe its occurrence and ascribed its formation only to strong electron-electron interaction. Moreover, the possible role of quantum anharmonicity has not been addressed. Here, by performing quantum anharmonic calculations, we show that semilocal functionals correctly describe the occurrence of a CDW in CuTe if ultradense electron momentum grids allowing for small electronic temperatures are used. The distortion is driven by the perfect nesting among 1D Fermi surface sheets extending in the k_y direction. Quantum anharmonic effects are important and tend to suppress both the distortion and T_CDW. The quantum anharmonic structural minimization of the CDW phase in the generalized gradient approximation leads, however, to distorted Te-Te bond lengths in the low temperature phase that are 21% of the experimental ones at T=20 K. This suggests that, even if the electron-electron interaction is not crucial for the mechanism of CDW formation, it is relevant to accurately describe the structural data for the low-T phase. We assess the effect of correlation on the CDW by using the DFT+U+V approximation with parameters calculated from first principles. We find that correlation enhances
the Te-Te distortion, T_CDW
and the total energy gain by the distortion.
Electron-phonon driven charge density wave in CuTe.
Matteo Calandra
Received 27 February 2023; accepted 23 May 2023
===================================================
§ INTRODUCTION
One dimensional (1D) and quasi 1D crystal are prone to charge density wave (CDW) instabilities due to their low dimensional, often point like, Fermi surfaces and the resulting divergence in the charge response. This is what is predicted by the Landau-Peierls theory that is characterized by three main features: (i) the transition is second order and manifests itself via a soft phonon going to zero at the transition temperature (T_CDW), (ii) the occurrence of an order parameter (the phonon displacement induced by the CDW) that is non-zero only for T<T_CDW and decreases by increasing temperature until it becomes zero at the transition and, finally, (iii) the opening of a gap in the electronic excitation spectrum whose magnitude should be of the same order of the total energy gain by the CDW distortion. The common belief is that most one dimensional systems are globally well modeled by the Landau-Peierls model.
The Landau-Peierls model is, however, incomplete as it does not account for quantum anharmonicity (i.e. the quantum nature of the ions and the anharmonicity in the ionic potential) that is crucially important for light atoms and in proximity of a second order structural instability. Moreover, it neglects strong electron-electron interaction. Recently the archetypal case of carbyne in vacuum was studied with a variety of density functional theory (DFT) and manybody approaches accounting for non-perturbative quantum anharmonicity and electron-electron interaction<cit.>. It was shown that the total energy gain by the distortion is two order of magnitudes smaller (25 meV) than the distortion-induced electronic gap (3 eV) and, most surprisingly, the order parameter increases with increasing temperature for T<T_CDW. This pathology of the carbon chain in vacuum is in part related to the light mass of the carbon atoms and its quantum nature and does not invalidate the applicability of Landau-Peierls theory in a broader spectrum of materials. Still, it implies that there could be remarkable exceptions to this theory, either because quantum anharmonic effects and the electron-electron interaction are crucially important or because the Fermi surface deviates from what expected for a 1D system. The applicability of the Landau-Peierls picture to higher dimensional material has been questioned in several works<cit.>.
Recently, the layered material CuTe (vulcanite) received considerable interest<cit.>. In this compound
Te chains run above and below a puckered
copper layer so that each copper atom has a distorted
tetrahedral environment (see Fig.<ref> and Ref. stolze2013cute).
At temperatures lower than T_CDW = 335 K,
CuTe undergoes a 5× 1× 2 CDW <cit.>. The distortion involves a Te-Te bond alternation with phonon displacements as shown in Fig. <ref>. The superstructure is
visible in ARPES data as a (partial) gapping of the Fermi surface<cit.>, the maximum size of the gap being approximately 192 meV<cit.>. ARPES data<cit.> in the high-T phase show the occurrence of quasi 1D Fermi surface sheets extending along the k_y direction and perfectly nested along k_x.
Resistivity <cit.> and optical<cit.> data confirm the quasi 1D character of the CDW as the temperature dependence of the resistivity along the b axis shows the classical behaviour of a metallic system and is not affected by the CDW, while the resistivity along the a axis ( i.e. the CDW direction) displays a marked hump at T=T_CDW. Interestingly, the Hall coefficient is enhanced by approximately a factor of two across the CDW transition (larger values of R_H are in the distorted phase) suggesting a carrier reduction but an incomplete gapping of the Fermi surface in the low-T one<cit.>. The constant pressure specific heat displays a marked jump at the transition albeit with no hysteresis, confirming the second order nature of the transition<cit.>.
Thus, this experimental picture seems to point to a second order Landau-Peierls transition mostly due to the perfect nesting of the quasi 1D Fermi surface sheets.
However, three recent theoretical works <cit.> calculated the harmonic phonon dispersion of the high-T phase of CuTe with semilocal functionals and found no tendency toward CDW (i.e. no imaginary phonon frequencies).
The difficulty in reproducing the occurrence of the CDW with semilocal functionals led some authors to speculate that the CDW in this system is exclusively driven by electron-electron correlation<cit.>. Indeed, by performing DFT+U calculations, the authors of Ref. <cit.> showed that very large values of U can induce a structural instability comparable with the experimental one. However the considered value for the Hubbard parameter (U=9 eV) is extremely large and not calculated ab initio. Furthermore, the role of anharmonicity was not discussed. Recently, a careful study of collective excitations in CuTe <cit.> pointed out the possible existence of acoustic plasmons, making the study of this compound even more appealing. Finally, CuTe has been reported to support a superconducting state at high pressures <cit.>
In this work we investigate the electronic, structural and vibrational properties of CuTe within density functional perturbation theory. We include the effect of non-perturbative quantum anharmonicity by using the Stochastic Self-Consistent Harmonic Approximation <cit.>.
We demonstrate that, contrary to what claimed in all published theoretical papers and in agreement with the experimental picture, the CDW is mostly driven by the electron-phonon coupling and Fermi surface nesting with relevant corrections related to quantum anharmonicity. Electron-electron interactions are not negligible but are not the driving force for the CDW transition: they are probably required to accurately describe the structural properties of the low-T phase.
The paper is structured as follows. In Sec. <ref> we give the technical details of the first principles calculations, in Sec. <ref> we address the electronic structure and the mechanism for CDW formation, in Sec. <ref> we describe the structural properties of the CDW phase and in Sec. <ref> we draw the main conclusions.
§ TECHNICAL DETAILS
Density-functional theory (DFT) and density functional perturbation theory (DFPT) calculations are carried out using the Quantum ESPRESSO package<cit.>. We use the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE)<cit.> parametrization.
The experimental measured lattice parameters for bulk CuTe a = 3.151 Å, b = 4.089 Å and c = 6.950 Å are adopted in all calculations, while we perform structural optimization of internal coordinates.
We use ultrasoft pseudopotentials<cit.> and a
50 Ry plane wave energy cutoff for the kinetic energy (500 Ry for the charge density).
As phonon dispersion curves in one dimensional materials are extremely sensitive to the k-point sampling and to the electronic temperature (T_e) used in the calculation, we perform extremely accurate convergence tests of the phonon frequency at the CDW phonon momentum q_CDW=[0.4,0,0.5] (square brakets means that the components are given with respect to the basis vectors of the reciprocal lattice).
In more details,
the harmonic phonon dispersion is calculated using Γ centered k-points meshes. We considered grids of the kind k_x×16× 4 with k_x values up to 150. We then calculate the phonon frequency for each mesh as a function of the Fermi temperature used in the calculations. The results of these calculations are
explained in more details in Sec. <ref>. At the end of these tests we adopted an 80×16×4 electron-momentum grid in the 1×1×1 cell and an electronic temperature T_e=200 K (Fermi Dirac smearing).
When using supercells, the k-points meshes are then rescaled according to the size of the supercells (e.g., we use a 8×16×2 k-points mesh on a 10×1×2 cell and a 16×16×4 k-points mesh on a 5×1×1 cell).
The quantum anharmonic calculation is performed with the Stochastic Self Consistent Harmonic Approximation (SSCHA)<cit.>.
The SSCHA is a stochastic variational technique that allows to access the non-perturbative quantum anharmonic free energy and its Hessian with respect to the atomic positions <cit.> (i.e., the phonon spectrum). The SSCHA technique requires the evaluation of forces in supercells with atoms displaced from their equilibrium positions following a suitably chosen Gaussian distribution. The forces can be calculated by using any force engine. In this work
we used DFT with the PBE functional for the force calculation.
We calculate the forces using the Quantum ESPRESSO package and supercells ranging from 5× 1×2 to 10× 1×2. In a
10×1×2 supercell (80 atoms) of the high-T phase structure the number of DFT force calculations needed to converge the free energy is
of the order of 800, while approximately 2000 forces are needed to converge the free energy Hessian at T = 0 K. The computational effort is substantial given the dense electron-momentum grids.
We determine the nature and the critical temperature T_CDW of the CDW transition by monitoring the positional free energy Hessian (second derivative of the free energy with respect to the atomic positions)<cit.>, as dictated by Landau theory of phase transitions.
§ HIGH-T PHASE
We first calculate the electronic structure of the high-T phase and compare the Fermi surface with that measured in ARPES (see Fig. <ref>).
Each panel refers to constant energy cuts from E_F to -0.5 eV from E_F (the value of the constant energy with respect to E_F is shown on the top of each panel) in the (k_x,k_y) plane and for k_z=0. Experimental ARPES data from Ref. zhang2018evidence
are also included for reference.
Globally the agreement between the experimental and measured constant energy scans is excellent. We are able to recover both the pockets extending along the k_x direction and the quasi-1D line segments along the k_y direction. These last dispersionless bands extending only along the k_y direction are clear fingerprints of the 1D physics in vulcanite.
The sharpness of these 1D Fermi surface portions suggests that a remarkably dense k-points mesh along the k_x direction may be required in order to correctly sample their contribution to the phonon dispersion at phonon momentum q= q_CDW.
We explicitly verified this point by performing careful convergence of the lowest energy phonon frequency at q= q_CDW as a function of k_x points and Fermi-Dirac electronic temperature T_e. The results are shown in Fig. <ref> (top) and unambiguously show that grids having k_x≈ 80 and electronic temperatures comparable to T_CDW must be used to see the CDW. By adopting an electronic temperature T_e=200 K and a k-point mesh of 80× 16× 4 we find converged results. As it can be seen, the lowest phonon frequency at q= q_CDW is imaginary and not positive as it has been reported in all published theoretical papers in the field<cit.>.
In these works, the difficulty in performing Brillouin zone sampling for CuTe has been completely overlooked. Much coarser grids, such as 30×16× 4, and, most likely, larger electronic temperatures have been used. We point out that the technical details reported in Refs. <cit.> are incomplete and the calculations are not reproducible (as an example in Refs. <cit.> the value of the electronic temperature is not reported).
From Fig. <ref>, it is also clear that by using the PBE semilocal functional and simply increasing the electronic temperature, i.e. neglecting quantum anharmonicity, the CDW critical temperature is in the range T_CDW=700-750 K. This is only a factor of two higher than the experimental one, suggesting that Fermi surface nesting is an important effect in this system.
The CuTe harmonic phonon dispersion is reported in Fig. <ref> (bottom panel). As it can be seen there are two sharp dynamical instabilities corresponding to the modulations q_CDW=[0.4,0,0.5] and
q_CDW^'=[0.4,0,0.0]. The planar instability at q_CDW^' leads to slightly more unstable phonons. However small changes in the simulations details (structural parameters, functional used,...) lead to a more unstable mode at q_CDW. These two instabilities are then almost degenerate. The local character in momentum space of the instability points at a crucial role of the Fermi surface.
In order to confirm this point we calculate the electron-phonon contribution to the phonon linewidth (FWMH), namely
γ_ qν=4 πω_ qν/N_k∑_ k,n,m|g_ k n, k+ q m^ν|^2δ(ϵ_ k n-E_F)δ(ϵ_ k+ q m-E_F)
where ω_ qν are the harmonic phonon frequencies, ϵ_ kn are the Kohn-Sham energy bands, E_F is the Fermi level and g_ k n, k+ q m^ν is the electron-phonon matrix element. We calculate γ_ qν for the lowest energy phonon mode along the ZU direction. The results are shown in Fig. <ref> and shows a strong enhancement of the phonon linewidth at the CDW wavevector mostly due to Fermi surface nesting. At the harmonic level and by using the PBE functional the instability is then electron-phonon driven.
The phonon patterns connected with these two instabilities are very similar in the CuTe ab-plane.
The only difference is that the distortion of momentum q= q_CDW shifts two parallel CuTe planes in antiphase.
The calculation of the energy gain obtained by displacing the ions along the directions of the imaginary phonon mode is approximately 1.29 meV per Cu atom in both cases.
The occurrence of imaginary phonon frequencies at the harmonic level is, however, not enough to demonstrate the presence of a CDW as quantum-anharmonic terms in the potential could remove the instability. In order to explore this possibility,
we investigate quantum anharmonic effects within the Stochastic Self-Consistent Harmonic Approximation (SSCHA)<cit.> that has been proven to be very effective in describing anharmonic quantum effects in a plethora of different systems<cit.>.
The quantum anharmonic phonon dispersion is obtained within the SSCHA by calculating the positional free energy (F) Hessian as a function of temperature. We define the temperature dependent dynamical matrix as:
D = M^-1/2∂^2 F/∂R^2| _R_eqM^-1/2
where M is the matrix of the ionic masses M_a with M_ab = δ_ab M_a and R is a cumulative variable for all the ionic positions (see Ref. bianco2017second for a detailed explanation). By Fourier transforming the matrix D and by diagonalizing it, we obtain as eigenvalues the squared quantum anharmonic phonon frequencies.
We perform the SSCHA calculation on a 10×1×2 supercell. The results are shown in Fig. <ref> (T=0, bottom panel) and in Fig. <ref> (top panel) as a function of temperature. At T=0 the main effect of quantum anharmonicity is an hardening of the CDW mode. However, the mode still remains imaginary signalling that at T=0 quantum anharmonicitiy does not remove the CDW.
The temperature dependence of the quantum anharmonic phonon dispersion is shown in Fig. <ref> (top panel). At T=200 K the phonon dispersion does not display any dynamical instability, meaning that the calculation is already in the undistorted high-T phase.
By plotting the square of the lowest phonon frequency as a function of temperature in Fig. <ref> (bottom panel) we estimate T_CDW≈ 60 K. This critical temperature is approximately 5.6 times smaller than the real one. As the transition occurs only via a change in the quantum free energy Hessian that becomes negative at the transition along the CDW pattern, we find that in our calculation the transition is purely second order, in agreement with experimental data <cit.>.
Two effects may be at the origin of the underestimation of T_CDW. The first one is that the supercell used in the calculation could be too small. However, we have carefully monitored the value of the phonon frequency at q= q_CDW and q= q_CDW^' for supercells of sizes 5× 1× 1, 5× 1× 2, 10× 1× 2
finding that the quantum anharmonic phonon frequency varies less than 1 cm^-1. This excludes that this reduced T_CDW is due to a finite supercell effect.
The second and most probable reason causing the underestimation of T_CDW is the treatment of the exchange and correlation used. In order to better understand this point we examine more in details the low temperature phase.
§ LOW TEMPERATURE CDW PHASE.
In order to study the structural and electronic properties of the CDW phase, we consider two supercells, the 5× 1× 1 and the 5×1×2, corresponding to instabilities at q= q_CDW^' and
q= q_CDW, respectively.
We first displace the atoms along the unstable phonon patterns and then perform structural optimization (we minimize the classical Born-Oppenheimer forces). The results of the optimization are shown in Tab. <ref> . As it can be seen, the structural distortion of the Te atoms is in good agreement with experiments at T=20 K, although the distortion is somewhat underestimated. Both the 5× 1× 1 and the 5×1×2 give comparable 1D distortion.
The fact that, as we have seen, quantum anharmonic effects are important in this system, as they reduce T_CDW more than a factor of 10 with respect to the harmonic calculation, suggests that the inclusion of quantum anharmonicity will reduce the distortion. As the quantum anharmonic minimization in this system is very expensive due to the very dense mesh needed, we perform the quantum anharmonic structural optimization with the SSCHA only in the 5× 1× 1 supercell. This is justified as we know that the two supercells lead to practically identical distortion of the Te-Te bond along the CDW direction.
The results of the quantum anharmonic minimization are again shown in Tab. <ref>. As expected the distortion is substantially reduced and the quantum anharmonic distortion is approximately 41% (0.21%) of the experimental one at T=295 K (T=20K).
As in low dimensional systems it is well known that the exchange interaction is not completely screened and semilocal functional usually underestimate the distortion<cit.>, this has to be somewhat expected.
Finally, for completeness, we address the pseudogap feature detected in ARPES<cit.> in the CDW phase. Previous calculations already showed that this feature can be fairly well reproduced if the distortion is large enough <cit.>. As it is typical for a Peierls distortion, the magnitude of the gap opening is linearly related to the CDW distortion.
This means that, as the magnitude of the distortion depends on the exchange and correlation approximation used in the calculation, the size of the pseudogap also will.
We then consider the experimental distorted structure on a 5×1× 2 supercell, calculate the electronic structure and unfold it <cit.> on the CuTe unit cell. A finite Lorentzian linewidth of 20 meV is added to the theoretical unfolded band structure in order to simulated the experimental broadening. The comparison with ARPES data from Ref. <cit.> is also shown. Our calculations reproduce the opening of the CDW with a pseudogap that is of the same magnitude of the experimental one. Small differences occur on the exact value of the experimental gap that are probably due in part to the ARPES matrix element, not explicitly considered in our calculation.
§ ESTIMATION OF CORRELATION EFFECTS VIA DFT+U+V
In order to account for correlation effects
on the electronic structure and the structural properties on equal footing, we model the system in the DFT+U+V formalism within the rotationally invariant scheme first proposed by Dudarev et al. in Ref. PhysRevB.57.1505. Following Ref. Leiria_Campo_2010, the DFT energy functional, E_DFT, is corrected to include on-site and inter-atomic interactions, by adding the term
E_UV = ∑_IU^I/2Tr[𝐧^IIσ (1 - 𝐧^IIσ)]
- ∑^*_I,J,σV^IJ/2Tr[𝐧^IJσ𝐧^JIσ]
where I and J represent atomic sites, the star in the sum operator denotes that for each atom I, J covers all its neighbors up to a given distance, while the on-site parameter U^I, the inter-site V^I,J and the occupation matrix 𝐧^IJσ are defined as in Ref. Leiria_Campo_2010.
The new total energy E_DFT+U+V is written as
E_DFT+U+V = E_DFT + E_UV
The on-site and intersite parameter U^I and V^IJ parameters are calculated from first principles, using the linear response method introduced by Timrov et al. in Refs. PhysRevB.98.085127,PhysRevB.103.045141.
We use the atomic wavefunctions (3d for Cu and 5p for Te) read from the pseudopotentials to build the Hubbard projectors. In the calculation, all the neighboring atoms up to the fourth shell were considered. A 8×4×1 momenta grid was necessary to converge the U and V values within 0.1 eV.
The calculated inter and on-site Hubbard values for CuTe in the normal phase are reported in Tab. <ref>.
We find large on-site repulsion parameter of 16.71 eV and 4.32 eV for Cu(3d) and Te(5p) sites, respectively. Furthermore, we observe that interatomic Cu-Te interactions are negligible, while a sizable first-neighbor Te-Te repulsive interaction (0.97 eV) exists. The inclusion of Hubbard parameters importantly modifies the electronic structure, resulting in the Fermi surface shown in Fig.<ref> (top panel, yellow lines). By looking at the comparison between the ARPES and the Fermi surface predicted by first principles calculations employing DFT+U+V, we conclude that the first principles on-site and inter-site parameters are not substantially improving the agreement between the theory and the experiment, especially in regard to the electron pocket around the Γ point.
Finally, we calculate the energy gain in the charge-density wave phase with respect to the normal state with the inclusion of Hubbard parameters, and compare the results to the predictions given by PBE. The results are depicted in Fig. <ref>. We find that the inclusion of correlation effects enhances the CDW energy gain by more than one order of magnitude, i.e. from 1.29 meV /f.u. in PBE to 32 meV/f.u. if both inter- and on-site parameters are included in the calculation, while we obtain an energy gain of 17 meV/.f.u. if only on-site terms on Cu and Te are included in the calculation. Correspondingly, the predicted structural distortion due to the charge-density wave is notably enhanced, with a maximum Te-Te dimerization Δ_ Te of the order of 0.9 Å , overestimating the measured values of Ref.stolze2013cute of a factor ≈ 2.4 at T=20 K. Moreover the free energy versus Δ profile becomes even more anharmonic, suggesting both an increase of T_CDW at the harmonic level as well as an enhancement of quantum anharmonic effects.
As it was already clear at the PBE level, the charge density wave temperature is the result of a delicate
compensation among the electron-phonon interaction (enhancing the tendency towards CDW) and anharmonicity (suppressing the CDW). Both effects are substantially enhanced by correlation effects and both effects are crucial and comparable in order. Within DFT+U+V at the harmonic level, we do indeed estimate T_ CDW as being as large as 6000 K, in stark disagreement with experiments, signalling once more the need of including anharmonicity to obtain results in better agreement with experiments.
§ CONCLUSION
In this work, by performing non perturbative quantum-anharmonic calculations, we studied the CDW formation in CuTe.
Contrary to all existing theoretical calculations in literature <cit.>, we find that semilocal functionals correctly describe the occurrence of CDW in this system.
Previous calculations where unable to describe the CDW instability most likely due to the use of a too large electronic temperature.
We find that the CDW is due to the almost perfect nesting among the quasi 1D Fermi surface sheets extending along the k_y direction resulting in a large electron-phonon interaction and a consequent phonon softening. Quantum anharmonicity reduces this softening but does not suppress the CDW at T=0.
Quantum anharmonic effects reduce the T_CDW by a factor of 10 with respect to the harmonic estimate based on the electronic temperature only.
The calculated T_CDW≈ 60 K, resulting from the combined effect of the electron-phonon interaction and anharmonicity, underestimates the experimental one by a factor ≈ 5.6. Similarly, the quantum anharmonic structural minimization of the CDW phase leads to distorted Te-Te bond lengths in the low temperature phase that are 40% smaller than the experimental ones. These two underestimations are related and suggest that, even if the electron-electron interaction is not crucial for the mechanism of CDW formation, it is relevant to accurately describe the structural data for the low-T phase.
In order to validate this statement we employ the DFT+U+V approximation with on-site and off-site Hubbard parameters calculated ab initio. Within this approximation, the CDW distortion is strongly enhanced and overestimate the experimental one by a factor 5.6. At the harmonic level T_CDW≈ 6000K, approximately 20 times larger than the experimental value. However, anharmonic effects also becomes substantially larger, underlying once more the need of including quantum anharmonic effects to obtain results in better agreement with experiments.
§ ACKNOWLEDGEMENTS
Co-funded by the European Union-NextGenerationEU, ICSC – Centro Nazionale di Ricerca in HPC, Big Data and Quantum Computing. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
We acknowledge the PRACE and CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. We acknowledge support from Seal of Excellence (SoE) fellowship promoted by University of Siena
elsarticle-num.bst
|
http://arxiv.org/abs/2307.06091v1 | 20230712113202 | AICT: An Adaptive Image Compression Transformer | [
"Ahmed Ghorbel",
"Wassim Hamidouche",
"Luce Morin"
] | cs.CV | [
"cs.CV",
"eess.IV"
] |
On the Galois covers of degenerations of surfaces of minimal degree
Email address: M. Amram: [email protected]; C. Gong: [email protected]; Jia-Li Mo: [email protected];
2020 Mathematics Subject Classification. 05E15, 14J10, 14J25, 14N20.
[
====================================================================================================================================================================================================================================================
Motivated by the efficiency investigation of the Tranformer-based transform coding framework, namely SwinT-ChARM, we propose to enhance the latter, as first, with a more straightforward yet effective Tranformer-based channel-wise auto-regressive prior model, resulting in an absolute ict.
Current methods that still rely on ConvNet-based entropy coding are limited in long-range modeling dependencies due to their local connectivity and an increasing number of architectural biases and priors. On the contrary, the proposed ict can capture both global and local contexts from the latent representations and better parameterize the distribution of the quantized latents.
Further, we leverage a learnable scaling module with a sandwich ConvNeXt-based pre/post-processor to accurately extract more compact latent representation while reconstructing higher-quality images.
Extensive experimental results on benchmark datasets showed that the proposed aict framework significantly improves the trade-off between coding efficiency and decoder complexity over the vvc reference encoder (VTM-18.0) and the neural codec SwinT-ChARM.
Neural Image Compression, Adaptive Resolution, Spatio-Channel Entropy Modeling, Self-attention, Transformer.
§ INTRODUCTION
Visual information is crucial in human development, communication, and engagement, and its compression is necessary for effective data storage and transmission. Thus, designing new lossy image compression algorithms is a goldmine for scientific research. The goal is to reduce an image file size by permanently removing less critical information, specifically redundant data and high frequencies, to obtain the most compact bit-stream representation while preserving a certain level of visual fidelity. Nevertheless, the high compression rate and low distortion are fundamentally opposed objectives involving optimizing the rd cost.
Conventional compression standards, including JPEG, JPEG2000, H.265/HEVC, and H.266/VVC, rely on hand-crafted creativity to present module-based encoder/decoder block diagram, i.e., Intra prediction, transform, quantization, arithmetic coding, and post-processing. Traditional coding algorithms have a lot of advantages, including mature technology with SW/HW-friendly implementations, low decoding complexity, and strong generalization on different contents.
Nevertheless, all of them mainly rely on hand-crafted coding techniques; thus, it is quite challenging to directly optimize rd cost for all types of image content due to the rapid development of new image formats and the growth of high-resolution mobile devices.
On the other hand, neural coding has gained wide attention from research and industry, yielding promising end-to-end nic solutions outperforming their conventional counterparts in coding efficiency. nic leverages ae to carry out a non-linear coding from the signal domain to a compact representation. Such ae-based system consists of three modular parts: transform, quantization, and entropy coding, which can be trained in an end-to-end fashion to minimize the distortion between a source image and its reconstruction, and the rate needed to convey the latent representation bit-stream.
Since the early rnn-based method <cit.> for lossy image compression, significant advancements have been made in integrating tailored modules for nic. Previous works use local context <cit.>, or additional side information <cit.> to capture short-range spatial dependencies, and others use non-local mechanisms <cit.> to model long-range spatial dependencies. Recently, Toderici <cit.> proposed a generative compression method achieving high-quality reconstructions, Minnen <cit.> introduced channel-conditioning taking advantage of an entropy-constrained model that uses both forward and backward adaptations, Zhu <cit.> replaced the ConvNet-based transform coding in the Minnen <cit.> approach with a Transformer-based one, Zou <cit.> combined the local-aware attention mechanism with the global-related feature learning and proposed a window-based attention module, Koyuncu et al. <cit.> proposed a Transformer-based context model, which generalizes the standard attention mechanism to spatio-channel attention, Zhu <cit.> proposed a probabilistic vector quantization with cascaded estimation under a multi-codebooks structure, Kim <cit.> exploited the joint global and local hyperpriors information in a content-dependent manner using an attention mechanism, and He <cit.> adopted stacked residual blocks as nonlinear transform and multi-dimension entropy estimation model.
In order to improve image-level prediction while minimizing computation costs, learned sampling techniques have been developed for several vision tasks. stn <cit.> introduce a layer that estimates a parametrized affine, projective, and splines transformation from an input image to recover data distortions. Based on the latter, Chen <cit.> proposed a straightforward learned downsampling module that can be jointly optimized with any neural compression kernels in an end-to-end fashion. Talebi <cit.> jointly optimize pixel value interpolated at each fixed downsampling location for classification. Jin <cit.> introduced a deformation module and a learnable downsampling operation, which can be optimized together with the given segmentation model.
One of the main challenges of nic is the ability to identify the crucial information necessary for the reconstruction, knowing that information overlooked during encoding is usually lost and unrecoverable for decoding. Another main challenge is the trade-off between coding performance and decoding latency. While the existing approaches improve the transform and entropy coding accuracy, they still need to be improved by the higher decoding runtime and excessive model complexity leading to an ineffective real-world use. To cope with those challenges, we present in this paper three contributions summarized as follows:
* We propose the ict, a nonlinear transform coding and spatio-channel auto-regressive entropy coding. These modules are based on swint blocks for effective latent decorrelation and a more flexible receptive field to adapt for contexts requiring short/long-range information.
* We further propose the aict model that adopts a scale adaptation module as a sandwich processor to enhance compression efficiency. This module consists of a neural scaling network and ConvNeXt-based pre/post-processor to jointly optimize differentiable resizing layers and a content-dependent resize factor estimator.
* We conduct experiments on four widely-used benchmark datasets to explore possible coding gain sources and demonstrate the effectiveness of aict. In addition, we carried out a model scaling analysis and an ablation study to substantiate our architectural decisions.
Extensive experiments reveal the impacts of the spatio-channel entropy coding, the sandwich scale adaptation component, and the joint global structure and local texture learned by the self-attention units through the nonlinear transform coding. These experiments validate that the proposed aict model achieves compelling compression performance, as illustrated in Fig. <ref>, outperforming conventional and neural codecs in both coding efficiency and decoder complexity.
The rest of this paper is organized as follows. First, the proposed aict framework is described in detail in Section <ref>. Next, we dedicate Section <ref> to describe and analyze the experimental results. Finally, Section <ref> concludes the paper.
§ PROPOSED AICT FRAMEWORK
§.§ Overall Architecture
The overall pipeline of the proposed solution is illustrated in Fig. <ref>. The framework includes three modular parts. First, the scale adaptation module, composed of a tiny rpn <cit.>, a ConvNeXt-based pre/post-processor, and a bicubic interpolation filter. Second, the analysis/synthesis transform (g_a,g_s) of our design consists of a combination of patch merging/expanding layers and swint <cit.> blocks. The architectures of hyper-transforms (h_a,h_s) are similar to (g_a,g_s) with different stages and configurations. Finally, a Transformer-based slice transform under a charm design is used to estimate the distribution parameters of the quantized latent. These resulting discrete-valued data (ŷ, ẑ) are encoded into bit-streams with an arithmetic coder.
§.§ Scale Adaptation Module
Given a source image x ∈ R^H × W × C, we first determine an adaptive resize factor M estimated by the rpn module, which consists of three stages of resblock. Indeed, the estimated resize parameter M is used to create a sampling grid τ_M following the convention stn, and used to adaptively down-scale x into x_d∈ R^H' × W' × C through the bicubic interpolation. The latter is then encoded and decoded with the proposed ict. Finally, the decoded image x̂_d∈ R^H' × W' × C is up-scaled to the original resolution x̂∈ R^H × W × C using the same, initially estimated, resize parameter M.
The parameterization of each layer is detailed in the rpn and resblock diagrams of Fig. <ref> (a) and (b), respectively. In addition, a learnable depth-wise pre/post-processor is placed before/after the bicubic sampler to mitigate the information loss introduced by down/up-scaling, allowing the retention of information. This neural pre/post-processing method consists of concatenation between the input and the output of three successive ConvNeXt <cit.> blocks. The ConvNeXt block diagram is also illustrated in Fig. <ref> (c). For a better complexity-efficient design, we decided to skip the scale adaptation module where M ≅ 1.
§.§ Transformer-based Analysis/Synthesis Transform
The analysis transform g_a contains four stages of patch merging layer and swint block to obtain a more compact low-dimensional latent representation y. In order to consciously and subtly balance the importance of feature compression through the end-to-end learning framework, we used two additional stages of patch merging layer and swint block in the hyper-analysis transform to produce an additional latent representation z.
During training, both latents y and z are quantized using a rounding function to produce ŷ and ẑ, respectively.
The quantized latent variables ŷ and ẑ are then entropy coded regarding an indexed entropy model for a location-scale family of random variables parameterized by the output of the charm, and a batched entropy model for continuous random variables, respectively, to obtain the bit-streams. Finally, quantized latents ŷ and ẑ feed the synthesis and hyper-synthesis transforms, respectively, to generate the reconstructed image. The decoder schemes are symmetric to those of the encoder, with patch-merging layers replaced by patch-expanding layers.
§.§ Transformer-based Slice Transform
Although there are strong correlations among different channels in latent space, the strongest correlations may come from the spatio-channel dependencies.
Thus, to better parameterize the distribution of the quantized latents with a more accurate and flexible entropy model and without increasing the compression rate, we propose a Transformer-based slice transform inside the charm. Unlike previous works, ours considers spatio-channel latent correlations for entropy modeling in an auto-regressive manner. As a side effect, it also leads to faster decoding speed.
The slice transform consists of two successive swint blocks with an additional learnable linear projection layer, used to get a representative latent slices concatenation. This charm estimates the distribution p_ŷ (ŷ | ẑ) with both the mean and standard deviation of each latent slice, and incorporates an auto-regressive context model to condition the already-decoded latent slices and further reduce the spatial redundancy between adjacent pixels.
§ RESULTS AND ANALYSIS
§.§ Experimental Setup
Baselines.[For a fair comparison, we only considered SwinT-ChARM <cit.> from the state-of-the-art models <cit.>, due to the technical feasibility of models training and evaluation under the same conditions and in an adequate time.]
We compare our solution with the state-of-art neural codec SwinT-ChARM proposed by Zhu <cit.>, and the Conv-ChARM proposed by Minnen <cit.> and conventional codecs, including bpg(4:4:4), and the vvc official Test Model VTM-18.0 in All-Intra configuration.
Implementation details.
We implemented all models in TensorFlow using tfc library, and the experimental study was carried out on an RTX 5000 Ti GPU and an Intel(R) Xeon(R) W-2145 @ 3.70GHz CPU. All models were trained on the same CLIC20 training set with 2M iterations using the ADAM optimizer with parameters β_1=0.9 and β_2=0.999. The initial learning rate is set to 10^-4 and drops to 10^-5 for the last 200k iterations, and L=D+λR is used as a loss function. L is a weighted combination of bitrate R and distortion D, with λ being the Lagrangian multiplier steering rd trade-off. Mean squared error (MSE) is used as the distortion metric in RGB color space. Each training batch contains eight random crops ∈ R^256 × 256 × 3 from the CLIC20 training set. To cover a wide range of rate and distortion points, for our proposed method and respective ablation models, we trained four models with λ∈{1000, 200, 20, 3}× 10^-5. We evaluate the image codecs on four datasets <cit.>, including Kodak, Tecnick, JPEG-AI, and the testing set of CLIC21. For a fair comparison, all images are cropped to the highest possible multiples of 256 to avoid padding for neural codecs.
§.§ Rate-Distortion Coding Performance
To demonstrate the compression efficiency of our proposed approach, we summarize, in Table <ref>, the BD-rate of our models and the baselines across four datasets compared to the VTM-18.0 as the anchor.
On average, aict is able to achieve 5.11% BD-rate reduction compared to VTM-18.0 and 3.93% relative gain from SwinT-ChARM. Also, we illustrate in Figure <ref> a comparison of compression efficiency on Kodak dataset.
Figure <ref> shows the BD-rate (with VTM-18.0 as an anchor) versus the decoding time of various approaches on the Kodak dataset. It can be seen from the figure that our ict and aict achieve a good trade-off between BD-rate performance and decoding time.
§.§ Models Scaling Study
We evaluated the decoding complexity of the four considered image codecs by averaging decoding time across 7000 images encoded at 0.8 bpp. We present the image codecs complexity in Table <ref>, including decoding time on GPU and CPU, codec flops, and codec total number of parameters.
Compared to the neural baselines, ict can achieve faster decoding speed on GPU but not on CPU, which proves the parallel processing ability to speed up compression on GPU and the well-engineered designs of both transform and entropy coding, highlighting an efficient and hardware-friendly compression model. This is potentially helpful for conducting high-quality real-time visual data streaming.
Our aict is on par with ict in terms of the number of parameters, flops, and latency, indicating that the scale adaptation module is not computationally heavy for real scenario applications.
§.§ Ablation Study
To investigate the impact of the proposed ict and aict, we conduct an ablation analysis according to the reported BD-rate results in Table <ref>.
The compression performance increases from Conv-ChARM to SwinT-ChARM on the considered datasets due to the inter-layer feature propagation across non-overlapping windows (local information) and self-attention mechanism (local information) in the swint.
With the proposed spatio-channel entropy model, ict is able to achieve, on average, -3.47% BD-rate reduction compared to SwinT-ChARM. Therefore, introducing the Transformer-based slice transform leads to significant improvement compared to the ConvNet-based entropy model using only short-range dependencies. In addition, our spatio-channel entropy model is more helpful when combined with the Transformer-based transform coding.
aict performs better than ict, indicating that the introduction of a scale adaptation module can further reduce spatial redundancies and alleviate coding artifacts, especially at low bitrate resulting in higher compression efficiency.
§ CONCLUSION
In this paper, we have proposed an up-and-coming neural codec aict, achieving compelling rd performance while significantly reducing the latency, which is potentially helpful to conduct, with further optimizations, high-quality real-time visual data compression.
We inherited the advantages of self-attention units from Transformers to effectively approximate both the mean and standard deviation for entropy modeling and combine global and local texture to capture correlations among spatially neighboring components for non-linear transform coding, achieving -4.65% BD-rate reduction over the VTM-18.0, by averaging over the benchmark datasets.
Furthermore, we presented a lightweight scale adaptation module to enhance compression ability, especially at low bitrates, reaching on average -5.11% BD-rate reduction over the VTM-18.0.
IEEEbib
|
http://arxiv.org/abs/2307.04588v1 | 20230710143024 | Extremal numbers and Sidorenko's conjecture | [
"David Conlon",
"Joonkyung Lee",
"Alexander Sidorenko"
] | math.CO | [
"math.CO"
] |
Reliable Devices Yield Stable Quantum Computations
The manuscript is authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan: https://www.energy.gov/doe-public-access-plan.
Samudra Dasgupta^1, 2^*, and Travis S. Humble^1,2^†
^1Quantum Science Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
^2Bredesen Center, University of Tennessee, Knoxville, Tennessee, USA
^*[email protected], ORCID: 0000-0002-7831-745X
^†[email protected], ORCID: 0000-0002-9449-0498
February 2023
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Sidorenko's conjecture states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko's conjecture does not hold for a particular r-partite r-uniform hypergraph H, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number ex(n,H), the maximum number of edges in an n-vertex H-free r-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all 3-partite 3-uniform tight cycles.
§ INTRODUCTION
An r-graphon W:[0,1]^r → [0,1] is an r-variable symmetric measurable function.[We note that this is different from the usual definition of hypergraphons (see, for instance, <cit.>), where 2^r-2 variables are used to model limits of r-uniform hypergraphs. Such an approach is required to make the space complete, which is not necessary for our purposes.] Given an r-uniform hypergraph (or r-graph) H, the homomorphism density t_H(W) of H in W is
t_H(W) := ∫∏_u_1⋯ u_r∈ E(H)W(x_u_1,x_u_2,…,x_u_r) dμ^ v(H).
An r-graph H is said to be Sidorenko if
t_H(W) ≥ t_K_r(W)^ e(H) = (∫ W dμ^r)^ e(H)
for all r-graphons W:[0,1]^r → [0,1], where K_r denotes the r-graph with one edge.
In graph-theoretic terms, an r-graph H is Sidorenko if quasirandom r-graphs contain asymptotically the minimum number of copies of H taken over all r-graphs with the same order and edge density.
A celebrated conjecture of Sidorenko <cit.> (see also the closely related conjecture of Erdős and Simonovits <cit.>) says that a graph H is Sidorenko if and only if it is bipartite. The necessity of the bipartiteness condition is straightforward to verify, but its sufficiency remains wide open despite significant attention in recent years <cit.>.
It is also tempting to make an analogous conjecture for r-uniform hypergraphs H, namely, that H is Sidorenko if and only if it is r-partite. Unfortunately, as already observed in <cit.>, this is false, with the 3-uniform loose triangle with vertex set {1, 2, …, 6} and edges {1, 2, 3}, {3, 4, 5}, {5, 6, 1} being a counterexample. However, as we will show, there is still something to be gained if the conjecture fails to hold, in that we can improve the lower bound for the extremal number of any r-uniform hypergraph H for which Sidorenko's conjecture is false.
Given a natural number n and an r-graph H, the extremal number (n,H) is the maximum number of edges in an n-vertex H-free r-graph.
It is known that for any fixed r-graph H, there exists a non-negative number π(H) such that (n,H) = (π(H) + o(1)) nr and that π(H) = 0 if and only if H is r-partite.
With very few exceptions (see, for example, <cit.> for classical results and <cit.> and its references for more recent developments),
the problem of estimating (n,H) more accurately in the degenerate case where H is r-partite is wide open. In general, the best known lower bound comes from a simple application of the probabilistic deletion method and says that for any fixed r-partite r-graph H there exists some constant γ > 0 such that
(n,H) ≥γ n^r- v(H)-r/ e(H)-1.
Our first result improves this estimate for non-Sidorenko r-graphs.
For any non-Sidorenko r-graph H, there exist constants c, γ >0 such that
(n,H) ≥γ n^r- v(H)-r/ e(H)-1+c.
One reason this result is interesting is that, by a result of Ferber, McKinley and Samotij <cit.>, any polynomial gain over the deletion bound for the extremal number of an r-graph H implies an optimal counting result for the number of H-free r-graphs on n vertices. Thus, we have the following corollary of <ref>.
For any non-Sidorenko r-graph H, there exists C > 0 and an infinite sequence of positive integers n such that
|ℱ_n(H)| ≤ 2^C ·(n, H),
where ℱ_n(H) is the set of all labelled H-free r-graphs with vertex set {1, 2, …, n}.
We note in passing that results similar to <ref> and <ref> were obtained recently by Conlon, Pohoata and Zakharov <cit.> for H = K_2,2,…,2, the complete r-partite r-graph with two vertices in each part. However, since Sidorenko's conjecture does hold for these graphs through some standard applications of the Cauchy–Schwarz inequality, their proof proceeds along very different lines, making use of a multilinear variant of Bukh's random algebraic method <cit.>.
Motivated by <ref> and its application <ref>, much of this paper is devoted to finding examples of r-partite r-graphs for which Sidorenko's conjecture is false. For instance, if we define the r-uniform loose triangle to be the r-graph with vertex set {1, 2, 3, …, 3r-3} and edges {1, 2, …, r}, {r, r+1, …, 2r-1}, {2r-1, …, 3r-3, 1}, then we have the following result. Note that here a linear r-graph is an r-graph where every pair of edges intersect in at most one vertex.
Any linear r-graph that contains a loose triangle is not Sidorenko.
By the celebrated (6,3)-theorem of Ruzsa and Szemerédi <cit.>, which states that dense linear r-graphs contain loose triangles, we have the following corollary.
For any integer r ≥ 3 and any c > 0, there exists k_0 such that any linear r-graph with k ≥ k_0 vertices and at least c k^2 edges is not Sidorenko.
While the extremal number is known exactly for some sparse linear r-graphs such as loose paths and cycles <cit.>, these results, applied in conjunction with <ref>, give the first polynomial improvement on the lower bound for the extremal number of a broad range of linear r-graphs.
In a somewhat different direction, we look at the tight cycles C_ℓ^(r) with vertex set {1, 2, …, ℓ} and edges {i, i+1, …, i+r-1} for all i = 1, 2, …, ℓ, where addition is taken mod ℓ. From an extremal viewpoint, these are some of the most closely studied hypergraphs (see, for example, <cit.>). We will show that, at least for certain choices of ℓ and r, they are again not Sidorenko. In the statement below, we also consider the r-graphs C_ℓ^(r) - e obtained by deleting a single edge e from C_ℓ^(r).
C_k^(3) is not Sidorenko for any k ≥ 4, C_k^(3) - e is not Sidorenko for any k ≥ 7 and C_2r^(r) is not Sidorenko for any odd r ≥ 3.
There are some recent results <cit.> that determine the Turán densities of C_k^(3) and C_k^(3)-e when k is sufficiently large and not divisible by 3.
<ref> gives the first non-trivial improvement on the lower bounds for the extremal numbers of C_k^(3) and C_k^(3)-e when k is divisible by 3.
We also give some examples of r-graphs with the stronger property that they are not common.
By saying that an r-graph H is common, we mean that t_H(W) + t_H(1-W) ≥ 2^1- e(H)
for every r-graphon W:[0,1]^r → [0,1].
In graph-theoretic terms, an r-graph H is common if the number of monochromatic copies of H in a two-colouring of the edges of K_n^(r) is asymptotically minimised by a quasirandom colouring. The study of such graphs is a central topic in Ramsey theory and we refer the interested reader to <cit.> for further context and additional references. For us, the important point is that
if an r-graph is Sidorenko,
it is automatically common, so non-common r-graphs are automatically not Sidorenko. As it involves some further notation,
we will hold off on giving a full description of our main result in this direction until Section <ref> and instead give an illustrative example.
For r odd, the grid r-graph G_r whose vertices are the points of the r × r grid
and whose edges are the 2r horizontal and vertical lines of the grid is not common.
Unlike our previous results, this does not allow us to give an improved bound for (n, G_r), since, by considering all of the edges containing a fixed vertex, we get the simple lower bound (n, G_r) ≥n-1r-1, which is considerably better than the deletion bound. However, the grid graphs are an interesting and well-studied family (see, for example, <cit.>), so we believe the fact that its odd members are not common is an interesting result in its own right.
§ LOWER BOUNDS FOR THE EXTREMAL NUMBER
In this short section, we will use the tensor power trick to prove Theorem <ref>, the statement that the deletion bound may be improved for counterexamples to Sidorenko's conjecture. We will need the following standard result from the theory of graph limits (see, for example, <cit.>), obtained by sampling n vertices v_1, v_2, …, v_n independently and uniformly at random from [0,1] and placing an edge on each v_i_1, v_i_2, …, v_i_r with i_1 < i_2 < … < i_r independently with probability W(v_i_1, v_i_2, …, v_i_r).
Let W be an r-graphon. Then there exists a sequence (G_n)_n=1^∞ of r-graphs such that |V(G_n)|=n and t_F(G_n) converges to t_F(W) for every fixed r-graph F.
The tensor product G⊗ H of two r-graphs G and H is the graph with vertex set V(G) × V(H) where ((x_1, y_1), (x_2, y_2), …, (x_r, y_r)) ∈ E(G⊗ H) if and only if (x_1, x_2, …, x_r) ∈ E(G) and (y_1, y_2, …, y_r) ∈ E(H). For N a positive integer, we may then define G^⊗ N inductively by G^⊗ 1 = G and G^⊗ N = G ⊗ G^⊗ N-1. A key property of these tensor powers, which we will need below, is that t_H(G^⊗ N) = t_H(G)^N for any graphs G and H and any positive integer N.
Let W be an r-graphon for which t_H(W) < t_K_r(W)^ e(H). If (G_m)_m=1^∞ is a sequence of r-graphs with |V(G_m)| = m given by <Ref>, then, provided m is sufficiently large, t_H(G_m) < t_K_r(G_m)^ e(H). Let G be an r-graph from this sequence for which this is the case and let α_0:=t_K_r(G)=r! e(G)/ v(G)^r and β_0:=t_H(G), so that β_0 < α_0^ e(H). We will assume that G is taken sufficiently large that β_0/α_0 ≥ v(G)^r - v(H).
Set n:= v(G)^N, α := α_0^N and β:=β_0^N. Then G^⊗ N is an n-vertex graph with α n^r/r! edges and at most β n^ v(H) labelled copies of H. Let c' := e(H)logα_0 - logβ_0/log v(G), so that c' > 0 and β = α^ e(H) n^-c'. Crucially, the number of copies of H in G^⊗ N is significantly smaller than the random count of roughly α^ e(H)n^ v(H), allowing us to apply the deletion method more efficiently.
Indeed, if we take a random subgraph (G^⊗ N)_p of G^⊗ N where every edge appears independently with probability p, the expected number of edges X in this subgraph is p α n^r/r! and the expected number of copies Y of H is at most (p α)^ e(H) n^ v(H) -c'. Note that the condition β_0/α_0 ≥ v(G)^r - v(H) is equivalent to α_0^ e(H)-1≥ v(G)^r+c'- v(H), which in turn implies that α^ e(H)-1≥ n^r+c'- v(H). Therefore, there is some p < 1 such that (p α)^ e(H)-1 = n^r+c'- v(H)/(2 r!). But then
𝔼[Y] ≤ (p α)^ e(H) n^ v(H) -c' = p α n^r/2 r! = 1/2𝔼[X],
so, by linearity of expectation, 𝔼[X-Y] ≥1/2𝔼[X] = p α n^r/2 r!. Therefore, there must exist a graph for which we can delete an edge from every copy of H and still leave at least p α n^r/2 r!≥γ n^r- v(H)-r/ e(H)-1+c edges, where γ>0 is an absolute constant and c = c'/( e(H)-1). This yields the required conclusion when n is a power of v(G), but, at the possible expense of replacing γ with a smaller number, we can easily interpolate between these values.
§ NON-SIDORENKO HYPERGRAPHS
§.§ Linear hypergraphs
Recall that an r-graph is linear if every pair of edges shares at most one vertex. The girth of a linear r-graph is the length of the shortest (loose) cycle in the graph. We shall prove the following statement that slightly generalises <Ref>.
In the proof, we will also need to know that, for s≤ r, the (s-1)-skeleton of an r-graph H is the s-graph obtained by replacing each r-edge of H by a copy of K_r^s, the complete s-graph on r vertices, and simplifying multiedges.
If H is a linear r-graph of odd girth, then H is not Sidorenko.
Consider the weighted r-graph on {-1, 1} where the edge (x_1, x_2, …, x_r) ∈{± 1}^r receives the weight f(x_1,…,x_r) = 1-c∑_i<jx_ix_j. For c ≤ 1/k2, f is a non-negative symmetric function with t_K_r(f) = 1.[For convenience, we phrase our example in terms of weighted r-graphs f rather than r-graphons W, but it is easy to convert between the two settings. We also allow f to take values larger than 1, but, since the inequality (<ref>) is homogeneous, it is sufficient for f to be bounded and non-negative.]
Observe that a monomial in the expansion of ∏_v_1… v_r ∈ E(H)f(x_v_1,…,x_v_r) has zero average whenever it contains a variable of odd degree. Thus, the non-vanishing terms in the expansion of t_H(f) correspond to `Berge' even subgraphs F of the 1-skeleton of H, those subgraphs F where every vertex has even degree and every (2-)edge e∈ E(F) extends to a unique r-edge in H. Moreover, every such F receives the weight (-c)^ e(F). Therefore, if g is the girth of H and g is odd, t_H(f) = 1 -Kc^g + O(c^g+1), where K denotes the number of shortest loose cycles. By choosing c>0 small enough, we see t_H(f)<1 = t_K_r(f)^ e(H), so H is not Sidorenko.
It is also possible to generalise <ref>, although we did not find any concrete applications of this more general result. Indeed, by replacing f with f(x_1,…,x_r) = 1-c∑_i_1< … < i_s x_i_1⋯ x_i_s for any s ≤ r, one can show that if the smallest subgraph F of the (s-1)-skeleton of H where every vertex has even degree and every s-edge e ∈ E(F) extends to a unique r-edge in H has an odd number of edges, then H is not Sidorenko. Since, in an s-uniform hypergraph F, ∑_v ∈ V(F) d(v) = s · e(F), such a subgraph F can only exist when s is even.
§.§ Tight cycles
Recall that C_ℓ^(r) denotes an r-uniform tight cycle of length ℓ. Since C_ℓ^(r) and C_ℓ^(r) - e can only be r-partite when ℓ is a multiple of r, in order to prove Theorem <ref>, it will suffice to study tight cycles of the form C_kr^(r).
Given an r-graph H, let κ_m(H) denote the number of subgraphs of H with m edges and no degree-one vertices.
As captured by the following proposition, the polynomial P_H(x) := ∑_i=1^ e(H)κ_i(H) x^i will play an important role in the proof of <Ref>.
Let r be odd and H be a subgraph of C_kr^(r).
If H is Sidorenko, then P_H(x) ≥ 0 for all x ∈ [-1,0].
Suppose that P_H takes a negative value on [-1,0].
Then there exists c ∈ (0,1)
such that P_H(-c) < 0.
For ε∈(0,1), let f_ε be the function on [0,1] defined by
f_ε(x) = ε if x≤1/1+ε
-1 otherwise.
Then ∫_0^1 f_ε dμ=0 and, for any fixed integer d>1 and ε sufficiently small,
∫_0^1 (f_ε)^d dμ = (-1)^d ε + O(ε^2).
Let
g_ε(x_1,…,x_r) := ∏_i=1^r f_ε(x_i), so that
t_G(g_ε)=0 whenever G has a vertex of degree one.
Moreover, for every n-vertex r-graph G with degree sequence d_1,…,d_n≥ 2,
t_G(g_ε) = (-1)^∑_i=1^nd_iε^ v(G) +O(ε^ v(G)+1) = (-1)^ e(G)ε^ v(G) +O(ε^ v(G)+1),
since r· e(G)=∑_i=1^n d_i and e(G) have the same parity.
Let h_ε,c := 1 + cg_ε, noting that this function is non-negative.
Then
t_H(h_ε,c) = 1 + ∑_G ⊆ H c^ e(G) t_G(g_ε),
where the sum is taken over all non-empty edge subsets of H, which can be seen as subgraphs G of H.
In any subgraph of the tight cycle C_kr^(r),
degrees of consecutive vertices of the cycle differ by at most one.
Thus, in a non-empty subgraph G with no vertices of degree one,
no isolated vertices exist and, hence, all but those G with minimum degree at least two vanish in (<ref>).
Therefore, κ_m(H) counts the number of m-edge subgraphs of H on kr vertices with minimum degree at least two. It then follows that
t_H(h_ε,c) = 1 + ∑_G ⊆ H, δ(G)≥ 2 c^ e(G) t_G(g_ε)
=1 + ε^kr∑_G ⊆ H, δ(G)≥ 2 (-c)^ e(G) + O(ε^kr+1) = 1+ε^krP_H(-c) +O(ε^kr+1).
Therefore, for sufficiently small ε>0,
t_H(h_ε,c) < 1. But ∫ h_ε,c dμ^r = 1, so this contradicts our assumption that H is Sidorenko.
We will now use <ref> to prove the following three results, which together make up <ref>.
C_3k^(3) is not Sidorenko for k ≥ 2.
C_3k^(3)-e is not Sidorenko for k ≥ 3.
C_2r^(r) is not Sidorenko for any odd r ≥ 3.
For the proofs, we will need to better understand the functions κ_m(H) for the r-graphs H under consideration.
κ_i(C_3k^(3))=0 for i<2k and
κ_2k+i(C_3k^(3)) = 3k/k+2ik+2i3i
for 0 ≤ i ≤ k.
A subgraph G of C_3k^(3) with i edges
such that each vertex has degree 2 or 3
must be obtained from C_3k^(3) by removing
3k-i disjoint edges.
But the number of disjoint edges cannot exceed k, and
the number of ways to select 1 ≤ j ≤ k
independent edges in C_3k^(3) is
3k/j3k-1-2jj-1.
Thus, κ_i(C_3k^(3))=0 for i<2k and
κ_2k+i(C_3k^(3)) =
3k/k-i3k-1-2(k-i)k-i-1
= 3k/k-ik+2i-1k-i-1
= 3k/k+2ik+2ik-i
= 3k/k+2ik+2i3i
for 0 ≤ i ≤ k-1.
Moreover, κ_3k(C_3k^(3))=1.
κ_i(C_3k^(3)-e)=0 for i<2k
and κ_2k+i(C_3k^(3)-e) = k+2i-13i
for 0 ≤ i ≤ k-1.
The statement follows from <ref> and
the fact that
κ_i(C_kr^(r)-e) = kr-i/krκ_i(C_kr^(r)).
We also need to verify some elementary inequalities.
For integers k ≥ 2 and i≥ 1,
(k+2i+1)(k+2i)(k-i)/(3i+3)(3i+2)(3i+1)≤k^3+k^2/60.
It is easy to check that each of the ratios
k+2i+1/3i+3,
k+2i/3i+2,
k-i/3i+1
decreases with i.
Hence,
(k+2i+1)(k+2i)(k-i)/(3i+3)(3i+2)(3i+1)≤(k+3)(k+2)(k-1)/120
= k^3 + 4k^2 + k - 6/120.
We therefore need to show that
k^3 + 4k^2 + k - 6 ≤ 2(k^3+k^2),
which is equivalent to
F(k) := k^3 - 2k^2 - k + 6 ≥ 0.
But this follows since F(2)=4
and F'(k) = 3 k^2 - 4k - 1 = 3k(k-2) + 2k - 1 > 0
for k ≥ 2.
For integers k ≥ 3 and i≥ 1,
(k+2i+1)(k+2i)(k-i-1)/(3i+3)(3i+2)(3i+1)≤7/600 (k^3-k).
It is easy to check that each of the ratios
k+2i+1/3i+3,
k+2i/3i+2,
k-i-1/3i+1
decreases with i.
Hence,
(k+2i+1)(k+2i)(k-i-1)/(3i+3)(3i+2)(3i+1)≤(k+3)(k+2)(k-2)/120
= k^3 + 3k^2 - 4k - 12/120.
We therefore need to show that
k^3 + 3k^2 - 4k - 12 ≤7/5 (k^3-k),
which is equivalent to
F(k) := 2k^3 - 15k^2 + 13k + 60 ≥ 0.
But this follows since F(3)=18, F(4)=F(5)=0
and F'(k) = 6k^2 - 30k + 13 = 6k(k-5) + 13 > 0
for k ≥ 5.
We are already in a position to prove Theorems <ref> and <ref>.
By <ref>, it will be sufficient to find some
x ∈ [-1,0] such that P_C_3k^(3)(x) < 0.
The coefficients of P_C_3k^(3) are given by <ref>.
It is easy to check that
P_C_6^(3)(x) = x^4 (3+6x+x^2)
is negative at x=-2/3.
Thus, we may assume that k ≥ 3.
For a fixed k and 1 ≤ i ≤ k, set
A_i := 3k/k+2ik+2i3i (30/k^3+k^2)^i .
By <ref>, for 1 ≤ i ≤ k-1,
A_i+1/A_i = (k+2i+1)(k+2i)(k-i)/(3i+3)(3i+2)(3i+1)·30/k^3+k^2 ≤ 1/2.
Set x=30/k^3+k^2.
As A_2j≤1/2 A_2j-1≤ A_2j-1,
we get
x^-2k P_C_3k^(3)(-x) = 3 + ∑_i=1^k (-1)^i A_i
≤ 3 - A_1 + A_2
≤ 3 - 1/2 A_1
= 3 - 1/2 3k/k+2 k+23 30/k^3+k^2 = 3 - 15/2 < 0,
as required.
By <ref>,
it will be sufficient to find some
x ∈ [-1,0] such that P_C_3k^(3)-e(x) < 0.
The coefficients of P_C_3k^(3)-e are given by <ref>.
It is easy to check that
P_C_9^(3)-e(x) = x^6 (1+4x+x^2)
is negative at x=-2/3.
Thus, we may assume k ≥ 4.
For a fixed k and 1 ≤ i ≤ k-1, set
B_i := k+2i-13i (300/7(k^3-k))^i.
By <ref>, for 1 ≤ i ≤ k-2,
B_i+1/B_i = (k+2i+1)(k+2i)(k-i-1)/(3i+3)(3i+2)(3i+1)·300/7(k^3-k) ≤ 1/2.
Set x = 300/7(k^3-k).
As B_2j≤1/2 B_2j-1≤ B_2j-1,
we get
x^-2k P_C_3k^(3)-e(-x) = 1 + ∑_i=1^k-1 (-1)^i B_i
≤ 1 - B_1 + B_2
≤ 1 - 1/2 B_1
= 1 - 1/2k+13 300/7(k^3-k) = 1 - 25/7 < 0,
as required.
For the proof of <ref>, we need to do a little more work.
Consider an m-element subset A⊆_2r and assume that its elements are cyclically ordered as
A=(x_0,x_1,…,x_m=x_0).
We say that x_i is good if
x_i+1-x_i-1∈{2,…,r}
and bad otherwise.
The number of m-element subsets A⊆_2r
that have at least one bad element is
2r(m-2)rm-1 for
m ≥ 4.
Suppose A=(x_0, x_1, …, x_m=x_0) is such a subset.
Notice that if x_i and x_j are two distinct bad points,
then i-j = ± 1.
Hence, there is either just one bad point or there are two consecutive bad points.
Thus, there exists a unique index j such that x_j is good
and x_j-1 is bad.
Without loss of generality, we may assume that j=1.
x_1 can have any of the 2r possible values.
We will assume that x_1=0 and show that there are then
exactly (m-2) rm-1 choices
for x_0,x_2,…,x_m-1.
As x_1 is good, x_2-x_0 ∈{2,…,r}.
As x_0 is bad, x_1-x_m-1∈{r+1,…,2r-1},
so x_m-1∈{1,…,r-1}.
If x_2=i, there are r-i choices for x_0
and (r-1)-im-3 choices for x_3,…,x_m-1.
Thus, the total number of choices is
∑_i=1^r-1 (r-i) (r-1)-im-3 = ∑_i=1^r-1 (m-2) r-im-2 =
(m-2) rm-1,
as required.
Note that a subgraph H of the tight cycle C_2r^(r) has no degree-one vertices if and only if the set A of initial vertices of edges in H contains no bad elements. Thus, we have the following immediate corollary of <ref>.
κ_i(C_2r^(r)) =
0 if i ≤ 3,
2ri - 2r(i-2)ri-1 if 4 ≤ i ≤ 2r.
By <ref>, since 2r(m-2)rm-1 = 0 for r+2 ≤ m ≤ 2r,
P_C_2r^(r)(x) = ∑_i=4^2r2ri x^i - ∑_i=4^r+1 2r(i-2)ri-1 x^i
= ∑_i=0^2r2ri x^i - (1 + 2rx + r(2r-1) x^2 + 2/3 r(r-1)(2r-1) x^3)
- 2r ∑_i=1^r+1 (i-2)ri-1 x^i
+ 2r (-x + 1/2 r(r-1) x^3)
=
(1+x)^2r + 2r ∑_i=1^r+1ri-1 x^i
- 2r ∑_i=1^r+1 (i-1)ri-1 x^i
- (1 + 4rx + r(2r-1) x^2 + 1/3 r(r-1)(r-2) x^3)
=
(1+x)^2r + 2rx(1+x)^r - 2r^2 x^2 (1+x)^r-1
- (1 + 4rx + r(2r-1) x^2 + 1/3 r(r-1)(r-2) x^3).
If r ≥ 16, set x = -1/r.
Then (1+x)^2r < e^-2 and (1+x)^r + (1+x)^r-1≥ 2e^-1, so that
P_C_2r^(r)(x) <
e^-2 - 4e^-1 + 4/3 + 2/3r^2 < -0.002849 + 2/3r^2 < 0.
If r ≤ 16, set x = -2/r.
Then (1+x)^2r < e^-4 and
4(1+x)^r + 8(1+x)^r-1 =
4((1+x)^r + (1+x)^r-1) + 4(1+x)^r-1 >
8e^-2 + 4e^-2 = 12e^-2,
so that
P_C_2r^(r)(x) <
e^-4 - 12e^-2 + 5/3 - 4/r + 16/3r^2 < 0.060959 - 4/r + 16/3r^2 < 0.
Therefore, by <ref>, C_2r^(r) is not Sidorenko for any odd r ≥ 3.
To conclude this section, we note that we have also used <ref> to show that C_kr^(r) is not Sidorenko for all values of k and r with r ≥ 5 odd and kr ≤ 30. This suggests, and we conjecture, that C_kr^(r) is not Sidorenko for any odd r≥ 5 and k ≥ 2.
§ NON-COMMON HYPERGRAPHS
Recall that an r-graph H is common if
t_H(W) + t_H(1-W) ≥ 2^1- e(H)
for any r-graphon
W:[0,1]^r → [0,1] and that every Sidorenko
hypergraph is automatically common.
By substituting W=1+f/2,
we can rewrite the requirement for H to be common as
t_H(1+f)+t_H(1-f) ≥ 2
for any r-variable symmetric measurable function
f:[0,1]^r → [-1,1].
By expanding out, this inequality is equivalent to
∑_G ⊆ H, e(G)
≡ 0 mod 2, e(G)>0
t_G(f)
≥ 0.
If <ref> fails for some function f,
then H is not common and, hence, is not Sidorenko.
To state the main result of this section, we need some definitions.
Following Camarena et al. <cit.>, we say that an r-graph H is positive if
t_H(W) ≥ 0 for any r-variable symmetric function W:[0,1]^r → [-1,1]. When
r ≥ 3, we say that an r-graph is 2-connected if the removal of a single vertex or a single edge
does not disconnect it, while, for r = 2, we just mean the usual notion,
that a graph is 2-connected if it is not disconnected by the removal of a single vertex.
Let r be odd or r=2.
If an r-graph H has a non-positive 2-connected subgraph with 2m edges
and every other subgraph with an even number of edges not exceeding 2m
is either non-positive and 2-connected or has a vertex of degree 1,
then H is non-common.
When r ≥ 3, examples coming from <ref> are quite plentiful.
To see this, we will make use of the following proposition
from <cit.>. Here the Levi graph of an r-graph H
is the bipartite graph L(H) with vertex set V(H) ∪ E(H)
where v ∈ V(H) and e ∈ E(H) are adjacent if and only if
v ∈ e in H. Note that for r ≥ 3 the r-graph H is 2-connected if and only if its
Levi graph L(H) is 2-connected in the usual sense.
If an r-graph H is positive,
then its Levi graph L(H) is positive.
When r is odd,
L(H) is positive if and only if H is positive.
Consider the half-octahedron G, the 3-graph
with vertices 1,2,3,4,5,6 and edges
{1,3,5}, {1,4,6}, {2,3,6}, {2,4,5}.
Its Levi graph L(G) has 10 vertices and 12 edges.
With a single exception, all positive graphs with at most 10 vertices
are classified in <cit.>
and L(G) is not one of them.
Hence, by <ref>, G is non-positive.
Therefore, if all 4-edge subgraphs of a 3-graph H
are either isomorphic to G or have a vertex of degree 1,
then, by <ref>, H is non-common and non-Sidorenko.
Recall that G_r is the grid r-graph
whose vertices are the points of the r × r grid
and whose edges are the 2r horizontal and vertical lines of the grid.
It was shown in <cit.> that G_r is not positive for odd r.
Since any proper subgraph of G_r has a vertex of degree 1,
<ref> implies that G_r is non-common for odd r.
Moreover, if we add more edges to G_r without creating new subgraphs
whose minimum vertex degree is at least 2,
then the resulting r-graph will remain non-common.
The next statement was proved in <cit.> for r=2,
but the proof can be repeated verbatim for an arbitrary r.
An r-graph G is positive if and only if every connected r-graph
that occurs among the connected components of G an odd number of times
is positive.
We also note the following result.
In the proof, we often consider the tensor product f⊗ g of r-variable symmetric functions f and g, defined by
(f⊗ g)((x_1,y_1),…,(x_r,y_r)) = f(x_1,…,x_r) g(y_1,…,y_r),
where we identify each ((x_1,y_1),…,(x_r,y_r)) with a point in [0,1]^r through a measure-preserving bijection from [0,1]^2r to [0,1]^r.
If the r-graphs G_1,…,G_k are not positive,
then there exists a function f
such that t_G_i(f) < 0 for all i=1,…,k.
We use induction on k.
The base case k=1 is trivial, so we consider the induction step going from k-1 to k ≥ 2.
For each j=1,…,k, by the induction hypothesis,
there exists a function f_j such that
t_G_i(f_j) < 0 for all i ≠ j.
If necessary, we perturb f_j by a little bit to ensure that t_G_j(f_j) ≠ 0
while preserving t_G_i(f_j) < 0 for all i ≠ j.
If t_G_j(f_j) < 0, then f_j is the function we need.
Thus, we may assume that t_G_j(f_j) > 0.
If k is even,
f=f_1 ⊗⋯⊗ f_k
satisfies t_G_i(f) < 0 for all i=1,…,k.
Suppose then that k is odd.
Let G be the union of disjoint copies of
G_1,…,G_k.
By <ref>, G is not positive,
so there exists a function f_0 such that t_G(f_0) < 0.
Notice that t_G(f_0) = ∏_i=1^k t_G_i(f_0).
We may assume that t_G_i(f_0) > 0 for 1 ≤ i ≤ m
and t_G_i(f_0) < 0 for m+1 ≤ i ≤ k,
where m is even.
If m=0, then f_0 is the function we need.
If m>0,
then f=f_0⊗ f_1 ⊗⋯⊗ f_m
satisfies t_G_i(f) < 0 for all i=1,…,k, so we again have the required function.
The following result and its corollaries are the key to proving <ref>.
Note that we call an r-variate symmetric measurable function f
zero-averaging if
∫ f(x_1,…,x_r-1,x_r) dx_r = 0
for any x_1,…,x_r-1.
If H is a non-positive 2-connected graph with an even number of edges, then
there exists a zero-averaging function f:[0,1]^2→ [-1,1]
such that t_H(f)<0.
Our plan is to construct a {± 1}-weighted graph Γ such that every vertex has a vanishing sum over the weights of its incident edges and t_H(Γ)<0.
Let U:[0,1]^2→ [-1,1] be a measurable symmetric function, i.e., a signed graphon, that satisfies t_H(U)<0.
By using the standard decomposition U=U^+ - U^- and applying <Ref> to find appropriate graphs approximating the graphons U^+ and U^-, one may obtain a {± 1}-weighted graph G such that t_H(G)<0.
Let d:= v(G) and s:= v(H) for brevity and write w_H(G) = d^s t_H(G). We may then assume that w_H(G)<-d^s-1/2 by replacing G with a blow-up by a sufficiently large factor. Note that, as H has an even number of edges, w_H(-G)=w_H(G), where -G denotes the graph with edge weights of opposite sign from G.
For any sufficiently large even n, there is a d-regular n-vertex bipartite graph F with girth greater than s (see, for example, <cit.>). Partition the edges of F into d perfect matchings, colouring the edges of each matching with one of the d colours from [d].
Now consider the line d-graph
of F whose vertices are the edges of F and whose d-edges are the collections of d edges in F incident to each vertex of F.
A {± 1}-weighted d-graph ℱ is defined by assigning +1 or -1 to each edge in this line d-graph, depending on which side of the bipartition the corresponding vertex of F lies.
The required zero-averaging weighted graph Γ is then obtained by replacing each d-edge in ℱ by a copy of G, where we map each vertex of V(G)=[d] to the corresponding coloured vertex (with the colouring inherited from the matchings) and multiply the weight on each edge of G by the {± 1}-weight on the corresponding edge of ℱ.
We claim that t_H(Γ)<0. To prove this, we say that a (weighted) homomorphism from H to Γ is good if the homomorphic image of H lies in one d-edge of ℱ. The weighted sum of good homomorphisms is a negative number less than n· w_H(G)< -n d^s-1/2, where we used the fact that w_H(G)=w_H(-G).
On the other hand, there may be some homomorphic images of H that are not entirely covered by a single d-edge, which we call bad.
As the girth of F is larger than the number of vertices in H, the unique minimal collection of d-edges whose union contains a fixed bad image of H must form a d-hypertree in ℱ, where uniqueness follows from the fact that ℱ is a linear hypergraph.
As ℱ is linear, deleting a vertex v lying in the intersection of two d-edges in a d-hypertree disconnects the subgraph of Γ induced on the vertex set of the d-hypertree. In particular, if a bad image of H contains v,
then the image of H is disconnected once v is deleted. As H is a 2-connected graph, the bad image of H must therefore be degenerate, i.e., there are at least two vertices of H that are mapped to v.
Suppose that there are t+1 edges in a d-hypertree 𝒯⊆ℱ for some t≥ 1. Then there are (t+1)d-t vertices in 𝒯, exactly t of which have degree two (here we used that each vertex in ℱ has degree exactly two). If 𝒯 is a minimal cover of a bad homomorphic image of H, then there are t disjoint pairs of vertices in H, each of which maps to a unique one of the t vertices of degree two.
Thus, there are at most s2^t((t+1)d-t)^s-2t bad homomorphic copies of H whose minimal cover is 𝒯.
Given a d-hypertree 𝒯 with t+1 edges, one can recover the tree in F corresponding to the union of the d-edges of 𝒯 in three steps: replace each d-edge by the corresponding vertex in F; connect those vertices that correspond to intersecting d-edges; and turn each degree-one vertex in a d-edge e of 𝒯 into a leaf adjacent to the unique vertex of F which corresponds to e.
Note that the leaves added in the last step are determined once the first two steps give a (t+1)-vertex tree in F. Thus, there are at most nd^t isomorphic images of 𝒯 in ℱ.
Therefore, there are at most s^2t(t+1)^s-2tnd^s-t bad homomorphic images of H whose minimal cover is isomorphic to 𝒯. Hence, as 1≤ t<s and the number of distinct d-hypertrees with t+1 edges and maximum degree two is bounded as a function of t, the number of bad homomorphic images of H is at most Cnd^s-1 for a constant C=C(s). This is asymptotically smaller than -nd^s-1/2, the upper bound for the weighted sum of good homorphisms provided d is sufficiently large, so that t_H(Γ) is negative, as required.
Let r ≥ 3 be odd. If H is a non-positive 2-connected r-graph with an even number of edges,
then there exists an r-variate zero-averaging function h
such that t_H(h) < 0.
By <ref>, the Levi graph L(H) of H is a non-positive 2-connected graph with an even number of edges.
Therefore, by <ref>,
there exists a 2-variate zero-averaging function f
such that t_L(H)(f) < 0.
Consider the r-variate symmetric measurable function h given by
h(x_1,…,x_r) = ∫∏_i=1^r f(x_i,y) dy.
It is easy to see that h is zero-averaging and
t_H(h) = t_L(H)(f) < 0.
If G_1,…,G_k are non-positive 2-connected graphs each with an even number of edges,
then there exists a zero-averaging function f
such that t_G_i(f) < 0 for all i=1,…,k.
Using <ref>, we proceed exactly as in the proof of <ref>.
Let r ≥ 3 be odd.
If G_1,…,G_k
are non-positive 2-connected r-graphs each with an even number of edges,
then there exists a zero-averaging function f
such that t_G_i(f) < 0 for all i=1,…,k.
Using <ref>, we proceed exactly as in the proof of <ref>.
We are now in a position to prove <ref>.
Let us assume that m in the statement of the theorem
is as small as possible,
that is, there are non-positive 2-connected
subgraphs G_1,…,G_k with 2m edges
and every other subgraph with an even number of edges not exceeding 2m
has a vertex of degree 1.
By <ref> (if r=2) or <ref> (if r is odd),
there exists a function f such that
S := t_G_1(f) + ⋯ + t_G_k(f) < 0 and
t_G(f)=0 for any r-graph G that has a vertex of degree 1.
Hence, for ε > 0 sufficiently small,
∑_G ⊆ H, e(G) ≡ 0 mod 2, e(G)>0 t_G(ε f)
= ε^2m S + O(ε^2m+1)
< 0,
so H is not common.
§ CONCLUDING REMARKS
We say that an r-graph H is locally Sidorenko if there exists ε > 0 such that t_H(W) ≥ t_K_r(W)^ e(H) for all r-graphons W with W-1/2_≤ε. That is, H is locally Sidorenko if the required inequality holds for all r-graphons which are sufficiently close to the uniform graphon 1/2, where closeness is measured in terms of the cut norm (see, for example, <cit.>). Since it is probably difficult to give a complete characterisation of those r-graphs which are Sidorenko, we instead conclude by asking for a characterisation of locally Sidorenko r-graphs.
Which r-graphs are locally Sidorenko?
It was shown by Lovász <cit.> that every bipartite graph is locally Sidorenko and later, by Fox and Wei <cit.>, that a graph is locally Sidorenko if and only if it is either a forest or has even girth. The results of Section <ref> are all proved by showing that the relevant r-graphs are not locally Sidorenko and may help give some hints as to what a full characterisation should look like. However, at present, we have no concrete conjectures, even in the r-partite case most relevant to us. Indeed, despite Theorem <ref>, it is already open to determine which tight cycles are locally Sidorenko for r ≥ 4.
plainurl
|
http://arxiv.org/abs/2307.04527v1 | 20230710125059 | Automatic Debiased Machine Learning for Covariate Shifts | [
"Michael Newey",
"Whitney K. Newey"
] | stat.ME | [
"stat.ME"
] |
Automatic Debiased Machine Learning for Covariate Shifts
Michael Newey and Whitney K Newey Research was sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. This research was supported by NSF Grant 1757140
August 12, 2023
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In this paper we address the problem of bias in machine learning of parameters following covariate shifts. Covariate shift occurs when the distribution of input features change between the training and deployment stages. Regularization and model selection associated with machine learning biases many parameter estimates. In this paper, we propose an automatic debiased machine learning approach to correct for this bias under covariate shifts. The proposed approach leverages state-of-the-art techniques in debiased machine learning to debias estimators of policy and causal parameters when covariate shift is present. The debiasing is automatic in only relying on the parameter of interest and not requiring the form of the form of the bias. We show that our estimator is asymptotically normal as the sample size grows. Finally, we evaluate the method using a simulation.
§ INTRODUCTION
In applications, machine learners trained on one data set may be used to estimate parameters of interest in another data set that
has a different distribution of predictors.
For example, training data could be a sub-population of a larger population or the training and estimation could take place at different times where the distribution of predictors varies between times.
A case we consider in this work is a neural network trained to predict on one data set and then used to learn average outcomes from previously unseen data on predictor variables.
Another case, from causal statistics, is a Lasso regression trained on outcome, treatment, and covariate data that is used to estimate counterfactual averages in another data set with a different distribution of covariates.
This is an important case of distribution shift that is known as covariate shift <cit.>.
Such covariate shifts are of interest in a wide variety of settings, including
estimation of
counterfactual averages and causal effects with shifted covariates
<cit.>.
Additionally, covariate shifts are interesting for classification where the training data may differ from
the field data <cit.>. There are many important
parameters that depend on covariate shifts, including average outcomes or average potential outcomes learned from shifted covariate data.
This paper concerns machine learning of parameters of interest in field data
that depend on regressions in training data. An important problem with
estimation is the bias that can result from regularization and/or model
selection in machine learning on training data. In this paper we address this
problem by giving automatic debiased machine learners of parameters of
interest. The debiasing is automatic in only requiring the object of interest
and in not requiring a full theoretical formula for the bias correction.
The debiased estimators given are obtained by plugging a training data
regression into the formula of interest in the field data and adding a debiasing
term. The debiasing term consists of an average product in the training data
of a debiasing function and regression residuals. The debiasing function is
estimated by linking the field data with certain features of the training data
in a way that only uses the formula for the parameter. The debiased estimators
have a form similar to those of <cit.>. The estimators here
differ from previous estimators in that training and field data come from
different sources, with field data being statistically independent of training data and the distribution of covariates being different in the field data than the training data.
In this paper we will describe the estimator, provide the underlying theory,
and report results of a simulation on artificial data.
§ PARAMETERS OF INTEREST AND DEBIASED ESTIMATING EQUATIONS
The parameters of interest we consider depend on a regression where Y is an
outcome variable of interest, X is a vector of regressors, and
γ_0(X)=E(Y|X).
Here γ_0(X) is the conditional expectation of the outcome variable
Y given regressors X. The parameter of interest will also depend on a
vector of random variables Z, that will often be a shifted vector of
regressors, with the same dimension as X but a different distribution than
X. This setup is meant to apply to settings where γ_0(X) is a
nonparametric regression for training data and the parameter depends on field
data Z. The parameter θ_0 we consider will be the expectation of a
functional m(Z,γ) of a vector of variables Z and a possible
regression γ, given by
θ_0=E[m(Z,γ_0)].
We focus on parameters where m(Z,γ) depends linearly on the
function γ.
An example of this type of parameter has m(Z,γ
)=γ(Z) so that
θ_0=E[γ_0(Z)].
In this example, θ_0 is the expectation of Y when the regressor
distribution is shifted from that of X to Z and the regression function. Here θ_0 quantifies what the
expectation of Y would be if the covariate distribution shifted from that of
X to that of Z. For example, if Y is a binary variable for classification
into one of two groups, then θ_0 is the classification probability
that would be produced by the field data.
Another example, from causal statistics, is an average potential outcome in
field data using a conditional mean from training data. Here X=(D,X_2) for
a discrete treatment D and covariates X_2 and Y=Y(D) where each
potential outcome Y(d) is independent of D conditional on X_2 in the
training data. Let Z be random vector in field data with the same dimension
as X_2 but with possibly a different distribution than X_2. The object
of interest is
θ_0=E[γ_0(d,Z)].
In this example, when treatment is independent of the potential outcomes
conditional on X_2, the θ_0 is an average of potential outcome
Y(d) when covariates X_2 have been shifted to Z. This object can be an
average potential outcome in the field data. Here m(z,γ)=γ(d,z)
for some specific possible treatment d.
For estimation we explicitly consider the case with distinct training and
field data, as in the motivating examples. Here Z and (Y,X) are from
distinct data sets that do not overlap. We will consider a regression learner
γ̂(X) that is computed from training data with T samples
(Y_1,X_1),...,(Y_T,X_T).
Estimators of the object of interest also make use of N samples of field
data
Z_1,...,Z_N.
A plug-in estimator of the parameter of interest can be constructed from a
learner γ̂ of the conditional mean based on the training data.
Replacing the expectation in equation (<ref>) with a sample
average over the field data Z_i and the true conditional expectation
γ_0 with an estimator γ̂ gives
θ̃=1/N∑_i=1^Nm(Z_i,γ̂).
This plug-in estimator of the parameter of interest is known to suffer from
biases resulting from regularization and/or model selection in γ̂,
as discussed in <cit.>. We will now describe an
alternative, debiased estimator that reduces the bias in θ̃
that comes from γ̂.
The debiased estimator is obtained by using the training data to form a bias
correction for the plug-in estimator. The bias correction is based on a
function of the training data given by
ϕ(Y,X,γ,α)=α(X)[Y-γ(X)],
where α(X) is a function that helps lower bias. We will assume that
there is some some true, unknown debiasing function α_0(X) with
finite second moment such that
E[m(Z,Δ)]=E[α_0(X)Δ(X)], for all Δ(X) with
E[Δ(X)^2]<∞.
From the Riesz representation theorem it is known that existence of such an
α_0 is equivalent to mean square continuity of E[m(Z,Δ)] in
Δ, meaning that there is a constant C with | E[m(W,Δ
)]| ^2≤ CE[Δ(X)^2] for all Δ with finite second
moment. To see how such an α_0(X) helps in debiasing, note that for any
γ(X), Δ(X)=γ(X)-γ_0(X), and α(X) taking
expectations we have
E[m(Z,γ)]-θ_0+E[ϕ(Y,X,γ,α)]
=E[m(W,Δ
)]+E[α(X)(Y-γ(X))]
=E[α_0(X)Δ(X)]-E[α(X)Δ(X)]
=E[(α_0
(X)-α(X))Δ(X)],
where the first equality follows by linearity of m(Z,γ) in γ and
the second equality by iterated expectations. In this way adding the training
data term E[ϕ(Y,X,γ,α)] to the identifying term E[m(Z,γ
)]-θ_0 makes the expectation differ from θ_0 by only the
expected product term following the last equality. In particular, when
α=α_0 the expression preceding the first equality is zero so
that adding E[ϕ(Y,X,γ,α)] to E[m(Z,γ)]-θ_0
exactly cancels the effect of γ. Also, when γ(X)=γ_0(X)
the expression preceding the first equality is zero, even when
α(X)≠α_0. Thus, the presence of the bias correction term
ϕ(Y,X,γ_0,α) does not affect the expectation even though
α≠α_0. The fact that the expectation preceding the first
equality is zero when either γ or α is not equal to its true
value (but one of them is), is a double robustness property shown by <cit.> for linear functions of a regression.
Using this bias correction to estimate the parameter of interest depends
crucially on being able to estimate the α_0(X) of equation
(<ref>). This α_0(X) can be identified as the
minimizing value of the expectation of a known function of α,
α_0=min_αE[-2m(Z,α)+α(X)^2].
To see that α_0 is identified in this way, we note that by adding and
subtracting C=E[α_0(X)^2] and completing the square we obtain
E[-2m(Z,α)+α(X)^2] =-C+E[α_0(X)^2-2α_0
(X)α(X)+α(X)^2]
=-C+E[(α_0(X)-α(X))^2].
This justification of of equation (<ref>) is similar to that in <cit.> where Z is understood to come from field data and Y and X from training data.
Here we see that the objective function of equation (<ref>) does
indeed have a unique minimum at the α_0(X) of equation
(<ref>). Consequently an estimator of α_0 can be
constructed by minimizing a sample version of the objective function in
equation (<ref>). Thus α̂ can be used to construct a
bias correction by adding a training sample average of ϕ(Y,X,γ̂,α̂) to the plug-in estimator. We describe this debiased machine
learner in the next section.
§.§ Estimation with Cross-Fitting
One kind of debiased machine learner can be based on cross-fitting, a form of
sample splitting. Cross-fitting is known to further reduce bias for some
estimators and to help obtain large sample inference results for a variety of
regression learners as in <cit.>. The cross-fitting will
average over different data than used in the construction of γ̂.
To describe the cross-fitting let I_ℓ,(ℓ=1,...,L) denote a partition
of the training set sample indices into L distinct subsets of about equal
size and let T_ℓ be the number of observations in I_ℓ. In
practice L=5 (5-fold) or L=10 (10-fold) cross-fitting is often used. Also
let γ̂_ℓ(x) and α̂_ℓ(x) respectively be
estimators of γ_0 and α_0 computed from all observations not
in I_ℓ, where α̂_ℓ(x) will be described in what
follows. For each fold ℓof the cross-fitting a debiased machine
learner can be constructed as the sum of a plug-in estimator and a bias correction
θ̂_ℓ=1/N∑_i=1^Nm(Z_i,γ̂_ℓ)+1/T_ℓ∑_t∈ I_ℓα̂_ℓ(X_t)[Y_t
-γ̂_ℓ(X_t)].
This estimator is the sum of a plug-in term and a bias correction term that is
motivated by the bias correction described above.
To estimate the asymptotic variance for each θ̂_ℓ we trim the estimator of the debiasing function to obtain α̃_ℓ(X)=τ_n(α̂_ℓ(X)) where
τ_n(a)=1(|a|<τ̅_n)a+1(|a|≥τ̅_n)sgn(a)τ̅_n
and τ̅_n is a large positive constant that grows with n.
The purpose of this trimming is guarantee consistency of the asymptotic variance estimator when we only have mean square convergence rates for the estimators of the regression and the debiasing function.
The asymptotic variance of √(N)(θ̂_ℓ-θ_0) can be estimated as
V̂_ℓ =ŝ_ℓ m^2+ŝ_ℓα^2,
ŝ_ℓ m^2 =1/N∑_i=1^N{m(Z_i,γ̂_ℓ)-m̅_ℓ}^2,
m̅_ℓ =1/N∑_i=1^Nm(Z_i,γ̂_ℓ)
ŝ_ℓα^2 =1/T_ℓ
∑_t∈ I_ℓα̃_ℓ(X_t)^2[Y_t-γ̂_ℓ(X_t)]^2
A single bias corrected estimator and asymptotic variance estimator can then
be obtained by a weighted average of the estimators across the sample splits,
θ̂=∑_ℓ=1^LT_ℓ/Tθ̂_ℓ,V̂=∑_ℓ=1^LT_ℓ/TV̂_ℓ.
An estimator α̂_ℓ(x) of the α_0(x) from equation
(<ref>) is needed for each θ̂_ℓ. Estimators
can be constructed by replacing the expectations of -2m(Z,α) and
α(X)^2 in equation (<ref>) by respective sample averages
over field and training data and minimizing over α in some feasible,
approximating class of functions. A penalty term or terms can be added to
regularize when the feasible class of functions is high dimensional.
We construct α̂_ℓ(x) using a feasible class of functions that
are linear combinations of a dictionary b(x)=(b_1(x),...,b_J(x))^' of approximating functions. We replace the expectation in equation
(<ref>) by sample averages over the test and training data
respectively and minimize a penalized version of the objective function over
all linear combinations of b(x). To describe this α̂_ℓ(x)
let
M̂_j=1/N∑_i=1^Nm(Z_i,b_j),M̂=(M̂_1,...,M̂_J)^',Q̂_̂ℓ̂=1/T-T_ℓ
∑_t∉ I_ℓb(X_t)b(X_t)^'.
We consider α̂_ℓ(x)=b(x)^'ρ̂_ℓ where
ρ̂_̂ℓ̂ =min_ρ{-21/N∑_i=1^N
m(Z_i,b^'ρ)+1/T-T_ℓ∑_t∉ I_ℓ
[b(X_t)^'ρ]^2+r∑_j=1^J|ρ_j|}
=min_ρ{-2M̂^'ρ+ρ^'Q̂_̂ℓ̂
ρ+r∑_j=1^J|ρ_j|}.
Here ρ̂_ℓ minimizes an objective function where the linear term
is obtained from the test data, the quadratic term from the training data, and
an absolute value penalty is included. This estimator differs from that of
<cit.> in the linear and quadratic terms coming
from different data.
§.§ Estimation with Lasso Regression and No Cross-fitting
The cross fitting used to construct the estimator θ̂ requires
repeatedly reusing the training data. In particular the debiasing requires
reusing the training data to estimate γ̂_ℓ and α̂_ℓ for all observations not in I_ℓ as and averaging α̂_ℓ(X_i)[Y_i-α̂_ℓ(X_i)] over observations in
I_ℓ for each split ℓ. When a single Lasso regression estimator
γ̂ is used, based on all the training data, it is possible to do
the bias correction without sample splitting and so reduce greatly the
requirement to reuse the training data. The only training data items needed
for the bias correction will be the second moment matrix of the Lasso
dictionary and the average product of the Lasso dictionary with the Lasso
residuals Y_i-γ̂(X_i).
To describe the estimator let b(x) be the same J×1
dictionary of functions used for construction of α̂ and let r>0 denote a penalty degree. The Lasso regression estimator
from the training data is
γ̂(x)=b(x)^'β̂=min_β1/T∑
_t=1^T[Y_t-b(X_t)^'β]^2+2r∑_j=1^p|β_j| .
A corresponding α̂ can be constructed using the Lasso estimator
describe in Section <ref> without cross-fitting. Let
M̂_j=1/N∑_i=1^Nm(Z_i,b_j),M̂=(M̂_1,...,M̂_J)^',Q̂=1/T∑_t=1
^Tb(X_t)b(X_t)^'.
A Lasso estimator α̂ can be obtained as α̂(x)=b(x)^'ρ̂ where
ρ̂=min_ρ{-2M̂^'ρ+ρ^'Q̂
ρ+2r_α∑_j=1^J|ρ_j|}.
A debiased machine learner without cross-fitting is
θ̂ =1/N∑_i=1^Nm(Z_i,γ̂)+1/T∑_t=1^Tα̂(X_t)[Y_t-γ̂(X_t)]
=1/N∑_i=1^Nm(Z_i,γ̂)+ρ̂^'[
1/T∑_t=1^Tb(X_t){ Y_t-γ̂(X_t)}] .
From equations (<ref>) and (<ref>) we see that the only features of the training data needed to
construct this estimator are the second moment matrix Q̂ of the
dictionary and the cross product ∑_t=1^Tb(X_t){ Y_t
-γ̂(X_t)} between the observations on the dictionary
b(X_t) and the Lasso residuals Y_t-γ̂(X_t).
An asymptotic variance estimator without cross-fitting is
V̂=1/N∑_i=1^N{m(Z_i,γ̂)-m̅}^2+1/T∑_t=1^Tα̃(X_t)^2[Y_t-γ̂(X_t)]^2,m̅=1/N∑_i=1^Nm(Z_i,γ̂),
where α̃(X_t)=τ_n(α̂(X_t)).
§ ASYMPTOTIC THEORY
In this Section we show asymptotic normality of the cross-fit estimator under weak regularity conditions and state a result on asymptotic normality of the Lasso estimator without cross-fitting.
The first condition is the following one:
AssumptionAssumption
LemmaLemma
Theorem[Lemma]Theorem
a) m(Z,γ) is linear in γ; b) E[m(Z,γ)^2]≤
C‖γ‖ _2^2; c) α_0(X) is bounded and
Var(Y|X) is bounded, d) (Z_1,...,Z_N) and W_1,...,W_T are i.i.d.
and mutually independent.
This condition requires that a) the object of interest be a linear functional of a regression; b) the functional m(Z,γ) be mean square continuous in γ; c) that the debiasing function α_0(X) and the conditional variance Var(Y|X) are bounded; and d) the training and test samples are i.i.d. and mutually independent.
For each ℓ the learners γ̂_ℓ and α̂_ℓ satisfy
a) ‖γ̂_ℓ-γ_0‖_2 =o_p(1) and ‖α̂_ℓ-α_0‖_2=o_p(1),
b) ‖α̂_ℓ-α_0‖_2‖γ̂_ℓ-γ_0‖
_2=o_p(1/√(T)),
This condition requires that both α̂ and α̂ are mean square consistent and that the product off their convergence rates is faster than 1/√(min(n,T)). Under these conditions the estimation of γ_0 and α_0 will not affect the large sample properties of the estimator, as shown by the following rueslt.
If Assumptions 1 and 2 are satisfied then
θ̂=θ_0+1/N∑_i=1^Nm(Z_i,γ_0)+1/T∑_t=1^Tα_0(X_t)[Y_t-γ_0(X_t)]+o_p
(min{N,T}^-1/2)
Proof of Lemma 1: For notational convenience we will drop the ℓ subscript
and replace the average over I_ℓ with the average over the entire
training sample while maintaining indpendence of α̂ and the
training data and γ̂ and the field and training data. Algebra
gives θ̂=1/N∑_i=1^Nm(Z_i,γ_0)+1/T∑_t=1^Tα_0(X_t)[Y_t-γ_0(X_t)]+R_1+R_2
+R_3,
R_1 =1/N∑_i=1^Nm(Z_i,γ̂-γ_0)+1/T∑_t=1^Tα_0(X_t)[γ_0(X_t)-γ̂
(X_t)],
R_2 =1/T∑_t=1^T[α̂(X_t)-α_0(X_t
)][Y_t-γ_0(X_t)],
R_3 =1/T∑_t=1^T[α̂(X_t)-α_0
(X_t)][γ_0(X_t)-γ̂(X_t)].
Define Δ̂:=∫ m(z,γ̂-γ_0)F_0(dz) and note
that by Assumption <ref> we have
Δ̂=∫α_0(x)[γ̂(x)-γ_0(x)]F_0(dx).
Then
R̂_̂1̂ =R̂_̂1̂-Δ̂+Δ̂=R_11+R_12,R_11=1/N∑_i=1^N[m(Z_i,γ̂-γ_0)-Δ̂]
R_12 =1/T∑_t=1^T[α_0(X_t){γ_0
(X_t)-γ̂(X_t)}+Δ̂].
Note that by γ̂ independent of training and field data,
E[α_0(X_t){γ_0(X_t)-γ̂(X_t)}|γ̂]=Δ̂,E[m(Z,γ̂-γ_0)|γ̂
]=Δ̂.
Also by Assumption 1 and α_0(X_t) bounded, |Δ̂|≤ C‖γ̂-γ_0‖
_2p⟶0,
E[m(Z_i,γ̂-γ_0)^2|γ̂] ≤ C‖γ̂-γ_0‖ _2^2p⟶0,
E[α_0(X_t)^2{γ_0(X_t)-γ̂(X_t)}^2
|γ̂] ≤ C‖γ̂-γ_0‖
_2^2p⟶0.
Then taking conditional expectations,
E[R_11^2|γ̂]≤C/T‖γ̂-γ
_0‖ _2^2=o_p(1/T),E[R_12^2|γ̂]≤C/N‖γ̂-γ_0‖ _2^2
=o_p(1/N).
It follows by the conditional Markov inequality that
R_11=o_p(1/√(T)),R_12=o_p(1/√(N)).
Next, note that
E[{α̂(X)-α_0(X)}{Y-γ_0(X)}|α̂]
=0,
E[{α̂(X)-α_0(X)}^2{Y-γ_0(X)}^2|α̂] =E[{α̂(X)-α_0(X)}^2Var(Y|X)]
≤ C‖α̂-α_0‖ _2^2
p⟶0.
It then follows similarly to R_11=o_p(1/√(T)) that R_2
=o_p(1/√(T)). Also,
E[|R_3||α̂,γ̂] ≤ E[(α̂(X)-α
_0(X))(γ_0(X)-γ̂(X))]
≤‖α̂-α_0‖_2‖γ̂-γ_0
‖_2=o_p(1/√(T)),
so R_3=o_p(1/√(T)) by the conditional Markov inequality. The
conclusion then follows by the triangle inequality. Q.E.D..
Asymptotic normality of the cross-fit estimator follows from Lemma 1 and the central limit theorem.
We next state a result for the Lasso estimator without cross-fitting that follows similarly to Corollary 9 of Bradic et al. <cit.>. For brevity we omit here a detailed statement of the conditions and the proof.
If Assumptions 1 and 2 are satisfied, N/T converges to ξ with 0<ξ<1, and the Assumptions of Corollary 9 of Bradic et al. <cit.> are satisfied then √(N)(θ̂-θ_0) converges in distribution to N(0,V) where V=Var(m(Z_i,γ_0)) + ξE[α_0(X_t)^2Var(Y_t|X_t)]
§ SIMULATION AND RESULTS
We evaluate the proposed method on a regression problem using a Monte-Carlo simulation. Along with providing useful analysis, this provides a simple case study of how the bias correction could be used in practice.
§.§ Simulation Description
We use a Monte-carlo simulation described by a high dimensional high order multivariate polynomial as follows.
g(𝐮) = α_0 + α_1 ∑_k=1^Kβ_1k u_k + α_2 (∑_k=1^Kβ_2k u_k )^2 + ... + α_3 (∑_k=1^Kβ_3k u_k )^Q
The polynomial is of order Q in K dimensions with cross terms, and defined by α and β, where 𝐮 is a K dimensional vector. The training data is denoted by the random variable X_t, which is also a K dimensional vector. The output of the simulation for an observation from the training data is then described by:
Y_t = g(X_t) + ϵ_t
Here ϵ is zero-centered normally distributed noise with standard deviation σ. Additionally, we can describe the true data curve simply as γ_0(X) = g(X).
For this simulation, both training data (denoted X) and validation data (denoted V) are drawn from a normal distribution with mean zero and standard deviation of one in each dimension. They are combined with a low weight uniform distribution that extends from -5 to +5 in each dimension. In order to represent the effects of distribution shifts, the test data (denoted Z) is drawn from similar distributions but with a normal distribution with a shifted mean.
The simulation outputs are scaled by a constant so that they stay near the range of -1 to 1. The standard deviation, σ, of the noise term ϵ is set to 0.1. The simulation uses a 3rd order polynomial with 6 dimensions. The distribution shift for z used here is 1.1 times σ.
Multiple specifications of the simulation are created by randomizing the simulation parameters (β), with some of elements set to be near zero, resulting in about 60% sparsity. Additionally, for ease of visualization, the β parameters in first dimension in the simulation are multiplied by about 1.71 relative to the rest of the parameters, and the last dimension is multiplied by about 0.29.
We evaluate for many samples and we also vary the simulation specification. For each retraining of our network a new sample of 10,000 observations of x and 10,000 observations of z were generated. For each simulation specification, the network is retrained over 60 different samples. 30 specifications are used resulting in 1800 total iterations of the simulation.
§.§ Bias Correction Applied To This Problem
Our aim is to estimate γ_0(X_t) with γ̂(X_t) obtained by minimizing the sum of squares of γ(X_t) - Y_t using least squares regression with regularization. We will construct the regression estimator, γ̂, using a neural network.
We then take the expected value of the outputs over a large number of values for both X and Z.
We use a relatively simple network model utilizing a fully connected neural network, sometimes called MLP (multilayer perceptron), or ANN (artificial neural network) <cit.>. The network has 4 hidden layers, each with 32 nodes, and ReLU activation. In total, We have about 3000 total learn-able parameters.
For this processing we used Matlab's trainNetwork with featureInputLayer's and fullyConnectedLayer functions. We used a learn rate of 0.01, a mini-batch size of 1024, and up to 500 epochs of training time. Additionally, the neural network is L2 (i.e. Tikhonov) <cit.> regularized with a regularization parameter of 0.0002.
The bias correction can be applied to this simulation. We utilize the dictionary vector b(x) (from (<ref>)) that is J polynomial functions of order 2 described by
b(x_t) = [1, x_1t, x_2t, ..., x_1t x_1t, x_1t x_2t, ..., x_Kt x_Kt]'
Using (<ref>), we first calculate ρ̂_̂ℓ̂ calculate from:
ρ̂ =min_ρ{-2M̂^'ρ+ρ^'Q̂_̂ℓ̂
ρ+r∑_j=1^J|ρ_j|}
We take m from (<ref>) to be the mean of the data Z so that:
M̂_j=1/N∑_i=1^N b_j(Z_i),M̂=(M̂_1,...,M̂_J)^',Q̂=1/T
∑_t=1^Tb(X_t)b(X_t)^'.
For cross fitting data, we simply used a second sample from the simulation, and create a second polynomial expansion, which we'll denote V here. The bias corrected mean of γ̂(Z) therefore is:
θ̂ =1/N∑_i=1^Nγ̂(Z_i)+ρ̂^'[
1/T_v∑_t=1^T_vb(V_t){ Y_t-γ̂(V_t)}]
where T_v is the number of observations in the second sample.
§.§ Results
In order to evaluate algorithm effectiveness, we estimate the algorithm bias for each simulation replication by averaging the difference of the estimated function γ̂(Z) with the average of a large set of 1 million independently selected observations, γ_0(Z) + ϵ, over the 60 samples (with network retraining). The true bias changes for each specification therefore to measure the bias we calculate the root mean square error from zero of the biases across specification. Additionally, we estimate the root mean square error of γ̂(Z) and γ_0(Z) + ϵ.
In figure <ref>, we plot the result as a function of training time. We can see right away how the bias correction lowers the expected average bias consistently across the data. The bias correction has the largest effect at lower training times. This can be explained as an increased bias due to increased regularization as the lower training times represent early stopping regularization.
Figure <ref> also includes error bars which shows the estimated standard deviation of the root mean square of the bias estimate and the standard deviation of the RMSE. As the two curves in each plot are all over about 3 standard deviations apart, the confidence interval between the bias corrected estimate and the original estimate is generally over 99.5%.
Inspecting the plots closely we see that the bias corrected data basically provides the same RMSE at 100 epochs as at 500 epochs, and much more consistency across all of the epochs. This indicates both allowed lowering of required training time and less precision required in determining the best training time.
We also note that the bias correction is shown to provide benefit on average across specifications. For a given specification, the benefit may be negligible or even negative or may be much larger than is shown.
§.§ Discussion
While a variety of possible cross fitting methods can work, the cross-fitting we implemented was done for ease of use in the following manner: The cross-fit data which we will call v is an entirely new sample from the same distribution as x. This represents well the case where we have enough data to have both very large training and validation sets.
It also is interesting to note that we observed during our processing that cross-fitting was generally not necessary for this specific problem. Therefore, if we ignore the need for cross-fitting, we can get an alternative description of the debiasing function that applies under the simple assumptions used in the bias correction for this simulation.
Consider that we have a set of functions b(x) where you are finding the weights ρ for each b(x) given a list of residuals.
Then given that Q̂ in equation <ref> is symmetric, the first order conditions to equation <ref> are a regularized solution of the relationship:
M̂ = ρ̂Q̂
Another solution technique for M̂ = ρ̂Q̂ would be to solve for ρ̂ using a generalized inverse of Q̂. We can re-write our variables with matrix notation:
𝐁=[b(X_1),...,b(X_T)]'
𝐲 = [y(X_1),..., y(X_T)]
γ̂ = [γ̂(X_1),..., γ̂(X_T)]
Thus we can write this generalized inverse as 1/TQ̂^+ = [ B' B]^+. Without cross-fitting, we then can rewrite (<ref>), using α̂(x)=b(x)^'ρ̂, as
1/N∑_i = 1^Nγ̂(Z_i) + [1/N∑_i = 1^Nb(Z_i)]'[ B' B]^+ B' [ 𝐲 - γ̂]
Inspecting this equation, we can set ϕ̂ = [ B'B]^+ B' [ 𝐲 - γ̂] to be the coefficients of a least squares regression of the residuals on B, with the caveat that Q̂ is likely not invertible and thus must be solved using regularization (e.g. a pseudo-inverse on Q̂ or Lasso regression on the whole equation). Then the debiased estimator is 1/N∑_i = 1^N [γ̂(Z_i) + b(Z_i)' ϕ̂], where the bias correction is now the coefficients ϕ̂ on the residuals applied to the test data dictionary b(Z_i).
The test provided here was evaluated with a few different algorithms and, perhaps surprisingly, we found that with appropriately tuned regularization parameters the debias function worked equally well regardless of the method used.
§ CONCLUSIONS
In this paper we have provided debiased machine learning estimators of parameters when there is a covariate shift. We developed a method to debias functionals of trained machine learning algorithms under covariate shift. With cross-fitting the methods were shown to be asymptotically normal as the sample size grows for a variety of regression learners, including neural nets and Lasso.
For Lasso regression it was shown that cross-fitting was not needed for these results.
We evaluated the cross-fit method in a relatively simple simulation generated using polynomial coefficients and a neural network with explicit regularization for fitting. The results strongly indicated that the debiased machine learner described here effectively removes the bias.
A significant caveat to consider is that the proposed methodology requires averaging over a reasonable large z sample, and a substantial amount of z data may be required for accurate results. Importantly, without enough averaging, the noise in the bias correction would likely make the results look worse. Because of this, the proposed methodology requires additional computation, which may become significant depending on the dataset sizes.
We believe these results show promise and in future work these methods could be extended to other settings, for example ’counterfactual averages', or data classification.
plain.bst
|
http://arxiv.org/abs/2307.04331v1 | 20230710035458 | On the Jets Induced by a Cavitation Bubble Near a Cylinder | [
"Yuxin Gou",
"Junrong Zhang",
"Akihito Kiyama",
"Zhao Pan"
] | physics.flu-dyn | [
"physics.flu-dyn"
] |
Quasicrystalline second-order topological semimetals
Dong-Hui Xu
August 12, 2023
====================================================
The dynamics of cavitation bubbles in the vicinity of a solid cylinder or fibre are seen in water treatment, demolition and/or cleaning of composite materials, as well as bio-medical scenarios such as ultrasound-induced bubbles near the tubular structures in the body.
When the bubble collapses near the surface, violent fluid jets may be generated.
Understanding whether these jets occur and predicting their directions—departing or approaching the solid surface—is crucial for assessing their potential impact on the solid phase.
However, the criteria for classifying the onset and directions of the jets created by cavitation near a curved surface of a cylinder have not been established.
In this research, we present models to predict the occurrence and directions of the jet in such scenarios.
The onset criteria and the direction(s) of the jets are dictated by the bubble stand-off distance and the cylinder diameter.
Our models are validated by comprehensive experiments.
The results not only predict the jetting behaviour but can serve as guidelines for designing and controlling the jets when a cavitation bubble collapses near a cylinder, whether for protective or destructive purposes.
§ INTRODUCTION
Cavitation is a phase transition process from liquid to gas, which is often observed when the pressure of the liquid experiences a significant drop within a short time.
The collapse and rebound of the bubble may generate shock waves, extreme heating, and high-velocity jets, resulting in damage to the solid boundaries nearby.
This process is detrimental in many scenarios, such as cavitation erosion to hydraulic machinery and destruction of human tissues (e.g., bone or brain, <cit.>).
On the other hand, some applications such as biomedical ultrasound and ultrasonic cavitation cleaning <cit.> take advantage of the force acting on the boundary.
Hence, the cavitation dynamics near the boundaries have been of interest to the community.
Studies on bubble dynamics near a wall and associated damaging mechanisms can be traced back to 1940's <cit.>, focusing on the cavitation phenomena near a flat surface (see, for example, <cit.>, and an illustration in figure. <ref>(a)).
When a bubble collapses near a flat solid wall, the bubble may migrate to the wall, and a directional liquid jet towards the wall is created.
The concentrated momentum impacts a small area on the wall, where the induced pressure and shear are considered to be one of the primary mechanisms for cleaning and/or damaging the surfaces <cit.>.
Therefore, the onset of the directional jet is the key factor determining the interaction between the bubble and the boundary.
The direction of the jet depends on a multitude of factors, especially the geometry of the boundaries.
<cit.> experimentally studied the direction of the jet generated upon the rebound of a bubble in a corner of two solid boundaries, where the angle between them was set to either 90 or less (figure. <ref>(b & c)).
<cit.> proposed a generalized formula that predicts the jet direction in a corner with an arbitrary opening angle α and proximity to the walls (figure. <ref>(d)).
They show that there exist analytic solutions that predict the jet direction for α = π/n, where n is a natural number.
Several studies reported that the fluid jet formed upon the bubble collapse near a solid wall with complex geometry does not always point to the wall.
<cit.> reported the dynamics of the bubbles near trapezoidal ridges and valleys (figure. <ref>(e)) and found that the fluid jet can appear in two different directions (i.e., a departing or approaching jet to the wall).
The departing jet may appear when a bubble collapses near the ridge, while a bubble near the valley can only form an approaching jet in their experiments.
The configuration might share some similarity to the bubble dynamic near a curved surface (e.g., the surface of a cylinder or a sphere, see figure <ref>(f & g)).
The morphology of the bubble in the neighbourhood of a curved surface has been studied <cit.>, and the curvature of the solid wall was found to be one of the primary parameters in addition to the stand-off distance <cit.>.
A departing jet may appear when the bubble collapses near a convex (positive curvature) surface.
However, extensive data or detailed discussions on the direction of the dual fluid jets were not reported.
An interesting feature of the bubble near a convex surface is the “mushroom” bubble before collapsing, which is almost always associated with the departing jet.
This observation has been reported in earlier studies (e.g., <cit.>) and recent research on cavitation near the tip of a thin cylinder also concurred with similar evidence.
<cit.> reported that the mushroom-shaped collapsing bubble could happen when a cavitation bubble was initiated near the tip of a thin cylinder (figure. <ref>(h)).
The fluid-gas interface resembling the `stem of the mushroom' (i.e., the interfaces close to the tip of the cylinder) contracts faster than the `mushroom cap', which results in a departing jet when the bubble fully collapses.
<cit.> also suggested that an optimal length scale of cylinder thickness exists, compared to a fixed bubble diameter, so that the jet becomes the most powerful.
<cit.> numerically approached this problem and revealed that the mushroom-shaped bubble near the tip of the cylinder might be linked to the reduction of the impact load on the surface.
It is perhaps because the not-yet-formed departing jet carries momentum away from the solid surface.
Beyond the distinct physics, this setup of bubbles near the tip of a thin cylinder can generate a high-speed departing jet (up to O(1000) m/s according to the simulations by <cit.>) and is of interest to applied research.
However, the direction of the jets and the criteria of the departing jet onset were not analyzed.
In the current work, we are interested in the dynamics of bubbles and jets next to the side surface of cylinders.
To the best of our knowledge, this scenario has not been reported except for <cit.> studying the micro-bubbles near a fiber, as well as <cit.> where bubble behaviour near a thick cylinder (inspired by cavitation near the hull of a ship) was investigated.
There are no detailed discussions on the direction of the jet(s) when the bubble collapses near a cylinder available in the current literature.
In this paper, we report a regime diagram, validated by vast experimental data, that classifies the onset and the direction of the jet(s), which is dictated by two non-dimensional parameters (i.e., bubble stand-off distance and the cylinder thickness relative to the bubble diameter).
Particularly, we find that when a large bubble is close to a thin cylinder, a departing jet is likely to form after collapsing and the cylinder is protected.
This discovery might be insightful for some applied scenarios.
For example, fibrous or tubular structures in the vicinity of a cavitation bubble could be free from severe damage and it is possible to design patterned surface <cit.> or fibrous structure to reduce cavitation erosion.
§ EXPERIMENTAL SETUP
The experimental setup is shown in figure <ref>(a).
The cavitation bubbles were generated by shorting adjustable direct current voltage carried by two thin wires of 0.14 mm in diameter.
The sizes of the bubbles varied from 5.45 to 24.58 mm in diameter by adjusting the voltage (within the range of 60 – 120 V).
The cylinders used in the experiments are made from stainless steel with a contact angle of around 60^∘.
The wires are at least one order of magnitude thinner compared to the size of the cylinders and the cavitation bubbles and thus the influence of the wires is negligible.
The wires and the cylinder were placed in the middle of a tank (20 × 20 × 20 cm^3) filled with degassed tap water.
The tank is large enough to ensure the bubble behaviour was not affected by either the free surface or the rigid wall.
The dynamics of the cavitation bubbles was filmed by a high-speed camera (FASTCAM SA-Z or NOVA S20, Photron, Tokyo, Japan) at 60,000 frames per second.
A schematic of the bubble and the cylinder overlaid on a high-speed image is shown in figure <ref>(b).
Two key non-dimensional parameters—the standoff distance γ and the non-dimensional cylinder diameter η—are defined as
γ =d_s/D_0 and η = D/D_0,
respectively, where d_s is the distance from the spark location, which can be considered as the nominal center of the bubble, to the closest cylinder surface, D_0 is the maximum bubble diameter (marked by a blue circle), and D is the cylinder diameter (marked by a red circle).
The distance between the nominal center of the bubble and the center line of the cylinder is written as d = d_s + D/2, which can be normalized by D_0 as
ζ = d/D_0 = γ +η/2.
This is an alternative non-dimensional length scale characterizing the distance between the bubble and the cylinder.
§ RESULTS
We carried out comprehensive experiments on spark-induced cavitation bubbles in the vicinity of a cylinder by varying η and γ.
The experiments revealed five distinct bubble behaviours for various conditions (demonstrated in figure <ref>).
The dimensional and non-dimensional parameters of these typical cases are listed in Table <ref>.
When the bubble is initiated far enough from the surface of a cylinder, it is expected that the bubble remains spherical when expanding and collapsing, and no jets are formed after the bubble collapses.
We refer to this observation as a “no jet (NJ)” case hereafter.
For example, in figure <ref>(a), a bubble is initiated by a spark (indicated by the apex of the green triangle at t=0 ms) at γ=1.44 from a cylinder (marked by the scarlet circle).
The bubble grows and reaches its maximum diameter D_0 at t = 0.46 ms, collapses at t = 0.87 ms for the first time, and rebounds to the maximum of the cloud at t=1.03 ms.
The direct observation of the jets (onset and directions) during collapse can be difficult, thus we use the displacement (δ_D) from the bubble onset location (marked by the green triangle in figure <ref>) to the centroid of the maximum bubble cloud of the second expansion (marked by the yellow triangle) as an indicator of the net momentum due to the bubble collapse.
The positive direction of δ_D points from the centerline of the cylinder to the center of the bubble).
A non-zero δ_D infers a liquid jet generated when the bubble collapses.
The non-dimensional displacement, δ = δ_D/D_0, in figure <ref>(a) was δ = 0.00 (note that NJ is classified for |δ|<δ_0, δ_0 = 0.03 is a small value as the measurement threshold in this work.)
As the center of the bubble moves closer to the cylinder, a jet shooting toward the cylinder is generated when the bubble collapses and we address this case as “approaching jet only (AJO)".
As shown in figure <ref>(b) as an example, the bottom of the bubble is deformed when approaching the cylinder from a standoff distance of γ = 0.45 (e.g., see two frames at t = 0.40 and 0.96 ms).
The centroid of the rebound bubble (marked by the yellow triangle at t = 2.24 ms) moves towards the cylinder (δ=-0.12 in this case), compared to the spark location (marked by the green triangle at t = 0 ms).
This footprint indicates a liquid jet approaching the cylinder is generated during the bubble rebound.
In addition, no other jet(s) were observed.
The bubble cloud formed during the second expansion cycle collapses and largely covers the cylinder (t = 2.72 ms), implying that the approaching jet may carry a large momentum.
This process that generates an approaching jet is similar to a bubble collapsing near a flat rigid surface.
Figure <ref>(c) presents a typical case where the mushroom bubble forms and a departing jet starts to appear.
In this work, we refer to this scenario as “departing jet emerging (DJE)”.
The stand-off distance γ = 0.26 and the non-dimensional cylinder size η=0.09 in this case were smaller than those of the case in figure <ref>(b).
In figure <ref>(c), when the bubble reaches its maximum volume (at t = 1.09 ms), the bubble partially warps the narrow cylinder and maintains its spherical shape in general.
The stem of the “mushroom" is formed due to the fast-retracting liquid jets pinching the bubble near the cylinder (indicated by the orange arrowheads at t = 2.02 ms).
While collapsing, the cap of the mushroom remains spherical as the gas-liquid interface (indicated by the purple arrowhead at t =2.02 ms) is far away from the cylinder and recedes slower compared to the pinching jets.
The dynamics are similar to the observations made by <cit.>.
It is noteworthy that the bubble cloud in the second expansion cycle moves in two directions.
The centroid of the rebound bubble moves toward the cylinder (δ= -0.05, comparing the location of the green and yellow triangles at t = 0 and 2.58 ms, respectively), similar to the case in figure <ref>(b), while there is a minor cloud bubble shooting away from the
cylinder (see t = 2.58 ms, marked by the short pink arrowhead in figure <ref>(c)).
This observation indicates that two jets exist after the collapse: one jet is approaching and the other one is departing from the cylinder.
The departing jet, which is an emerging feature compared to the case in figure <ref>(b), however, does not yet dominate the entire jetting process.
When the bubble is close to a relatively thin cylinder, the departing jet may dominate over the approaching jet and we denote this scenario as “departing jet dominant (DJD)”.
A typical case is shown in figure <ref>(d) for γ = 0.06 and η = 0.09.
The bubble completely wraps the cylinder when it expands to the maximum diameter (t= 1.15 ms) and then collapses.
Similar to the case shown in figure <ref>(c), the elongated rebound bubble cloud covering the cylinder meanwhile moving away from the (t=2.53 ms) indicates the existence of both approaching and departing jets.
Noting centroid of the bubble cloud (t= 2.53 ms, marked by the yellow triangle) is further away from the cylinder than the center of bubble onset (green triangle at t= 0 ms) and the corresponding displacement δ = +0.04, we argue that the jet forming at collapse is mainly departing.
Figure <ref>(e) shows another “no-jet (NJ)” case.
A bubble is initiated right next to a thin cylinder, where the size of the bubble is much larger than that of the cylinder (η = 0.05).
The bubble behaviour in this case is similar to a free bubble.
The centroid of the bubble (cloud) does not show any apparent movement upon rebound, indicating that no jet was generated.
Despite the NJ outcome that is similar to the case shown in figure <ref>(a), we emphasize that the phenomenon shown in figure <ref>(e) is due to vanishing cylinder diameter (η→ 0) whereas the NJ case in figure <ref>(a) is associated with the standoff distance in the limit of γ→∞.
§ MECHANISMS
The observations in figure <ref> imply that when a bubble collapses near a cylinder, depending on the relative position as well as the size of the bubble and the cylinder (γ and η), the cylinder may affect the liquid flow in two ways (i.e., blocking and focusing).
First, the cylinder can block the liquid behind it from directly moving to the center of the bubble, while the liquid on the other side of the bubble is free to move to fill the cavity during collapsing.
This causes a pressure gradient and, in turn, the collapsing bubble generates a jet approaching the cylinder <cit.>.
This often happens when the cylinder is relatively large and/or the bubble is not too close to the cylinder (e.g., see the case in figure <ref>(b)).
This mechanism is similar to the well-known jet formation from a bubble collapsing next to a solid flat surface.
Second, when the cylinder is relatively small and the bubble is initiated close enough to the cylinder, the bubble can be significantly deformed during its growth.
In figure <ref>(c), for example, the bubble partially wraps the cylinder while achieving its maximum volume (at t = 1.09 ms), leaving two regions of the gas-liquid interface having a higher curvature than other parts of the bubble.
The higher curvature is corresponding to a smaller equivalent local bubble radius, which is associated with a shorter time for a local collapse.
This mechanism has also been argued by <cit.> based on the Rayleigh's collapse time, T ≃ 0.915D̃_0√(ρ /p_∞), where T is the collapse time, ρ is the liquid density, p_∞ is the ambient pressure, and D̃_0 is the equivalent bubble size reflecting the local curvature of the bubble.
Over the initial stage of the collapse, the advantage of the high-speed flows driven by the high curvature interface accumulates, which results in two jets pinching the bubble (see the orange arrowheads in figure <ref>(c) for instance).
The two pinching jets forms the stem of the mushroom-shaped bubble before collapsing.
After pinch-off, the two pinching jets merge and the momentum is focused upward, pointing away from the cylinder, which can dominate the retracting liquid near the cap of the mushroom-shaped bubble (see the purple arrowhead in figure <ref>(c)).
This focusing mechanism is similar to the shaped charge effect.
The competition between these two mechanisms dictates the onset and direction(s) of the jet(s), and some typical results as shown in figure <ref>.
§ REGIME DIAGRAMS AND VALIDATION
Based on the above experimental observations and analysis on the mechanisms, we hypothesize that the direction(s) of the jet(s) caused by the bubble collapsing near a cylinder are dictated by two parameters.
One is the standoff distance γ = d_s/D_0 measuring the distance from the bubble to the cylinder, and the other is the non-dimensional cylinder diameter η = D/D_0.
Several critical states regarding γ and η are proposed below and illustrated in figure <ref>.
When a bubble wraps about half of the cylinder, the virtual circle enclosing the bubble passes the center of the cylinder (see figure <ref>(a)).
We conjecture that this is a state separating the blocking and focusing mechanisms and determines if a departing jet would emerge.
The corresponding geometric relationship for the circles representing the bubble and cylinder is d_s=1/2(D_0-D), and the non-dimensional form is γ = 1/2 - 1/2η.
If the standoff distance is smaller than this threshold, that is to say
γ < 1/2 - 1/2η,
high curvature on the sufficiently deformed bubble leads to the evident focusing effect and a departing jet is expected.
When the bubble is even closer to the cylinder, especially when the bubble is relatively large, the focusing effect is more pronounced than the blocking and the departing jet starts to dominate.
This condition translates to d < κ_1 D_0, where κ_1 is a coefficient that can be determined by experimental data (see figure <ref>(b) for illustration).
Invoking d = d_s+1/2D, the non-dimensional form of this criterion is
γ < κ_1 - 1/2η.
When the bubble is far enough from a sufficiently small cylinder, d_s > 1/2D_0 + κ_2 D, where κ_2 is another constant to be determined (see figure <ref>(c)), the effect of the cylinder (blocking or focusing) is negligible and thus no jet is expected.
The corresponding non-dimensional form is
γ > 1/2 + κ_2 η.
This criterion considers the combined effects of the relative size and position of a bubble and cylinder.
The asymptotic behaviours (i.e., small η→ 0 and large γ→∞) of such a setup are also of interest.
When the cylinder is significantly smaller than the bubble (see figure <ref>(d) for illustration), for example, D < κ_3 D_0 ≪ D_0 with corresponding non-dimensional form
η < κ_3 ≪ 1,
the relative placement of the bubble and cylinder is not important anymore.
Jets are not expected when the bubble collapses due to the diminishing impact of the cylinder of a small length scale. κ_3 ≪ 1 is a small constant that can be found by experiments.
When the bubble is too far away from the cylinder (see figure <ref>(e)), the size of the cylinder does not matter.
We expect there exists a critical value κ_4 so that if d_s > κ_4 D_0 ≫ D_0/2, no jet would be generated when the bubble collapse.
The non-dimensional form of this criterion is
γ > κ_4 ≫1/2.
Recall (<ref>) again, the above criteria can also be expressed using ζ instead of γ.
We use γ to be consistent with the current literature, however, ζ is practical to investigate some of the critical states regarding the directions of the jets.
The directions of the jets after bubble collapsing can be qualitatively observed by the direction of the moving bubble cloud in the high-speed videos.
For example, when a departing jet appears, the bubble cloud tends to move away from the cylinder over the collapsing-rebound cycles.
This can be quantitatively identified using the value of δ = δ_D/D_0 as a measure, which is a characteristic displacement of the bubble cloud.
If there is only an approaching jet appears after the first collapse, the momentum of the jet would carry the bubble cloud towards the cylinder (e.g., see figure <ref>(b)) and we expect δ < -δ_0<0.
Similarly, when the departing jet dominates the approaching one, δ > +δ_0>0 (see figure <ref>(d) for instance).
However, if the departing and approaching jets cannot dominate one to the other, the direction of δ_D and the `sign' of δ are not necessarily determined.
We present δ as a function of ζ in figure <ref> to show our argument above is valid.
Viewing the δ–ζ phase diagram vertically, we can see that all the AJO cases (orange upside-down triangle in figure <ref>) are located in the region of δ<-δ_0, whereas DJD cases (pink upright triangles) are in δ>+δ_0.
NJ cases (black crosses) are distributed along δ = 0 (-δ_0 < δ < +δ_0 to be more specific) whereas the DJE cases (blue diamond symbols) are scattered on both sides of δ = 0.
Interrogating the experimental data on the δ – ζ phase diagram ( figure <ref>) horizontally is useful for verifying the aforementioned models and identifying the coefficients such as κ_1.
It is visible that the jet direction evolves from approaching to departing as ζ decreases.
In the region of ζ > 0.5 (yellow-shaded, to the right of the blue chain line in figure <ref>), almost only AJO cases exist.
Recalling (<ref>), ζ= 0.5 is an alternative expression of γ=1/2-1/2η, thus, (<ref>) is validated.
The departing jets emerge when ζ < 0.5,
and further reducing ζ, the departing jet eventually becomes dominant for ζ < 0.25, which is equivalent to (<ref>) for κ_1 = 1/4.
This is supported by observing that in the red-shaded region to the left of the magenta line (corresponding to ζ = 0.25), almost only DJD cases exist.
The DJE cases (blue diamond symbols) are located in the transient region for 0.25<ζ < 0.5.
The black symbols represent the data extracted from <cit.>, where a laser-induced micro-bubble collapsing near a micro-fibre was studied.
This work did not focus on the direction of jets, and the bubble dynamics after the first collapse was not reported.
Instead, the location of the bubble near collapsing was recorded.
Comparing the displacement from the location of the bubble onset to the center of the bubble at the first collapse, one could still infer the directions of jets.
Despite being a different measure of δ than we used for our data, this qualitative classification is sufficient to tell the AJO, DJE, and DJD cases apart in <cit.>, and we see that the experimental data by <cit.> agree with our model.
To validate equations (<ref>) and (<ref>), we plot the non-dimensionalized experimental data on the γ – η plane (figure <ref>).
The blue chain line indicates equation (<ref>) separating the AJO and the DJE cases.
The magenta line in figure <ref> is based on equation (<ref>) that separates most DJD cases from the DJE cases.
Experimental data on the γ – η plane also provides quantitative insights into the NJ cases due to different reasons.
κ_2=0.5 for (<ref>) separates the NJ cases and the AJO cases for 5×10^-2≲η≲ 7 (see the orange dotted line in figure <ref>).
For η < 5×10^-2, a sufficiently thin cylinder cannot affect the dynamics of the bubble and almost no jets were observed in our experiments.
Thus, κ_3 = 5×10^-2 in (<ref>) allows our model to establish the criterion of a thin cylinder.
For the other extreme, κ_4 = 4 for equation (<ref>) was suggested by our experiments, which is the criterion for a large stand-off distance.
We note that κ_4=4 agrees with the established data about a cavitation bubble near a flat surface (<cit.>), which can be considered as a thick cylinder with vanishing curvature (i.e., η→∞).
In figure <ref>, criteria based on equations (<ref>) – (<ref>) separate the γ – η phase diagram into four regimes.
Regime I (yellow shade) covers most of the AJO cases (orange upside-down triangles).
In regime III (pink shade), almost only pink triangles (associated with DJD cases) appear.
The transient cases for the directional jet(s) (DJE cases, marked by the blue diamond symbols in blue-shaded regime II) are in between Regimes I and III.
Regime IV (different shades of green for three sub-regimes) indicates NJ cases rooted in different mechanisms.
In Regime IV-1, NJ happens as a cylinder is too thin (small η).
In Regime IV-3, NJ is expected as the bubble is too far away from the solid surface (large γ).
Regime IV-2 can be thought of as the transient region between Regime IV-1 and IV-3, where the combined effect of η and γ must be considered and is governed by
(<ref>).
Again, the data extracted from <cit.> falls in our regime diagram, and provides additional validation based on the interaction of micro-bubbles.
§ CONCLUDING REMARKS
In the current work, we carried out systematic experiments to investigate a cavitation bubble collapsing near a cylinder.
We find that the onset and the direction of the jet(s) are dictated by the relative positioning and the size of the bubble and the cylinder (i.e., the standoff distance γ and the normalized cylinder diameter η).
When the cylinder is too thin and/or too far away from the bubble, a bubble does not expel any visible jets.
Once the bubble starts interacting with the cylinder—when γ and/or η are small enough—a jet approaching the cylinder occurs, as one might expect, which is similar to that for a bubble collapsing in the vicinity of a flat wall.
When the cavitation bubble is onset closer to an even smaller cylinder within a particular range, the bubble possesses a mushroom-like collapse followed by a departing jet.
Given a certain maximum bubble size, the departing jet carries the energy
away from the cylinder, which might result in a reduction of the cavitation-induced damage.
In this sense, the cylinder is protected by being thin and staying close to the cavitation.
We proposed models to classify these phenomena including transition into four regimes on the γ – η phase diagram, which are validated by experiments.
The experimental results and criteria
shown in this work may be of interest to applications where cavitation bubbles interact with (thin) cylinders and fibres.
For example, a direct implication based on our result is that the demolition of thin fibres and fibrous materials could be challenging, and small bubbles are more effective than bigger ones.
When a cylinder near a cavitation bubble needs protection, our regime diagram provides a guideline: one may want to manage the standoff distance and bubble size to avoid the jet onset or staying in the departing jet dominant regime.
§ ACKNOWLEDGMENTS
We thank Drs. S. Peterson and M. Worswick for lending us equipment and J. Beginner and J. Imbert-Boyd for manufacturing and technical support.
|
http://arxiv.org/abs/2307.05640v1 | 20230711095454 | A GH-compactification of CAT$(0)$-groups via totally disconnected, unimodular actions | [
"Nicola Cavallucci"
] | math.MG | [
"math.MG",
"math.GR"
] |
Multi-index Importance Sampling for McKean-Vlasov Stochastic Differential Equation
Nadhir Ben Rached Department of Statistics, School of Mathematics, University of Leeds ([email protected])., Abdul-Lateef Haji-Ali Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK ([email protected])., Shyam Mohan Subbiah Pillai Corresponding author; Chair of Mathematics for Uncertainty Quantification, Department of Mathematics, RWTH Aachen University, Aachen, Germany([email protected]).
and Raúl Tempone Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia ([email protected]). Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Aachen, Germany ([email protected])..
This work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. URF/1/2584-01-01 and the Alexander von Humboldt Foundation. This work was also partially performed as part of the Helmholtz School for Data Science in Life, Earth and Energy (HDS-LEE) and received funding from the Helmholtz Association of German Research Centres. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.
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We give a detailed description of the possible limits in the equivariant-Gromov-Hausdorff sense of sequences (X_j,G_j), where the X_j's are proper, geodesically complete, uniformly packed, CAT(0)-spaces and the G_j's are closed, totally disconnected, unimodular, uniformly cocompact groups of isometries. We show that the class of metric quotients G/X, where X and G are as above, is compact under Gromov-Hausdorff convergence. In particular it is a geometric compactification of the class of locally geodesically complete, locally compact, locally CAT(0)-spaces with uniformly packed universal cover and uniformly bounded diameter.
§ INTRODUCTION
The author, together with A.Sambusetti, studied in <cit.> the possible Gromov-Hausdorff limits of locally geodesically complete, locally CAT(0)-spaces with bounded diameter and with uniformly packed universal cover, with particular attention to the collapsing case, i.e. when the dimension of the limit space is smaller than the dimension of the approximating ones.
In order to synthetize part of the results of <cit.> we fix some notations. Let 𝒪-CAT_0^disc(P_0,r_0,D_0) be the class of quotient metric spaces M:=Γ\ X, where X is a proper, geodesically complete, (P_0,r_0)-packed, CAT(0)-space and Γ is a discrete group of isometries of X with diam(Γ\ X) ≤ D_0. Here 𝒪 stays for orbispace. The numbers P_0,r_0,D_0 are structural constants that have to be thought arbitrary, but fixed once for all. The packing condition on X is a very weak synthetic lower bound on the curvature of the metric space (cp. Section <ref> and <cit.> for more details) and is a necessary and sufficient condition for precompactness under Gromov-Hausdorff convergence (cp. Lemma <ref>, <cit.> and <cit.>). In case Γ is torsion-free then M is actually a compact, locally geodesically complete, locally CAT(0)-space of diameter at most D_0.
One of the main results of <cit.> is
Let { M_j }⊆𝒪_0^(P_0,r_0,D_0) be such that M_j converges to some space M_∞ in the Gromov-Hausdorff sense. Then M_∞ is the quotient of a proper, geodesically complete, (P_0,r_0)-packed, (0)-space by a closed, totally disconnected group of isometries, and (M_∞) ≤ D_0.
The proof of Theorem <ref> is based on a careful description of the convergence of the group actions (X_j,Γ_j) to a limit group action (X_∞, G_∞) in the equivariant pointed Gromov-Hausdorff sense (cp. Section <ref> and <cit.>), where M_j = Γ_j \ X_j and M_∞ = G_∞\ X_∞. The main issue is that in general G_∞ is not discrete, so it is not clear a priori the relation between X_∞ and M_∞. The analysis was divided in two cases: the collapsed and the non-collapsed one. The two cases can be summarized in the following equivalences (<cit.>):
Non-collapsed ⇔ G_∞^o = {𝕀}⇔(M_∞) = (M_j) for j≫ 0;
Collapsed ⇔ G_∞^o ≠{𝕀}⇔(M_∞) < (M_j) for j≫ 0,
where G_∞^o is the connected component of the identity of the topological group G_∞. From (<ref>) we can see the validity of Theorem <ref> in the non-collapsed case. The same result in the collapsed case requires a deep understanding of G_∞^o which is a non trivial result of <cit.>. We remark that in the generality of Theorem <ref>, it can happen that M_∞ does not belong to 𝒪-CAT_0^disc(P_0,r_0,D_0), i.e. that the group G_∞ is not discrete, even in the non-collapsed case (cp. <cit.>). Therefore totally disconnected, non-discrete, actions appear in this framework as limit case of discrete actions and it is natural to study them in order to find a compactification of the space 𝒪-CAT_0^disc(P_0,r_0,D_0).
With this in mind we denote by 𝒪-CAT_0^td,u(P_0,r_0,D_0) the class of quotients M:=G \ X, where X is again a proper, geodesically complete, (P_0,r_0)-packed, CAT(0)-space and G is a closed, totally disconnected and unimodular group of isometries of X with diam(G\ X) ≤ D_0. We refer to Section <ref> for the definition of unimodularity. It naturally holds
𝒪-CAT_0^disc(P_0,r_0,D_0) ⊆𝒪-CAT_0^td,u(P_0,r_0,D_0)
since every discrete group is totally disconnected and unimodular. The unimodular conditions is justified by the following result, which is the main theorem of the paper. It extends Theorem <ref> to totally disconnected, unimodular actions, that is to sequences belonging to 𝒪_0^(P_0,r_0,D_0).
The class 𝒪_0^(P_0,r_0,D_0) is closed (and compact) under Gromov-Hausdorff convergence. In other words let { M_j }⊆𝒪_0^(P_0,r_0,D_0) be such that M_j converges to some space M_∞ in the Gromov-Hausdorff sense. Then M_∞ is the quotient of a proper, geodesically complete, (P_0,r_0)-packed, (0)-space by a closed, totally disconnected, unimodular group of isometries, and (M_∞) ≤ D_0.
Observe that in particular Theorem <ref> shows that the space M_∞ appearing in Theorem <ref> is not only the quotient by a totally disconnected group, but actually by a unimodular one. A more important consequence is that the class 𝒪_0^(P_0,r_0,D_0) is a compactification of 𝒪_0^(P_0,r_0,D_0) with respect to the Gromov-Hausdorff topology. This fact is useful in case we want to find uniform geometric estimates holding for all the spaces in 𝒪_0^(P_0,r_0,D_0). Examples of such uniform estimates will be proved especially in Section <ref> and will be used for the proof of Theorem <ref>. To clarify this idea we present here a particular case of one of them.
Let P_0,r_0,D_0 > 0. Then there exists M_0 = M_0(P_0,r_0,D_0) such that the following holds.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space. Let G<(X) be closed, totally disconnected, unimodular with (G\ X) ≤ D_0. Let g∈ G be a finite order isometry and write its order as (g) = p_1^α_1⋯ p_k^α_k, where the p_i's are prime numbers.
Then max_i p_i ≤ M_0.
It is not difficult to check the validity of this statement on a regular tree. In that case the full isometry group is totally disconnected and unimodular. Theorem <ref> says that a similar phenomenon happens for every totally disconnected, unimodular group acting on a space as in the assumptions. The author does not know a proof of Theorem <ref> for discrete groups that does not use totally disconnected actions. As remarked after (<ref>) any argument by contradiction provides limit groups that can be totally disconnected, or worse, but in general not discrete.
Using the fact that if a group G acts cocompactly on a metric space X, then X is (P_0,r_0)-packed for some P_0,r_0 > 0 (cp. <cit.>), we also get the following, non-uniform, estimate.
Let X be a proper, geodesically complete, (0)-space and let G < (X) be closed, totally disconnected, unimodular and cocompact. Then there exists M ∈ℕ such that every prime number appearing in the prime decomposition of the order of every finite order isometry of G is at most M.
The proof of Theorem <ref> follows a scheme similar to the one we used in <cit.>, but with two additional difficulties. First of all we have to deal with the fact that the groups under consideration are no more discrete, but just totally disconnected. This impacts every key argument in <cit.>: indeed the discussion therein starts from the Margulis Lemma. That is why in Section <ref> we prove the following, new version of the Margulis Lemma holding for totally disconnected actions on proper metric spaces. It generalizes <cit.> and it is proved by similar techniques.
Let K ≥ 1. There exists ε = ε(K) such that the following is true. Let X be any proper metric space and let x∈ X be a point such that (B(x,4), 1) ≤ K. Let G be a closed, totally disconnected group of isometries of X. Then the group G_ε(x) = ⟨ S_ε(x)⟩ contains an open, compact, normal subgroup N such that G_ε(x)/N is discrete, finitely generated and nilpotent.
The expression (B(x,4), 1) refers to the minimal cardinality of a 1-net inside B(x,4), see Section <ref> for a precise definition and its relation with the packing. The set S_ε(x) is the set of elements of G moving x less than ε, sometimes it is called the almost stabilizer of x. For more properties of these sets see Section <ref>. Observe that if G is discrete then N is finite, and Theorem <ref> becomes the known Margulis Lemma <cit.>.
In general the information provided by Theorem <ref> is not sufficient to get a good geometric description of the action of the almost stabilizers. The situation is different if the space X is CAT(0) and the group G is cocompact. Under these assumptions the group G_ε(x)/N is actually abelian and a geometric description of the action of such groups can be provided (cp. Section <ref>). It resembles the description we gave in <cit.> for virtually abelian groups acting on CAT(0)-spaces.
With these tools in our hands we will adapt the proofs of the Splitting Theorem D and the Renormalization Theorem E of <cit.> to our setting. This will be done in Sections <ref> and <ref>.
In Section <ref>, and in particular in Section <ref>, we use the unimodularity assumption, that is not used until this point, to attack the proof of Theorem <ref>. This is enough to prove (<ref>) in this more general setting. To be more precise we can prove that if M_∞ is the Gromov-Hausdorff limit of spaces M_j ∈𝒪-CAT_0^td,u(P_0,r_0,D_0) then M_∞ = G_∞' \ X_∞' for some proper, geodesically complete, (P_0,r_0)-packed, CAT(0)-space X_∞' and some closed, totally disconnected, D_0-cocompact group of isometries G_∞'. Observe that this is exactly the statement of Theorem <ref> if the M_j's belong to 𝒪-CAT_0^disc(P_0,r_0,D_0).
The second additional difficulty in the proof of Theorem <ref> with respect to <cit.> consists in the last step, namely in proving that the group G_∞' above is actually unimodular. We sketch here the argument in the non-collapsed case, the collapsed one being more involved but conceptually similar. In the non-collapsed case, calling M_j = G_j\ X_j, we know that (X_j,G_j) converges in the equivariant Gromov-Hausdorff sense to some (X_∞, G_∞). Moreover G_∞ is totally disconnected by the equivalent of (<ref>), so M_∞ = G_∞\ X_∞, in particular we can take X_∞ = X_∞' and G_∞ = G_∞'. It is here that the non-collapsed assumption simplifies the argument. We need to prove that G_∞ is unimodular. First of all we prove that G_j converges in the pointed Gromov-Hausdorff sense to G_∞ with respect to suitable metrics (cp. Proposition <ref>). Then we give a sufficient condition for the convergence of the Haar measures of G_j to a Haar measure of G_∞ along this pointed Gromov-Hausdorff convergence (cp. Proposition <ref>). Lastly we use uniform quantitative estimates in the spirit of Theorem <ref> to prove that this sufficient condition for the convergence is satisfied.
We end the introduction by highlighting a result of independent interest that we will use in the paper: it is a characterization of cocompact, totally disconnected subgroups of isometries of a proper, geodesically complete, CAT(0)-space in terms of the properties of the action (cp. Theorem <ref>).
Let X be a proper, geodesically complete, (0)-space and let G be a closed, cocompact group of isometries of X. Then the following are equivalent:
(i) G is totally disconnected;
(ii) G is semisimple and inf{ℓ(g) s.t. g hyperbolic} > 0;
(iii) the orbit Gx is discrete for every x∈ X.
(iv) the orbit Gx is discrete for one point x∈ X.
We recall that G is semisimple if it does not contain parabolic isometries, while the quantity ℓ(g) appearing in (ii) is the translation length of the isometry g, i.e. ℓ(g) = inf_x∈ X d(x,gx). The theorem is false even for proper, CAT(0)-spaces that are not geodesically complete (see Example <ref>).
Acknowledgments. The author thanks A.Lytchak for the interesting discussions during the preparation of this paper.
§ PRELIMINARIES
We start recalling several facts we will need along the paper. We start with notions on metric spaces.
§.§ CAT(0)-spaces
Throughout the paper X will be a proper metric space with distance d.
The open (resp. closed) ball in X of radius r, centered at x, will be denoted by B_X(x,r) (resp. B_X(x,r)); we will often drop the subscript X when the space is clear from the context.
A geodesic in a metric space X is an isometry c [a,b] → X, where [a,b] is an interval of ℝ. The endpoints of the geodesic c are the points c(a) and c(b); a geodesic with endpoints x,y∈ X is also denoted by [x,y]. A geodesic ray is an isometry c [0,+∞) → X and a geodesic line is an isometry cℝ→ X. A metric space X is called geodesic if for every two points x,y ∈ X there is a geodesic with endpoints x and y.
A metric space X is called CAT(0) if it is geodesic and every geodesic triangle Δ(x,y,z) is thinner than its Euclidean comparison triangle Δ (x̅,y̅,z̅): that is, for any couple of points p∈ [x,y] and q∈ [x,z] we have d(p,q)≤ d(p̅,q̅) where p̅,q̅ are the corresponding points in Δ (x̅,y̅,z̅) (see for instance <cit.> for the basics of CAT(0)-geometry).
A CAT(0)-space is uniquely geodesic: for every x,y ∈ X there exists a unique geodesic with endpoints x and y.
A CAT(0)-metric space X is geodesically complete if any geodesic c [a,b] → X can be extended to a geodesic line.
The boundary at infinity of a CAT(0)-space X (that is, the set of equivalence classes of geodesic rays, modulo the relation of being asymptotic), endowed with the Tits distance, will be denoted by ∂ X, see <cit.>.
A subset C of X is said to be convex if for all x,y∈ C the geodesic [x,y] is contained in C. Given a subset Y⊆ X we denote by Conv(Y) the smallest convex, closed subset of X containing Y.
If C is a convex subset of a CAT(0)-space X then it is itself CAT(0), and its boundary at infinity ∂ C naturally and isometrically embeds in ∂ X.
We will denote by HD(X) and TD(X) the Hausdorff and the topological dimension of a metric space X, respectively.
By <cit.> we know that if X is a proper and geodesically complete CAT(0)-space then every point x∈ X has a well defined integer dimension in the following sense: there exists n_x∈ℕ such that every small enough ball around x has Hausdorff dimension equal to n_x. This defines a stratification of X into pieces of different integer dimensions: namely, if X^k denotes the subset of points of X with dimension k, then
X= ⋃_k∈ℕ X^k.
The dimension of X is the supremum of the dimensions of its points: it coincides with the topological dimension of X, cp. <cit.>.
§.§ Packing and covering
Let X be a metric space and r>0.
A subset Y of X is called r-separated if d(y,y') > r for all y,y'∈ Y, while it is a r-net if for all x∈ X there exists y∈ Y such that d(x,y)≤ r.
Given x∈ X and 0<r≤ R we denote by Pack(B(x,R), r) the maximal cardinality of a 2r-separated subset of B(x,R), and by Cov(B(x,R), r) the minimal cardinality of a r-net of B(x,R). Moreover we denote by Pack(R,r) (resp. Cov(R,r)) the supremum of Pack(B(x,R), r) (resp. Cov(B(x,R), r)) among all points of X. The packing and covering quantities defined above are classically related as follows (cp. <cit.>):
Pack(B(x,R), r) ≤Cov(B(x,R), r) ≤Pack(B(x,R), r/2),
Pack(R, r) ≤Cov(R, r) ≤Pack(R, r/2).
Given P_0,r_0 > 0 we say that X is (P_0,r_0)-packed if Pack(3r_0,r_0) ≤ P_0.
The packing condition should be thought as a metric, weak replacement of a Ricci curvature lower bound: for more details and examples see <cit.>. Also remark that every metric space admitting a cocompact action is packed (for some P_0, r_0), see the proof of <cit.>.
The packing condition has many interesting geometric consequences for complete, geodesically complete CAT(0)-spaces, as showed in <cit.>, <cit.> and <cit.>. For instance it provides an upper bound on the dimension.
Let X be a complete, geodesically complete, (P_0,r_0)-packed, (0)-space. Then X is proper and
(i) (R,r) ≤ P_0(1+P_0)^R/min{ r,r_0} - 1 for all 0 <r≤ R;
(ii) the dimension of X is at most n_0 := P_0/2;
In particular, for a geodesically complete, CAT(0) space X which is (P_0, r_0)-packed, the assumptions complete and proper are interchangeable.
§.§ Topological groups
We recall now some terminology about topological groups. A topological group is a group G endowed with a topology for which the operation and the inverse are continuous. Every topological group we will consider will be locally compact and σ-compact, in particular second-countable. Any such group G admits a left invariant Haar measure, as well as a right invariant Haar measure. By definition they are Radon measures on the Borel σ-algebra of G that are preserved by the multiplication on the left (resp. on the right). The next lemma characterizes Haar measures.
Let G be a locally compact, second countable group. Then a left (resp. right) invariant measure on the Borel σ-algebra of G is a left (resp. right) invariant Haar measure if and only if it is finite on compact subsets and positive on open subsets.
The Haar measure of every open set is positive by left (resp. right) invariance. Viceversa since G is locally compact and second countable every open set is σ-compact. Therefore by <cit.> every left (resp. right) invariant measure which is finite on compact subsets and positive on open subsets is Radon, and so a Haar measure.
A topological group is called unimodular if it admits a left and right invariant Haar measure. For instance every discrete group is unimodular.
The connected component of the identity of a topological group G will be denoted by G^o. It is always a normal (actually characteristic), closed subgroup of G. The quotient group G/G^o endowed with the quotient topology is still locally compact and totally disconnected as a topological space. Therefore a topological group G is totally disconnected if and only if G^o = { 1 }.
A topological group G is compactly generated if there exists a compact subset S⊆ G such that ⟨ S ⟩ = G. The notation ⟨ S ⟩ stays for the group generated by S. For instance a discrete group is compactly generated if and only if it is finitely generated.
§.§ Isometry groups
For us the main source of topological groups is the full isometry group of a proper metric space.
Let Isom(X) be the group of isometries of X, endowed with the compact-open topology: as X is proper, it is a topological, locally compact, σ-compact group.
Let G be a closed subgroup of Isom(X). For x∈ X and r,R≥ 0 we set
S_r(x, R, X) := { g∈ G s.t. d(y,gy) ≤ r for all y ∈B(x,R)},
S_r(x, R, X) := { g∈ G s.t. d(y,gy) < r for all y ∈B(x,R)},
G_r(x, R, X) := ⟨S_r(x, R, X) ⟩, G_r(x, R, X) := ⟨ S_r(x, R, X) ⟩.
For R=0 we simply write S_r(x, 0, X) =: S_r(x, X) and similarly for the others. When r = 0 we use the notation S_0(x, R, X) =: _G(x,R): it is the pointwise stabilizer of the ball B(x,R). In the same way when both R = r = 0 we use tha notation S_0(x, 0, X) =: _G(x): it is the stabilizer of the point x. When the context is clear we will omit to write the dependence on the metric space X. The sets S_r(x, R) are compact by Ascolì-Arzelà Theorem, since X is proper. In particular the groups G_r(x, R) are compactly generated. Instead the sets S_r(x, R) are open. It follows from the definition that the sets { S_r(x, R) }_r,R > 0 form a neighbourhood basis of the identity.
Let x∈ X be a fixed base point. The left and right invariant L^∞-pseudometrics at scale R of center x are by definition
d_x,R^ℓ(g,h) := max_y∈B(x,R) d(gy,hy), d_x,R^(g,h) := max_y∈B(x,R) d(g^-1y,h^-1y).
As the name suggests they are left (resp. right) invariant pseudometrics on Isom(X). The expressions
d_x^ℓ (g,h) := inf_R > 0{1/R + d_x,R^ℓ(g,h) }; d_x^ (g,h) := inf_R > 0{1/R + d_x,R^(g,h) }
define proper, left (resp. right) invariant metrics on Isom(X) inducing the compact-open topology (cp. <cit.>). This follows also from the next result whose proof is a direct consequence of the definitions. The closed ball of center g and radius r with respect to the metric d_x^ℓ (resp. d_x^r) will be denoted by B_x^ℓ(g,r) (resp. B_x^r(g,r)).
Let X be a proper metric space, G< (X) be a closed subgroup, x∈ X and r>0. Then
S_r/2(x, 2/r) ⊆B_x^ℓ(𝕀,r), B_x^(𝕀,r) ⊆S_r(x, 1/r).
A subgroup G is totally disconnected if it is totally disconnected as a subset of (X) (with respect to the compact-open topology).
A closed group G < (X) is said to be cocompact if the quotient metric space G \ X is compact; in this case, we call codiameter of G the diameter of the quotient, and we will say that G is D_0-cocompact if it has codiameter at most D_0.
Notice that the codiameter of G coincides with
inf{ r>0 s.t. G ·B(x,r) = X ∀ x∈ X}.
It is well-known that if X is geodesic and G < (X) is closed and D_0-cocompact, then the subset S_3D_0(x) is a generating set for G, that is G_3D_0(x)=G, for every x∈ X. If the orbit Gx is discrete then the same is true for 2D_0 in place of 3D_0.
The translation length of g∈Isom(X) is by definition ℓ(g) := inf_x∈ Xd(x,gx).
When the infimum is realized, the isometry g is called elliptic if ℓ(g) = 0 and hyperbolic otherwise. The minimal set of g, Min(g), is defined as the subset of points of X where g realizes its translation length; notice that if g is elliptic then Min(g) is the subset of points fixed by g. An isometry is called semisimple if it is either elliptic or hyperbolic; a subgroup G of Isom(X) is called semisimple if all of its elements are semisimple. The displacement function d_g X → X defined by d_g(x) = d(x,gx) is convex (<cit.>). For λ≥ 0 we call the set M_λ(g) := d_g^-1([0,λ]) the λ-level set of g. It is a closed, convex subset of X. For instance M_ℓ(g)(g) = Min(g).
The free-systole of a group G < (X) at a point x∈ X is
sys^♢(G,x) := inf_g∈ G ∖ G^♢ d(x,gx),
where G^♢ is the subset of all elliptic isometries of G. The free-systole of G is accordingly defined as
sys^♢(G,X) = inf_x∈ Xsys^♢(G,x).
Similarly, the free-diastole of G is defined as
dias^♢(G,X) = sup_x∈ Xsys^♢(G,x).
By definition, we have the trivial inequality
sys^♢(G,X) ≤dias^♢(G,X).
The following result, which extends <cit.> to general groups acting on geodesically complete, CAT(0)-spaces, shows that the free systole and the free diastole are for small values quantitatively equivalent, provided one knows an a priori bound on the diameter of the quotient. Therefore it is equivalent to consider actions with small free-systole or small free-diastole. In the rest of the paper we will focus on the free-systole.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space and let G<(X) be D_0-cocompact. Then
^♢(G, X) ≥( 1+P_0 )e^-(2D_0+1)/min{^♢(G,X), r_0 }.
The proof is the same of <cit.>, we report it for completeness. It is based on the following important fact we will use also later.
Let P_0,r_0,R> 0 and 0<ε≤ r_0. Then there exists δ(P_0,r_0, R, ε) > 0 with the following property. Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space. If g is a non-elliptic isometry of X and x is a point of X such that d(x,gx) ≤δ(P_0,r_0, R, ε), then for every y∈ X with d(x,y)≤ R there exists m∈ℤ^* such that d(y,g^m y) ≤ε.
(We can choose, explicitely, δ(P_0,r_0, R, ε)= (1+P_0)e^-(2R+1)/ε).
Assume that
min{dias^♢ (G , X), r_0 } > ε.
By definition there exists x_0 ∈ X such that for every hyperbolic isometry g ∈ G one has d(x_0,gx_0) >ε.
Now, if sys^♢ (G, X) ≤ (1+P_0)e^-(2D_0+1)/ε =: δ, we could find x ∈ X and a hyperbolic g ∈ G such that d(x,gx) ≤δ.
By Proposition <ref>, for every y ∈B(x,D_0) there would exists a non trivial power g^m satisfying d(y,g^my) ≤ε.
But then, since the action is D_0-cocompact we could find a conjugate γ of g^m (thus, a hyperbolic isometry) such that d(x_0,γ x_0)≤ε, a contradiction. The conclusion follows by the arbitrariness of ε.
§.§ Totally disconnected, cocompact groups of CAT(0)-spaces
We collect here some important properties of cocompact, totally disconnected groups acting on a proper, geodesically complete, CAT(0)-space that we will use extensively along the paper. We suggest the reader to consult also <cit.> and <cit.>.
Let X be a proper, geodesically complete, (0)-space and let G be a closed, cocompact group of isometries of X. Then the following are equivalent:
(i) G is totally disconnected;
(ii) G is semisimple and ^♢(G,X) > 0;
(iii) the orbit Gx is discrete for every x∈ X.
(iv) the orbit Gx is discrete for one point x∈ X.
Moreover if any of the conditions above hold then:
(a) the sets _G(x,R) are open for every x∈ X and every R ≥ 0. They form a fundamental system of neighbourhoods of {}.
(b) The sets S_r(x, R) are open for every x∈ X and every r, R≥ 0.
(c) The set { gx s.t. g∈ K} is finite for every x∈ X and every precompact K⊆ G.
We need the following fact.
Let X be a geodesically complete, (0)-space and let G < (X) be a closed, cocompact group. Then G^o is either the identity or it contains an infinite cyclic discrete subgroup.
Every locally compact group either contains a compact open subgroup or an infinite cyclic discrete subgroup. We apply this statement to G^o: if it contains a compact open subgroup then it must be the whole G^o since it is connected. Therefore G^o is a compact, normal subgroup of G. In particular G stabilizes the non-empty (cp. <cit.>) closed, convex set Fix(G^o). By minimality of G (cp. <cit.>) we have that Fix(G^o) = X, so G^o = {id}. Thus either G^o = {𝕀} or G^o contains an infinite cyclic discrete subgroup.
We will use some basic facts about convergence recalled in Section <ref>.
First of all we observe that, given x∈ X, the orbit Gx is discrete if and only if the group Stab_G(x) is open. Indeed suppose Gx is discrete and take g_j ∈ G converging to g, with gx = x. Since g_jx converges to gx = x and the orbit Gx is discrete, then g_jx = x for j big enough. On the other hand, suppose Stab_G(x) is open and that Gx is not discrete. The set Gx is closed and not discrete, so it has an accumulation point gx. This means that there are pairwise distinct g_jx converging to gx with g_j ∈ G. Up to multiply on the left by g^-1 we can suppose that g_jx converges to x, and it is still composed by pairwise distinct points. The sequence g_j converges, up to a subsequence, to some g' ∈Stab_G(x). Since the latter is open then g_j ∈Stab_G(x) for j big enough, i.e. g_jx = x, which is a contradiction.
Now, (i) implies (ii) by <cit.>, while (i) implies (a) by <cit.> and so (i) implies also (iii) by the first observation. It is clear that (iii) implies (iv).
Now we show that (iv) implies (i). As in the proof of <cit.> we denote by C the subset of X of points whose stabilizer is open. Then C is convex (if x,y∈ C then Stab_G(x)∩Stab_G(y) is an open subgroup that stabilizes [x,y]) and G-invariant. It is also non-empty because of (iv) and the first observation. Since the action of G is minimal (cp. <cit.>) we conclude that C is dense. We use this fact to provide a neighbourhood basis of the identity made of compact open subgroups. This is enough to show that G is totally disconnected. We fix a basepoint x_0∈ X, R≥ 0 and ε > 0 and we choose a finite subset { c_1,…,c_n} of C∩B(x,R) which is ε-dense in B(x,R). We set U(R,ε) := ⋂_i=1^n Stab_G(c_i). It is clear that U(R,ε) is open and compact, being a finite intersection of open and compact subgroups. We claim that the family { U(R,ε) } form a neighbourhood basis of {𝕀}. For that it is enough to show that ⋂_R≥ 0, ε > 0 U(R,ε) = {𝕀}. Let g be in this intersection. For any R≥ 0 and any ε > 0 we know that g acts trivially on a ε-net of B(x_0,R), so it displaces every point of B(x_0,R) by at most 2ε. Since this is true for every ε we deduce that g fixes pointwise the whole B(x_0,R). By arbitrariness of R we conclude that g fixes pointwise every point of X, i.e. g=𝕀.
It remains to show that (ii) implies (i). We suppose that G has only elliptic and hyperbolic elements and that sys^♢(G,X) =: s > 0. We claim that the set of elliptic isometries G^♢ is open and closed in G. Suppose g_j ∈ G converges to g ∈ G^♢ and let x∈Fix(g). Then, for j big enough, we have d(x,g_jx) < s, so g_j∈ G^♢. This shows that G^♢ is open.
Suppose now to have g_j ∈ G^♢ converging to a hyperbolic isometry g. Let x be a point such that d(x, g x) = ℓ(g).
We denote by z_j the projection of x on the closed, convex set M_s/4(g_j). It must hold d(x, z_j) → +∞, otherwise ℓ(g) ≤s/4. Let y_j be the point along the geodesic [z_j, x] such that d(g_jy_j, y_j) = s/2. Observe that by convexity of the displacement function d_g_j along the geodesic [z_j,x] we have d(z_j,y_j) → +∞. We consider the sequence of isometric actions (X, y_n, G) and we fix a non-principal ultrafilter ω (see Section <ref> for the definitions). Since G is cocompact, by Lemma <ref> we have that the ultralimit (X_ω, y_ω, G_ω) of this sequence is naturally equivariantly isometric to (X,x,G). Observe that the sequence g_j defines an element g_ω of G_ω. The condition d(z_j,y_j) → +∞ forces the inequality ℓ(g_ω) ≥s/4, so g_ω is hyperbolic. On the other hand d(g_ω y_ω, y_ω) = s/2, so ℓ(g_ω) ≤s/2. This contradicts the assumption sys^♢(G_ω,X_ω) = sys^♢(G,X) = s. Therefore the set G^♢ is open and closed, implying G^♢∩ G^o = G^o. By Lemma <ref> either G^o = {𝕀} or it contains an infinite cyclic discrete subgroup. Every element of G^o is elliptic, so if it generates a discrete subgroup then it must be of finite order. This forces to have G^o={id}, i.e. G is totally disconnected.
This concludes the proof of the equivalences (i)-(iv).
These conditions imply (a) by <cit.>. Now (b) is a consequence of (i). Indeed suppose S_r(x, R) is not open for some x∈ X and r, R≥ 0. Then
there exists a sequence g_j ∈ G ∖S_r(x, R) converging to g∈S_r(x, R).
Then g^-1g_j converges to ∈_G(x, R). By (a) it must hold g^-1g_j ∈_G(x, R) for j big enough, i.e. g_j = g on B(x,R). This contradicts the assumption g_j ∉S_r(x, R). Finally if K is a compact subset of G then Gx is a compact subset of X which is also discrete by (iii). Therefore it must be finite.
The equivalences (i)-(iv) are no more true if the space is proper, CAT(0), but not geodesically complete. In the following example we show that (i) does not imply (iii).
We construct a tree in the following iterative way. We start with X_1 = [-1,1]. The set V_1 = {± 1 } is the set of free vertices of X_1, i.e. those vertices with only one edge issuing from them. To construct X_2 we glue a segment of length 1 at the point 0 ∈ X_1. X_2 is composed by three segments of length 1 glued at a point. Its set of free vertices V_2 has 3 elements and contains V_1. To construct X_3 we glue segments of length 1/2 to the midpoints of all the segments of length 1 composing X_2, so that X_3 is composed by 9 segments of length 1/2. Its set of free vertices is denoted by V_3: it contains V_2. Given X_j, made of 3^j-1-segments of length 2^-j + 2, we construct X_j+1 by gluing segments of length 2^-j+1 to each midpoint of the segments of X_j. We denote by V_j+1 the set of free vertices of X_j: it contains V_j. We repeat the procedure inductively and we define the space X_∞: it is a tree, so a CAT(0)-space, which is also compact, so proper, but not geodesically complete. Let us call V_∞ its set of free vertices. For every j and every two free vertices v,w ∈ V_j we can find an isometry of X_j sending v to w. Such isometry extends naturally to an isometry of X_∞, by construction of X_∞. Therefore Isom(X_∞) acts transitively on the set of free vertices V_∞. This implies that its action on X_∞ is minimal. The same proof of Lemma <ref> shows that either Isom(X_∞) is totally disconnected or it contains an infinite cyclic discrete subgroup. The second possibility cannot occur because Isom(X_∞) is compact, so it contains only elliptic isometries and a discrete group generated by an elliptic isometry is finite. This shows that Isom(X_∞) is totally disconnected. However the orbit by Isom(X_∞) of every free vertex is not discrete.
§.§ Crystallographic groups in the Euclidean space
We will denote points in ℝ^k by a bold letter v, and the origin by O. Among CAT(0)-spaces, the Euclidean space ℝ^k and its discrete groups play a special role.
A crystallographic group is a discrete, cocompact group Γ of isometries of some ℝ^k. The simplest and most important of them, in view of Bieberbach's Theorem, are Euclidean lattices: i.e.
free abelian crystallographic groups.
It is well known that a lattice must act by translations on ℝ^k (see for instance <cit.>); so, alternatively, a lattice ℒ can be seen as the set of linear combinations with integer coefficients of k independent vectors b_1, …, b_k (we will make no difference between a lattice and this representation).
A basis ℬ = { b_1,…, b_k} of a lattice ℒ is a set of k independent vectors that generate ℒ as a group. There are many geometric invariants classically associated to a lattice ℒ, we will need three of them:
– the covering radius, which is defined as
ρ(ℒ) = inf{ r > 0 s.t. ⋃_ v ∈ℒB( v,r) = ℝ^k};
– the shortest generating radius, that is
λ(ℒ) = inf{ r > 0 s.t. ℒ contains k independent vectors of length≤ r};
– the shortest vector, that is
τ(ℒ) = inf{‖ v‖ s.t. v∈ℒ∖{ O}}.
Notice that, by the triangle inequality, any lattice ℒ is 2ρ(ℒ)-cocompact. Moreover τ(ℒ) coincides with the free-systole of ℒ.
It is always possible to find a basis ℬ= { b_1,…, b_k} of ℒ such that τ(ℒ) = ‖ b_1 ‖≤⋯≤‖ b_k ‖ = λ(ℒ);
this is called a shortest basis of ℒ.
The shortest generating radius and the covering radius are related as follows:
ρ(ℒ) ≤√(k)/2·λ(ℒ).
In particular every lattice ℒ is (√(k)·λ(ℒ))-cocompact.
The proof of the Svarc-Milnor lemma in <cit.> gives the following estimate. We recall that given a generating set S of a lattice ℒ then ℓ_S( v) denotes the length of the shortest word in the alphabet S needed to write v.
Let ℒ be a lattice of ℝ^k. Recall it is (√(k)·λ(ℒ))-cocompact. Let S = S_2√(k)·λ(ℒ)( O) as defined in (<ref>) and recall it is a generating set. Then ℓ_S( v) ≤‖ v‖/τ(ℒ) + 1.
A property we will need in Section <ref> is the existence of nice bases.
Let ℒ be a lattice of ℝ^k. Then there exists a basis ℬ = { b_1,…, b_k} of ℒ such that
(i) max_i ‖ b_i‖≤ 2^k-1·λ(ℒ);
(ii) ∠( b_i, ∑_j≠ iℝ b_j) ≥ϑ_k > 0, where ϑ_k depends only on k.
The notation ∑_j≠ iℝ b_j means the (k-1)-dimensional subspace of ℝ^k generated by { b_j}_j≠ i.
It follows by the combination of <cit.> and <cit.>.
A set of vectors { v_1, …, v_k} satisfying condition (ii) of Lemma <ref> for some ϑ > 0 are called ϑ-linearly independent.
From any lattice we can always extract a sublattice with controlled geometry.
Let ℒ be a lattice of ℝ^k. Then there exists a sublattice ℒ' of ℒ such that λ(ℒ') = λ(ℒ) and λ(ℒ') ≤ 2·τ(ℒ').
We proceed recursively. If λ(ℒ) ≤ 2·τ(ℒ) there is nothing to prove. Otherwise we take a shortest basis { b_1, …, b_k}. In particular ‖ b_1‖ = τ(ℒ). Let λ_1 ∈ℕ be such that λ_1 ‖ b_1‖≤λ(ℒ) ≤ 2 λ_1 ‖ b_1‖. We replace ℒ with the sublattice ℒ' generated by {λ_1 b_1, …, b_k}. We have λ(ℒ') = λ(ℒ) and τ(ℒ') ≥τ(ℒ). We repeat the procedure up to arrive to a lattice satisfying the thesis. The algorithm stops because we need to do at most N steps, where N= #{ v∈ℒ s.t. ‖ v‖ < λ(ℒ)/2} < +∞.
The content of the famous Bieberbach's Theorems can be stated as follows.
There exists J(k), only depending on k, such that the following holds true.
For any crystallographic group Γ of ℝ^k
the subgroup ℒ(Γ) = Γ∩(ℝ^k) is a normal subgroup of index at most J(k), in particular a lattice.
Here (ℝ^k) denotes the normal subgroup of translations of (ℝ^k).
The subgroup ℒ(Γ) is called the maximal lattice of Γ.
As in <cit.> we will use the next fact in the study of collapsing sequences: it follows directly from Proposition <ref>.
Let Γ be a crystallographic group of ℝ^k, and let 0<r < √(2 sin( π/J(k) )). If g ∈Γ moves all points of B_ℝ^k ( O, 1/r) less than r, then g is a translation.
§ MARGULIS LEMMA AND ALMOST ABELIAN GROUPS
Our aim is to extend the Margulis' Lemma to totally disconnected group actions. In order to do so we need to introduce a new terminology.
§.§ Almost nilpotent and almost abelian groups
A topological group A is almost nilpotent (resp. almost abelian) if there exist a compact, open, normal subgroup N◃ A such that A/N is discrete, finitely generated and virtually nilpotent (resp. virtually abelian).
Observe that the existence of a compact, open subgroup N implies that A is locally compact.
Recall that for locally compact groups it holds the open mapping theorem, and in particular the classical isomorphism theorems provide topological isomorphisms.
The notion of being almost nilpotent or abelian passes to closed subgroups.
Let A be an almost nilpotent (resp. abelian) group. Then any closed subgroup B < A is almost nilpotent (resp. abelian).
Let N be a compact, open, normal subgroup of A such that A/N is discrete, finitely generated and virtually nilpotent (resp. virtually abelian). The group N' = N ∩ B is open, compact and normal in B because B is closed. The group B / N' is topologically isomorphic to a subgroup of A/N, thus it is discrete, finitely generated and virtually nilpotent (resp. virtually abelian).
The rank of an almost abelian group is well defined.
Let A be an almost abelian group. Then there exists k≥ 0 such that for every compact, open, normal subgroup N ◃ A, the group A/N is discrete, finitely generated and virtually abelian of rank k ≥ 0. Such k is called the of A and it is denoted by (A).
Let N_0 be a compact, open, normal subgroup of A such that A/N_0 is discrete, finitely generated and virtually abelian. Every virtually abelian, finitely generated group contains a free abelian subgroup of finite index of some rank k. Let A_0 < A be the preimage under the projection map π A → A/N_0 of such a free abelian group. Observe that the index of A_0 in A is finite and A_0 / N_0 ∩ A_0 is topologically isomorphic to ℤ^k.
Let N◃ A be an arbitrary compact, open, normal subgroup.
The index of N∩ N_0 in both the compact groups N and N_0 is finite because it is an open subgroup.
So A_0/N∩ N_0 ∩ A_0 is a finite extension of A_0/N_0 ∩ A_0 ≅ℤ^k. Therefore A_0/N∩ N_0 ∩ A_0 is discrete, finitely generated and virtually abelian of rank k. Since A_0/N ∩ A_0 is the quotient of A_0/N∩ N_0 ∩ A_0 by a finite group, then it is still virtually abelian of same rank k. Finally A/N contains A_0/N ∩ A_0 as a finite index group, so it is still discrete, finitely generated and virtually abelian of rank k.
We recall that a closed subgroup H of a locally compact group G is said cocompact if the topological space G/H is compact. We will use the two definitions of cocompactness, this one and the one for group actions on metric spaces, in different contexts. We hope the reader will not be confused.
Observe that any finite index subgroup is automatically cocompact, and that for discrete groups the two notions coincide. Moreover the following holds.
Let G be a locally compact group and H,K<G be closed subgroups.
(i) If K<H<G with K cocompact in H and H cocompact in G then K is cocompact in G;
(ii) if H is cocompact in G then H∩ K is cocompact in K;
(iii) if H and K are cocompact in G then H∩ K is cocompact in G.
It is classical that if G is locally compact then a closed subgroup H<G is cocompact if and only if G= C· H for some compact subset C. Therefore (i) follows. Instead (ii) is straightforward from the definition and (iii) follows by (i) and (ii). Indeed H∩ K is cocompact in K by (ii), but K is cocompact in G, so H∩ K is cocompact in G by (i).
We end this introduction on almost abelian groups with a useful result.
Let A be an almost abelian group and let B<A closed. Then (B) ≤(A). Moreover (B) = (A) if and only if B is cocompact in A.
The first inequality is obvious from Lemma <ref>, so we pass to the second part of the statement. As in the beginning of the proof of Lemma <ref> we take a compact, open, normal subgroup N_0 of A and a finite index subgroup A_0 < A such that A_0/N_0 is topologically isomorphic to ℤ^k. Let π A → A/N_0 be the projection map.
Suppose first that B is cocompact in A. In particular B_0 = B ∩ A_0 is cocompact in A_0 by Lemma <ref>.(ii). The image of B_0 through π is therefore cocompact in the image of A_0. Since the image of A_0 is ℤ^k, then all its cocompact subgroups have finite index. Moreover every finite index subgroup of ℤ^k is isomorphic to ℤ^k. Therefore B_0/B_0 ∩ N_0 is topologically isomorphic to ℤ^k. By Lemma <ref> we conclude that rk(B) = k = rk(A).
Suppose now rk(B) = rk(A) = k. Observe that also B_0 = B∩ A_0 is almost abelian of rank k, being a cocompact subgroup of B since it has finite index. Lemma <ref> says that B_0/N ∩ B_0 is discrete, finitely generated and virtually abelian of rank k, and moreover it is topologically isomorphic to a subgroup of A_0/A_0 ∩ N_0 ≅ℤ^k. In particular B_0/B_0∩ N_0 is free abelian of rank k, so it has finite index in A_0/A_0∩ N_0. We deduce that π^-1(π(B_0)) has finite index in A_0. Moreover π^-1(π(B_0)) = B_0· N_0, so B_0 is cocompact in π^-1(π(B_0)). Since π^-1(π(B_0)) has finite index in A_0, so in A, we deduce that B_0, and so B, is cocompact in A.
§.§ The Margulis Lemma for totally disconnected group actions
For discrete groups Γ < (X) of a proper metric space X the following version of the Margulis' Lemma, due to Breuillard-Green-Tao, holds.
Let K ≥ 1. There exists ε(K) such that the following is true. Let X be any proper metric space and let x∈ X be a point such that (B(x,4), 1) ≤ K. Let Γ<(X) be discrete. Then the almost stabilizer Γ_ε'(x) is virtually nilpotent for all 0≤ε' ≤ε(K).
Our aim is to extend this result to totally disconnected groups of isometries, and in particular to prove Theorem <ref>.
The proof of Proposition <ref> is based on the precise description of the structure of K-approximate subgroups of discrete groups developed in <cit.>.
Let K > 1. A K-approximate subgroup of a group G is a symmetric subset of G containing the identity for which there exists a symmetric subset X ⊆ G of cardinality at most K such that A · A ⊆ X · A.
The more general structure theory for open precompact approximate subgroups of locally compact groups was studied in <cit.> and partially improved in <cit.>. We report here a slightly modified version of <cit.> which adapts better to our situation.
Let K ≥ 1. Then there exists K' =K'(K), depending only on K, such that the
following holds. Assume G is a totally disconnected, locally compact group generated by a compact symmetric set S containing the identity. Let A be an open precompact K-approximate subgroup of G such that S^K'⊆ A. Then there are subgroups
N',L ◃ G with N' ⊆ A^6 ∩ L such that N' is compact, L is open and has
index O_K(1) in G, and such that L/N' is a Lie group of dimension at most
O_K(1).
Here the notation O_K(1) stays for a quantity which is bounded above by a constant depending only on K. It is the same notation used in <cit.>, <cit.> and <cit.>.
The proof is a straightforward modification of <cit.>, where one uses A in place of S^n therein.
This gives the analogue of <cit.> for totally disconnected, locally compact groups that are compactly generated.
Let K > 1. Then there exists K' = K'(K), depending only on K, such
that the following holds true. Assume G is a totally disconnected, locally compact group generated by a compact symmetric set S containing the identity. Let A be an open precompact K-approximate subgroup of G such that S^K'⊆ A. Then there exist G_1 < G of index ≤ O_K(1) and a compact, open, normal subgroup N◃ G, with N⊆ G_1, such that G_1/N is discrete, finitely generated and nilpotent of rank and step ≤ O_K(1). In particular G is almost nilpotent.
Let K' be the maximum between the constant of Proposition <ref> and the one provided by <cit.>.
By Proposition <ref> we can find subgroups N',L ◃ G such that
(a) L is open and it has finite index in G;
(b) N' is compact and L /N' is a Lie group.
Condition (a) implies L is also closed, so in particular it is totally disconnected. The quotient of a totally disconnected, locally compact group by a closed, normal subgroup is again totally disconnected. Therefore by (b) we deduce that L /N' is discrete. In particular N' is open in L, so also in G, because it is the continuous preimage of the open set { 1}⊆ L/N'.
Let π G → G/N' be the projection map. The projections of the precompact sets S and A are precompact, hence finite. Clearly π(S) generates G/N', π(A) is a K-approximate group in the sense of <cit.> and π(S)^K' = π(S^K')⊆π(A). Therefore G/N' is finitely generated and it satisfies the assumptions of <cit.>. Hence we can find G_1 < G/N' of index at most O_K(1) and a finite subgroup N◃ G/N', with N < G_1, such that G_1/N is nilpotent of rank and step ≤ O_K(1). In order to get the thesis it is enough to take G_1 = π^-1(G_1) and N = π^-1(N). Indeed N is clearly normal, it is open because it contains N' and it is compact because N/N' ≅N is finite.
Finally we can argue as in <cit.> to get the announced version of the Margulis lemma.
Let K ≥ 1. There exists ε = ε(K) such that the following is true. Let X be any proper metric space and let x∈ X be a point such that (B(x,4), 1) ≤ K. Let G < (X) be closed and totally disconnected. Then the open almost stabilizer G_ε'(x) is almost nilpotent for all 0≤ε' ≤ε(K).
The set A:= S_2(x) is open and it is precompact because G is closed and X is proper.
Take K points x_i ∈B(x,4) such that B(x,4) ⊆⋃_i=1^K B(x_i,1). Suppose that for i=1,…, k ≤ K there is g_i∈ S_4(x) such that d(x_i,g_ix) ≤ 1, while this does not happen for k+1≤ i ≤ K. For every g∈ A^2 ⊆ S_4(x) we find i such that d(gx,x_i) ≤ 1, so d(gx,g_ix) ≤ 2, i.e. g_i^-1g∈ S_2(x). In other words A^2 ⊆ S_4(x) ⊆⋃_i=1^k g_i· S_2(x). Therefore A is an open, precompact, K-approximate subgroup of G. Now let K'=K'(K) be the constant of Proposition <ref> and take ε(K) := 2/K'. The thesis holds for this choice of ε(K).
The conclusion of Theorem <ref> can be improved for totally disconnected, cocompact groups acting on geodesically complete, CAT(0)-spaces.
Let P_0,r_0 > 0. Then there exists ε_0 = ε_0(P_0,r_0) > 0 such that the following holds. Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space. Let G be a closed, totally disconnected, cocompact group of isometries of X. Then both the open and closed almost stabilizers G_ε(x) and G_ε(x) are almost abelian, for all x∈ X and all 0≤ε≤ε_0.
First of all Cov(4,1) is bounded in terms of the packing constants by Proposition <ref> and (<ref>). Therefore by Theorem <ref> we can find ε_0 depending only on P_0,r_0 such that every open stabilizer G_ε(x) with 0≤ε≤ε_0 is almost nilpotent. Moreover the same is true for the closed stabilizers because by Theorem <ref>.(b) the sets S_r(x) are open and the proof of Theorem <ref> holds also for these open, compact, approximate subgroups.
Let us prove that the open almost stabilizers above are actually almost abelian. The same proof will apply for the closed ones. The action of G is by semisimple isometries by Theorem <ref>.(ii). Let N be a compact, open, normal subgroup of G_ε(x) such that G_ε(x) /N is discrete, finitely generated and virtually nilpotent. We can find a finite index subgroup G' < G_ε(x) such that G'/G' ∩ N is nilpotent. The subset Min(N) is closed, convex and non-empty because N is compact (<cit.>): in particular it is a CAT(0)-space. The group G/N acts by semisimple isometries (<cit.>) on the CAT(0)-space Min(N). The kernel of this action is another normal subgroup N_0 of G which is open (because it contains N) and compact (because it fixes some point of X). Moreover Min(N) = Min(N_0). Therefore the group G/N_0 acts freely on Min(N_0), in particular so does the group G' / G'∩ N_0 which is discrete, finitely generated and nilpotent. It is not restrictive, up to take a finite index subgroup of G', to suppose that G' / G'∩ N_0 is torsion-free. By <cit.> we conclude that G'/G'∩ N_0 is actually abelian. Therefore G_ε(x)/N_0 is discrete, finitely generated and virtually abelian, i.e. G_ε(x) is almost abelian.
We will often refer to the constant ε_0=ε_0 (P_0,r_0) as the Margulis' constant.
From the proof of Theorem <ref> we can extract the following fact that will be used in Section <ref>. For simplicity it will be presented for geodesically complete, CAT(0)-spaces, but a similar version holds for proper metric spaces.
Let P_0,r_0 > 0. Then there exists M_0 = M_0(P_0,r_0) such that the following holds.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space, let G < (X) be closed and let μ be a left-invariant Haar measure of G.
Then
μ(S_r(x)^p) ≤ M_0^p-1·μ(S_r(x))
for every x∈ X, every r ≥ 0 and every p∈ℕ.
As in the proof of Theorem <ref> we find that S_r(x)^2 ⊆S_2r(x) ⊆⋃_i=1^k g_i S_r(x), for some k≤(2r, r/2). By (<ref>) and Proposition <ref>.(i) we get k ≤ P_0(1+P_0)^2r/min{r/4,r_0 } - 1 =: M_0. By induction on p we have
S_r(x)^p = S_r(x)^p-1·S_r(x) ⊆⋃_i_1,…,i_p-1 = 1^k g_i_1⋯ g_i_p-1S_r(x).
Therefore μ(S_r(x)^p) ≤ M_0^p-1·μ(S_r(x)).
§.§ Almost abelian groups acting on CAT(0)-spaces
Corollary <ref> suggests that almost abelian groups acting on CAT(0)-spaces play a special role. In this section we will describe the geometric properties of such groups, generalizing <cit.>.
Let X be a proper (0)-space and let A<(X) be closed, semisimple and almost abelian of rank k. Then k is the unique integer for which the following holds.
(i) There exists a closed, convex, A-invariant subset C(A) of X that splits as W×ℝ^k such that
(i.a) each g∈ A preserves the product decomposition and acts as the identity on the first component;
(i.b) the image A_ℝ^k of A under the projection A →(ℝ^k) is a crystallographic
group;
(i.c) if S is a compact, symmetric, generating set for A containing the identity, then there exists a finite subset Σ⊆ S^4J(k)+2 whose projection Σ_ℝ^k on (ℝ^k) generates
ℒ(A_ℝ^k), where J(k) is the constant of Proposition <ref>.
(ii) There exists ∂ A ⊆∂ X, called the trace at infinity of A, such that
(ii.a) ∂ A = ∂(Ax) for every x∈ X;
(ii.b) ∂ A is closed, convex, A-invariant and isometric to 𝕊^k-1;
(ii.c) if g is an isometry of X then ∂ (gAg^-1) = g ∂ A;
(ii.d) if B<A is closed then ∂ B ⊆∂ A with equality if and only if (B) = (A).
Let N be an open, compact, normal subgroup of A. The set (N) is closed, convex, and A-invariant because N is normal in A. It is also non-empty because N is compact (cp. <cit.>). Let N_0 be the kernel of the action of A on (N). It is again a compact, open, normal subgroup of A, because it contains N. Moreover Min(N_0)⊆Min(N) because N ⊆ N_0, and by definition we have Min(N_0)= Min(N). It follows that the induced action of A/N_0 on the CAT(0)-space Min(N_0) is faithful. Moreover A/N_0 is a discrete, finitely generated, virtually abelian group of rank k by Lemma <ref>, whose action on Min(N_0) is by semisimple isometries (<cit.>). Then (i) follows by the classical statement for discrete groups (cp. <cit.>, <cit.>).
Suppose h is another integer for which (i) holds. Let N be the kernel of the action of A on C(A) = W ×ℝ^h. It is clearly compact and normal. It is also open because the action of A on C(A) is discrete by (i.a) and (i.b). Therefore the group A/N must be discrete, finitely generated and virtually abelian of rank k by Lemma <ref>. However the group A/N acts faithfully and cocompactly on ℝ^h, so it must have rank h, as a discrete, virtually abelian group. Therefore k=h.
Let C(A) = W ×ℝ^k be as in (i). Fix w∈ W and call Z = { w }×ℝ^k: it is again an A-invariant, convex subset because of (i.a). We claim that ∂ Z = ∂Conv(Ax) for every x∈ X. Fix x∈ X and set R = d(x,Z). By A-invariance of Z we have that d(ax, Z) = R for every a∈ A. So Ax ⊆B(Z,R). The latter is a convex set, so Conv(Ax) ⊆B(Z,R) and ∂Conv(Ax) ⊆∂B(Z,R). By convexity of the distance function every geodesic ray of B(Z,R) must be at constant distance from Z, i.e. parallel to it. This shows that ∂Conv(Ax) ⊆∂B(Z,R) = ∂ Z. For the other inclusion we call z∈ Z the projection of x on Z. We fix any ξ∈∂ Z and we consider the geodesic ray [z,ξ]. The action of A on Z is cocompact by (i.b) so we can find some D>0 and elements a_j ∈ A such that d(a_jz,[z,ξ]) ≤ D and d(z,a_jz) tends to +∞. They belong to Conv(Ax). Since d(x,a_jx) ≥ d(z,a_jz) - 2R tends to +∞, then by properness of X the segments [x,a_jx] subconverge to a geodesic ray [x,ζ]. This ray is contained in Conv(Ax) because each segment [x,a_jx] is. Moreover d(a_jx, [z,ξ]) ≤ R + D, so ξ = ζ. This shows that ∂Conv(Ax) = ∂ Z. We set ∂ A := ∂ Z = ∂Conv(Ax) and we claim it satisfies (ii). The fact that ∂ A is closed, convex, A-invariant and isometric to 𝕊^k-1 is clear since ∂ Z has these properties.
In order to get (ii.c) we fix x∈ X and we use twice the characterization in (ii.a) to get
∂(gAg^-1) = ∂Conv((gAg^-1) gx) = g∂Conv(Ax) = g∂ A.
Moreover (ii.a) gives directly the inclusion in (ii.d). The equality case follows again from (ii.a) applied to some z∈ Z ≅ℝ^k on which both A and B act as discrete groups.
The following is an immediate consequence of Proposition <ref>.
Under the same assumptions of Proposition <ref> we have:
(i) (A) ≤(X), for any notion of dimension of X (topological, Hausdorff, geometric as in <cit.>);
(ii) for every C(A) = W×ℝ^k as in (i) and every subset S ⊆ A whose projection on (ℝ^k) generates a cocompact group of ℝ^k then ⟨ S ⟩ < A has rank k.
§ THE SPLITTING THEOREM
In this section we will prove the analogue of the splitting Theorem <cit.> or more precisely of <cit.> for sufficiently collapsed totally disconnected group actions, i.e. for actions with free-systole sufficiently small. We will prove a slightly improved version that will be useful in Section <ref>, the improvement is in items (v) and (vi). After our work on the Margulis Lemma and on the properties of almost abelian groups acting on CAT(0)-spaces we can essentially mimic the proof of <cit.>.
To set the notation, recall that for any proper, geodesically complete, (P_0, r_0)-packed, CAT(0)-space X we have the upper bound (X)≤ n_0 = P_0/2 provided by Proposition <ref> and the Margulis' constant ε_0 (only depending on P_0, r_0) given by Proposition <ref>.
For a totally disconnected subgroup G < (X)
recall the definition (<ref>) of the subgroup G_r (x) < G generated by S_r (x) given in Section <ref>.
Given positive constants P_0,r_0,D_0,
there exists a function
σ_P_0,r_0,D_0: (0,ε_0] → (0,ε_0] (depending only on the parameters P_0,r_0,D_0) such that the following holds.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space, and G < (X) be closed, totally disconnected and D_0-cocompact.
For any chosen ε∈ (0, ε_0], if ^♢(G,X) ≤σ_P_0,r_0,D_0(ε) then:
(i) the space X splits isometrically as Y ×ℝ^k, with k≥ 1, and this splitting is G-invariant;
(ii) there exists ε^∗∈ (σ_P_0,r_0,D_0 (ε), ε)
such that the rank of the almost abelian subgroups G_ε^∗(x) is exactly k, for all x∈ X;
(iii) the traces at infinity ∂G_ε^∗(x) equal the boundary 𝕊^k-1 of the convex subsets { y }×ℝ^k, for all x ∈ X and all y∈ Y;
(iv) for every x∈ X there exists y∈ Y such that G_ε^∗(x) preserves { y }×ℝ^k. The closure of the projection of G_ε^∗(x) on (Y) is compact;
(v) the projection
of G_ε^∗(x) on (ℝ^k) is a crystallographic group, whose maximal lattice ℒ_ε^*(x) satisfies λ(ℒ_ε^*(x)) ≤ε^*/2√(n_0) and is (ε^*/2)-cocompact;
(vi) the groups G_ε^*(x) are almost commensurated for all x∈ X.
Here, by G-invariant splitting we mean that every isometry of G preserves the product decomposition. By <cit.> we can see G as a subgroup of (Y) ×(ℝ^k). In particular it is meaningful to talk about the projection of G_ε^*(x) on Isom(Y) and Isom(ℝ^k).
We call the integer 1 ≤ k≤ n_0 the ε^∗-splitting rank of X. The fact that k is at most n_0 follows by Corollary <ref>. Observe that we do not need any unimodularity assumption on X. However we will say even more on the projection of G_ε^*(x) on Isom(Y) in Section <ref> in case G is unimodular. The almost commensurability condition of item (vi) is introduced below: it plays a fundamental role in the proof of Theorem <ref>.
§.§ Almost commensurability
Let G be a topological group. We say that two subgroups H,K <G are almost commensurable if H∩ K is cocompact in both H and K. A subgroup H<G is almost commensurated in G if H and gHg^-1 are almost commensurable for every g∈ G. Observe that if G is discrete then this notion coincides with the classical notion of commensurated subgroup.
Suppose S is a generating set for G. If H and gHg^-1 are almost commensurable for all g∈ S then H is almost commensurated.
Let g=g_1⋯ g_k be an element of G, with g_i ∈ S. We prove that H and gHg^-1 are almost commensurable by induction on k. The case k=1 is the assumption. In the general case, by inductive assumption we have
(a) g_2⋯ g_k H g_k^-1⋯ g_2^-1∩ H is cocompact in both g_2⋯ g_k H g_k^-1⋯ g_2^-1 and H;
(b) g_1Hg_1^-1∩ H is cocompact in both g_1Hg_1^-1 and H.
Conjugating (a) by g_1 we also get
(c) g H g^-1∩ g_1Hg_1^-1 is cocompact in both g H g^-1 and g_1Hg_1^-1.
Combining (b) and (c) we conclude that g H g^-1∩ g_1Hg_1^-1∩ H is cocompact in g_1Hg_1^-1∩ H and so in H, by Lemma <ref>. A fortiori g H g^-1∩ H is cocompact in H. In a similar way, using again Lemma <ref>, we also have that g H g^-1∩ H is cocompact in gHg^-1.
Lemma <ref> implies the following property.
Let G be a locally compact group and let K<H < G be closed subgroups, with K cocompact in H. Then H is almost commensurated if and only if K is.
We will show how an almost abelian subgroup which is almost commensurated provides an associated metric splitting of the space, generalizing the discrete case (cp. <cit.>, <cit.>). We just need to recall an additional definition. Given Z⊆∂ X we say that a subset Y⊆ X is Z-boundary-minimal if it is closed, convex, ∂ Y = Z and Y is minimal with these properties. The union of all the Z-boundary-minimal sets is denoted by (Z).
Let X be a proper (0)-space and let Z be a closed, convex subset of ∂ X which is isometric to 𝕊^k-1. Then each Z-boundary-minimal subset of X is isometric to ℝ^k and (Z) is a closed, convex subset of X which splits isometrically as Y×ℝ^k. Moreover Z coincides with the boundary at infinity of all the slices { y }×ℝ^k, for y ∈ Y.
We can now generalize <cit.>.
Let X be a proper, geodesically complete, (0)-space, and G < (X) be closed, totally disconnected and cocompact.
If A < G is a semisimple, almost abelian, almost commensurated subgroup of G of rank k then we have X=(∂ A) and X splits isometrically and G-invariantly as Y ×ℝ^k. Moreover, the projection of A on (ℝ^k) is a crystallographic group and the closure of the projection of A on (Y) is compact. The splitting X=Y ×ℝ^k satisfies the following properties:
(i) the trace at infinity ∂ A is G-invariant and coincides with the boundary of each slice { y }×ℝ^k, for all y ∈ Y;
(ii) if A' < A is another almost abelian, almost commensurated subgroup of G of rank k' then the splittings X = Y ×ℝ^k and X= Y' ×ℝ^k' associated respectively to A and A' are compatible, i.e. Y' is isometric to Y ×ℝ^k-k'.
The trace at infinity ∂ A of A is closed, convex and isometric to 𝕊^k-1, by Proposition <ref>.(ii.b). We claim it is G-invariant. If g∈ G then A∩ gAg^-1 is closed and cocompact in both A and gAg^-1. Moreover A∩ gAg^-1 is almost abelian by Lemma <ref>, and rk(A∩ gAg^-1)= rk(A) = rk(gAg^-1) = k by Lemma <ref>.
By (ii.c) and (ii.d) of Proposition <ref> we have
∂ A = ∂ (A∩ gAg^-1) = ∂ (gAg^-1) = g∂ A,
for all g∈ G.
By Lemma <ref> applied to Z=∂ A we deduce that Bd-Min(∂ A) is a closed, convex subset of X which splits isometrically as Y×ℝ^k, and that ∂ A coincides with the boundary at infinity of all sets { y }×ℝ^k. Each element of G sends a ∂ A-boundary-minimal subset into a ∂ A-boundary-minimal subset because ∂ A is G-invariant, therefore Bd-Min(∂ A) itself is G-invariant. Since G is cocompact, the action of G on X is minimal (<cit.>). We deduce that X=Bd-Min(∂ A), and so X splits isometrically and G-invariantly as Y×ℝ^k, which proves the first assertion and (i).
The fact that the projection of A on (ℝ^k) is a crystallographic group follows from Proposition <ref>. Indeed any set C(A) = W×ℝ^k as in <ref>.(i) is compatible with the splitting we just proved in the sense that W⊂ Y, because W ×ℝ^k is union of ∂ A-boundary minimal sets as follows by (i.a).
Moreover A acts as the identity on W. Therefore the projection of A on Y is precompact since it fixes a point.
Suppose now to have another almost abelian subgroup A'<A of rank k' which is commensurated in G.
Let X = Y' ×ℝ^k' be the splitting associated to A'. Let x∈ X be a point and write it as (y, v) ∈ Y ×ℝ^k and (y', v') ∈ Y'×ℝ^k'. Then, by the first part of the proof and by Proposition <ref> we have
∂ ({ y' }×ℝ^k') = ∂ A' ⊆∂ A = ∂ ({ y }×ℝ^k).
It follows that { y' }×ℝ^k'⊆{ y }×ℝ^k, so the parallel slices associated to A' are contained in the parallel slices associated to A. Decomposing ℝ^k as the orhogonal sum of ℝ^k' and ℝ^k - k', we also deduce that the sets { y }×ℝ^k - k' are parallel for all y∈ Y (since the slices of A' are all parallel). Therefore, X is also isometric to (Y ×ℝ^k - k') ×ℝ^k', which implies that Y' is isometric to Y ×ℝ^k - k' and proves (ii).
The same proof shows that the statement holds for proper (0)-spaces, not necessarily geodesically complete, and for closed groups of isometries acting minimally (see <cit.> for more details on this property).
§.§ Proof of Theorem <ref>
The main step consists in showing an almost abelian subgroup of rank k ≥ 1 which is almost commensurated in G. Observe that it will be automatically semisimple by Theorem <ref>.(ii).
Recall the constant J(k) given by Proposition <ref>, and define
J_0 := max_k∈{ 0,…,n_0}J(k) +1
which also clearly depends only on n_0, so ultimately only on P_0. Recall also that for every x∈ X the set S_2D_0(x) generates the whole G, by the discussion in Section <ref> and Theorem <ref>.(iii).
We fix 0<ε≤ε_0 as in the assumptions of Theorem <ref>, and we define inductively the sequence of subgroups G_ε_i (x) associated to positive numbers
ε_1 := ε > ε_2 > … > ε_3n_0 + 1 > 0
as follows:
– first, we apply Proposition <ref> to ε=ε_1 and R=2D_0 to obtain a smaller δ_2 := δ(P_0,r_0,2D_0,ε_1), and we set ε_2 := δ_2/8√(n_0)J_0;
– then, we define inductively δ_i+1 := δ(P_0,r_0,2D_0,ε_i) by repeatedly applying Proposition <ref> to ε_i and R=2D_0, and we set ε_i+1 = δ_i+1/8√(n_0)J_0.
Notice that, by construction, each ε_i depends only on P_0, r_0, D_0 and ε. By Proposition <ref>, the subgroups G_ε_i (x) form a decreasing sequence of almost abelian, semisimple subgroups for every x∈ X.
In <cit.> we defined ε_i+1 = δ_i+1/4J_0, here we are using the formula above in order to get the improved version of (v) in Theorem <ref>. Another difference, for the same scope, is that we define the sequence up to 3n_0 + 1 instead of 2n_0 + 1.
We set σ_P_0,r_0,D_0 (ε):= ε_3n_0 + 1 and we will show that this is the function of ε
for which Theorem <ref> holds; it clearly depends only on P_0,r_0,D_0 and ε. In what follows, we will write for short σ := σ_P_0,r_0,D_0 (ε).
If
(G_σ( x)) ≥ 1
then there exists
i∈{ 1,…, 3n_0 - 2 } such that
(G_ε_i( x)) = (G_ε_i+1(x )) = (G_ε_i+2(x )) = (G_ε_i+3(x )) ≥ 1.
The groups G_ε_i(x) are semisimple and almost abelian. Moreover by Lemma <ref> they satisfy
(G_ε_i(x))≥ 1, for all 1≤ i ≤ 3n_0 +1, since they contain G_σ(x).
By Corollary <ref>, the rank of each G_ε_i(x) cannot exceed the dimension of X, which is at most n_0. Since, again by Lemma <ref>, the rank decreases as i increases we conclude that for some 1≤ i≤ 3n_0 - 2 we have (G_ε_i( x)) = (G_ε_i+1(x )) = (G_ε_i+2(x )) = (G_ε_i+3(x )) ≥ 1.
If (G_σ(x)) ≥ 1 then there exists i ∈{ 1,…, 3n_0 - 2 } such that G_ε_i+j(x) is almost commensurated for all j∈{ 0,1,2,3}, and all these subgroups have the same rank k≥ 1.
Consider the almost abelian groups G_ε_i+3(x) < G_ε_i+2(x) < G_ε_i+1(x) < G_ε_i(x) which have the same rank k≥ 1,
given by Lemma <ref>, for some
1 ≤ i ≤ 3n_0 - 2.
Let C_i+3=W_i+3×ℝ^k be a G_ε_i+3(x)-invariant, convex subset of X as in Proposition <ref>.(i), applied to G_ε_i+3(x),
and call Ĝ_i+3 the image of G_ε_i+3(x) under the projection π_i+3:G_ε_i+3(x) →Isom(ℝ^k) on the second factor of C_i+3.
Finally, denote by ℒ_i+3 the maximal Euclidean lattice of the crystallograhic group Ĝ_i+3.
By Proposition <ref>(i.c) we can find a finite subset Σ of
S_ε_i+3(x)^4J_0⊆S_4J_0·ε_i+3(x)
whose projection Σ_ℝ^k=π_i+3(Σ) on Isom(ℝ^k) generates the lattice ℒ_i+3.
In particular every non-trivial element of Σ is hyperbolic.
Moreover, by the definition of ε_i+3 = δ_i+3 /8√(n_0) J_0,
the following holds:
∀ g∈Σ
and ∀ h∈S_2D_0(x) there exists m >0
such that h^-1g^m h ∈S_ε_i+2(x)
(in fact, d(x,gx) < δ_i+3=δ(P_0,r_0,2D_0, ε_i+2) for every g ∈Σ, so by Proposition <ref> there exists m> 0 such that d(x, h^-1g^m hx) = d(hx,g^mhx) ≤ε_i+2, that is h^-1g^m h ∈S_ε_i+2(x)).
Theorem <ref>.(c) implies that the cardinality of the orbit S_2D_0(x) x is finite, so we can fix h_1,…,h_ℓ∈S_2D_0(x) such that for every h∈S_2D_0(x) there exists j∈{ 1,…, ℓ} with hx=h_jx. The discussion above says that for every j ∈{ 1,…,ℓ} and every g∈Σ there exists an integer m_j,g>0 such that h_j^-1g^m_j,gh_j ∈S_ε_i+2(x). Moreover if hx=h_jx then also h^-1g^m_j,gh ∈S_ε_i+2(x), because
d(h^-1g^m_j,ghx, x) = d(g^m_j,ghx, hx) = d(g^m_j,gh_jx, h_jx)
= d(h^-1_jg^m_j,gh_jx, x) ≤ε_i+2.
Therefore, denoting by M = ∏_j,g m_j,g > 0, we have that hg^Mh^-1∈G_ε_i+2(x) for all g∈Σ and all h∈S_2D_0(x). Observe that M is finite because Σ is finite and j∈{ 1,…,ℓ}.
Let A < G_ε_i+3 (x)
be the subgroup generated by the subset { g^M | g ∈Σ}.
We claim that A is almost commensurated in G.
Actually, notice first that A is an almost abelian group of rank k because of Corollary <ref>, since its projection π_i+3(A) is a subgroup of finite index of the lattice ℒ_i+3 of ℝ^k.
Therefore, for all h∈S_2D_0(x), the almost abelian group hAh^-1 has also rank k, and is contained in the almost abelian group G_ε_i+2(x) of same rank. This implies, again by Lemma <ref>, that A and hAh^-1 are both cocompact in G_ε_i+2(x). In particular A∩ hAh^-1 is cocompact in both A and hAh^-1 by Lemma <ref>. Hence A and hAh^-1 are almost commensurable for every h∈S_2D_0(x).
Lemma <ref> shows that therefore A is almost commensurated in G. Since A<G_ε_i+j(x) for j=0,…,3 and all these groups have same rank, we deduce that A is cocompact also in G_ε_i +j(x), again by Lemma <ref>. The conclusion follows by Lemma <ref>.
Putting the ingredients all together we can give the
Since ^♢(G,X)≤σ, then there exists x_0 ∈ X with ^♢(G,x_0)≤σ; in particular, G_σ(x_0) contains a hyperbolic isometry, hence rk(G_σ(x_0)) ≥ 1. This follows by the characterization of the rank given in Proposition <ref>. Then we can apply Proposition <ref> to find
1 ≤ i ≤ 3n_0 - 2 such that the groups G_ε_i+j(x_0), j=0,…,3, have all rank k≥ 1 and are all almost commensurated.
Proposition <ref> now implies that X splits isometrically and G-invariantly as Y×ℝ^k, proving (i).
Moreover, we know that ∂G_ε_i+j(x) coincides with the boundary at infinity of each slice { y }×ℝ^k, for j=0,…,3. Let us call Z ⊆∂ X this common trace at infinity.
We show that the statements (ii)-(vi) hold for
ε^∗=ε_i+1∈ (σ, ε).
We start proving them for any x∈ X such that d(x, x_0) ≤ D_0. Denote by k_x the rank of G_ε_i+1(x).
Let Σ⊆S_ε_i+1^4J(k) + 2(x) ⊆S_4J_0·ε_i+1(x) be the finite subset provided by Proposition <ref>.(i.c). Observe that by Corollary <ref> the group generated by Σ has rank k_x. By Proposition <ref> and by definition of ε_i+1, for every g∈Σ there exists m_g >0 such that d(x_0,g^m_gx_0)≤ε_i. Let A be the group generated by { g^m_g s.t. g∈Σ}. The rank of A is still k_x by Corollary <ref>. Moreover A< G_ε_i(x_0), therefore by Lemma <ref>
k_x = rk(G_ε_i+1(x)) = rk(B) ≤rk(G_ε_i(x_0)) = k.
By Proposition <ref> we also have
∂G_ε_i+1(x) = ∂ A ⊆∂G_ε_i(x_0) = Z.
Reversing the roles of x and x_0 and starting from G_ε_i+3(x_0) we obtain the estimate
k=rk(G_ε_i+3(x_0)) ≤rk(G_ε_i+2(x)) ≤rk(G_ε_i+1(x)) = k_x
and Z= ∂G_ε_i+3(x_0)⊆∂G_ε_i+2(x) ⊆∂G_ε_i+1(x).
This proves proves (ii) and (iii) in this case. Observe that it also shows that for all x with d(x,x_0) ≤ D_0 we have rk(G_ε_i+2(x)) = rk(G_ε_i+1(x)) = k and ∂G_ε_i+2(x) = ∂G_ε_i+1(x) = Z.
The same group A above is cocompact in both G_ε_i +1(x) and G_ε_i(x_0) by Lemma <ref>. A double application of Lemma <ref> gives that G_ε_i +1(x) is almost commensurated, that is (vi). Moreover Proposition <ref> implies that the closure of the projection of G_ε_i(x_0) on Isom(Y) is compact, so it is the closure of the projection of A and automatically also the closure of the projection of G_ε_i + 1(x).
Since the the closure of the projection of G_ε_i+1(x) on Isom(Y) is compact then it fix some point y∈ Y (cp. <cit.>). Hence G_ε_i+1(x) preserves { y }×ℝ^k. This shows (iv).
As in the proof of Proposition <ref> we have that any splitting W×ℝ^k as in Proposition <ref> associated to G_ε_i+2(x) is compatible with the splitting X=Y×ℝ^k, in the sense that W⊆ Y. This is just because ∂G_ε_i+2(x) = Z.
Therefore the projection of G_ε_i+2(x) on Isom(ℝ^k) is a crystallographic group by Proposition <ref>.(i.b). By (i.c) of the same proposition we have that the maximal lattice ℒ_i+2 of this projection is generated by elements of S_ε_i+2(x)^4J_0⊆S_4J_0·ε_i+2(x). Observe that 4J_0·ε_i+2≤ε_i+1/2√(n_0), by definition.
The same argument shows that the projection of G_ε_i + 1(x) on Isom(ℝ^k) is crystallographic too. Since the maximal lattice ℒ_i+1 of the projection of G_ε_i+1(x) on Isom(ℝ^k) contains ℒ_i+2, then λ(ℒ_i+1) ≤λ(ℒ_i+2) ≤ε_i+1/2√(n_0). Therefore ℒ_i+1 is ε_i+1/2-cocompact. This concludes (v).
Finally, assume that x' is any point of X, say x'=gx with d(x,x_0) ≤ D_0.
Observe that
G_ε_i+1(x') = g G_ε_i+1(x)g^-1, so the rank does not change and
the conditions (ii) and (vi) continue to hold since
the splitting is G-invariant.
Moreover
∂G_ε_i+1(x')
=g ·∂G_ε_i+1(x)
=g · Z = Z, because Z is G-invariant, so (iii) still holds.
Finally, if the projection of G_ε_i+1(x) on Isom(Y) preserves { y }×ℝ^k, then the projection of G_ε_i+1(x') on Isom(Y) preserves { gy }×ℝ^k, which proves (iv).
§ THE RENORMALIZATION THEOREM
In this section we prove the analogue of the Renormalization Theorem <cit.> for totally disconnected group actions. This is a useful tool that that can be used to rescale a collapsing sequence in order to obtain a non-collapsing one. We will not use it here but it is essential for other applications in future works. The main point is to use the Splitting Theorem <ref> to show that, up to increasing in a controlled way the codiameter, we can always suppose that the free-systole is bounded away from zero by a universal positive constant.
Given P_0,r_0,D_0>0, there exist s_0 = s_0(P_0,r_0,D_0)>0 and Δ_0 = Δ_0(P_0,D_0) such that the following holds.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space and let G < (X) be closed, totally disconnected and D_0-cocompact.
Then G admits another faithful, continuous, Δ_0-cocompact action by isometries on a (0)-space X' isometric to X, i.e. G can be seen as a closed, totally disconnected, Δ_0-cocompact subgroup of (X'), with the additional property that
^♢(G, X') ≥ s_0.
With the tools we have developed so far the proof is exactly the same of <cit.>, we will report it for completeness.
Recall the constants n_0=P_0/2, which bounds the dimension of any proper, geodesically complete, CAT(0)-space X which is (P_0, r_0)-packed, and J_0, introduced at the beginning of Section <ref>. Also recall the Margulis constant ε_0=ε_0(P_0, r_0) given by Proposition <ref>, which we will always assume smaller than 1 in the sequel.
Recall the function σ_P_0,r_0,D_0(ε) of Theorem <ref>.
Then we define inductively D_1 = 2D_0 + √(n_0), σ_1 = σ_P_0,r_0,D_1( ε_0) and
D_j = 2D_j-1 + √(n_0) ,σ_j = σ_P_0,r_0,D_j (σ_j-1) > 0.
We claim that Δ_0 := D_n_0 -1 and s_0 := σ_n_0 satisfy the thesis; notice that both depend only on P_0,r_0 and D_0.
We define an algorithm which takes the CAT(0)-space X_0:=X and produces a new proper, geodesically complete, (P_0,r_0)-packed CAT(0)-space X_1 on which G still acts faithfully by isometries, still satisfying all the assumptions of the theorem, except that diam(G \ X_1) ≤ D_1; and we will show that, repeating again and again this process, we end up with a CAT(0)-space X_j with ^♢(G,X_j) > σ_n_0, for some j≤ n_0.
If ^♢(G,X_0) > σ_n_0, there is nothing to do, and we just set X'=X_0.
Otherwise, ^♢(G,X) ≤σ_n_0 < σ_1 = σ_P_0,r_0,D_0 (ε_0), and we apply
Theorem <ref>
with ε=ε_0.
Then, there exists ε^∗_0:=ε^∗∈ (σ_1 ,ε_0) such that the groups
G_ε^∗_0(x,X_0) have all rank k_0 ≥ 1 for every x∈ X_0. We then fix x_0 ∈ X_0.
By Theorem <ref> the group G_ε^∗_0(x_0,X) of rank k_0 ≥ 1 is almost commensurated in G and X_0 splits isometrically and G-invariantly as Y_0 ×ℝ^k_0.
Moreover, always by Theorem <ref>, there exists y_0∈ Y_0 such that G_ε^∗_0(x_0, X_0) preserves { y_0 }×ℝ^k_0, and the maximal lattice ℒ_ε^∗_0(x_0,X_0) of the crystallographic group π_ℝ^k_0(G_ε^∗_0(x_0, X_0)) satisfies λ(ℒ_ε^*_0(x)) ≤ε^*_0/2√(n_0). Here π_ℝ^k_0 G →Isom(ℝ^k_0) is the projection map.
So, we can find a shortest basis ℬ_0 = { b^0_1, …, b^0_k_0} of the lattice ℒ_ε^∗_0(x_0,X_0) whose vectors have all length at most ε^∗_0/2√(n_0) < 1; without loss of generality, we may suppose that ‖ b^0_1‖_0 ≤…≤‖ b^0_k_0‖_0 = λ(ℒ_ε^∗_0(x_0,X_0)) =:ℓ_0 < 1, where ‖‖_0 denotes the Euclidean norm of ℝ^k_0.
By (<ref>), we also know that ρ(ℒ_ε^∗_0(x_0,X_0)) ≤√(k_0)/2·ℓ_0 ≤√(n_0)/2·ℓ_0.
Now, we define the metric space
X_1 := Y_0 ×( 1/ℓ_0·ℝ^k_0). This is again a proper, geodesically complete, CAT(0)-space, still (P_0,r_0)-packed, on which G still acts faithfully by isometries (because the splitting of X_0 is G-invariant).
We claim that the action of G on X_1 is D_1-cocompact. In fact, let x = (y, v) be any point of X_1. Since the action of G on X_0 is D_0-cocompact, we know that there exists g∈ G such that d_X_0(x , g·(y_0, O) ) ≤ D_0; moreover,
as G_ε^∗_0(x_0, X) preserves { y_0 }×ℝ^k_0, up to compose g with elements of this group we can suppose that g = (g',g”) ∈ G satisfies d_X_0(x, g · (y_0, O)) ≤ D_0 and
‖ v - g”· O‖_0 ≤√(n_0)/2·ℓ_0. Therefore
d_X_1((x, g · (y_0, O)) ≤√(d_Y_0(y, g' · y_0)^2 + (1/ℓ_0)^2·‖ v - g”· O‖_0^2)
≤√(D_0^2 + n_0/4)≤ D_0 + √(n_0)/2 = D_1/2.
As (y, v) ∈ X_1 was arbitrary, we then deduce that
X_1 = G·B_X_1(( y_0, O), D_1/2),
so G < Isom(X_1) is D_1-cocompact.
If now ^♢(G,X_1) > σ_n_0, we stop the process and set X' = X_1: this space has all the desired properties.
Otherwise, we have
^♢(G,X_1) ≤σ_n_0
< σ_2 = σ_P_0,r_0,D_1 (σ_1)
and we can apply again Theorem <ref> to X_1, with ε=σ_1.
Then, there exists ε^∗_1 ∈ (σ_2 ,σ_1)
such that the groups
G_ε^∗_1(x,X_1) have rank k_1 ≥ 1 for every x∈ X_1, in particular rk(G_ε^∗_1(x_0,X_1))=k_1. Moreover G_ε^∗_1(x_0,X_1) is almost commensurated in G, the space X_1 splits isometrically and G-invariantly as Y_1 ×ℝ^k_1, and the trace at infinity ∂G_ε^∗_1(x_0, X_1) coincides with the boundary at infinity of all the sets { y }×ℝ^k_1. Moreover the maximal lattice ℒ_ε^∗_1(x_0,X_1) of π_ℝ^k_1(G_ε^∗_1(x_0, X_1)) satisfies λ(ℒ_ε^∗_1(x_0,X_1)) ≤ε_1^*/2√(n_0), where the notation is as before. Therefore, we can find
a shortest basis ℬ^1 = { b_1^1, …, b_k_1^1 } of ℒ_ε^∗_1(x_0,X_1) with lengths (with respect to the Euclidean norm ‖‖_1 of ℝ^k_1)
‖ b_1^1‖_1 ≤…≤‖ b_k_1^1‖_1 = λ(ℒ_ε^∗_1(x_0,X_1)) =:ℓ_1 ≤ε^∗_1/2√(n_0) < 1.
Observe that, as
by construction the factor (1/ℓ_0 )^2 is bigger than 1, we have
G_ε^∗_1( x_0, X_1) < G_σ_1( x_0, X_1) < G_σ_1( x_0, X_0) < G_ε^∗_0( x_0, X_0),
so
k_1= rk(G_ε^∗_1( x_0, X_1)) ≤rk(G_ε^∗_0( x_0, X_0)) = k_0. Moreover the metric splittings of X_1 as Y_0 ×( 1/ℓ_0·ℝ^k_0) and Y_1 ×ℝ^k_1 determined, respectively, by the groups G_ε^∗_0( x_0, X_0) and G_ε^∗_1( x_0, X_1) satisfy ℝ^k_1⊂1/ℓ_0·ℝ^k_0 and Y_0 ⊂ Y_1, because of the second part of Proposition <ref>.
We will now show that k_1 ⪇ k_0.
Actually, suppose that k_1 = k_0=:k.
Then Y_0= Y_1 and the two splittings coincide.
The lengths of the basis ℬ^1 with respect to the Euclidean norm ‖‖_0 of the Euclidean factor ℝ^k_0 of X_0 are ‖ b_i^1‖_0 = ℓ_0 ·‖ b_i^1‖_1 < ℓ_0
for every i=1,…,k.
But then, since ℒ_ε^∗_1(x_0,X_1) < ℒ_ε^∗_0(x_0,X_0) by (<ref>), we would be able to find k independent vectors of ℒ_ε^∗_0(x_0,X_0) of length less than its shortest generating radius, which is impossible.
This shows that k_1 < k_0.
We can now define X_2:= Y_1 ×(1/ℓ_1·ℝ^k_1), on which G acts faithfully by isometries and D_2-cocompactly, for D_2 = 2D_1 + √(n_0) computed as before.
We can repeat this process to get a sequence of proper, geodesically complete, (P_0,r_0)-packed, CAT(0)-spaces X_j on which G always acts faithfully and D_j-cocompactly by isometries. Moreover at each step
either ^♢(G,X_j) ≥σ_n_0 or the ε^∗_j-splitting rank k_j of X_j provided by Theorem <ref> is strictly smaller than the ε^∗_j-1-splitting rank k_j-1 of X_j-1. Since the splitting rank of X_0 is at most n_0 there must exist j ∈{ 1,…, n_0 } such that ^♢(G,X_j) ≥σ_n_0. The proof then ends by setting X' = X_j. It is clear from the construction of X' that it is isometric to X.
The proof actually produces a sequence of almost abelian commensurated subgroups of G
{}≨ A_0 ≨ A_1…≨ A_m
for some m ≤ n_0-1, such that:
(a) denoting by k_j the rank of A_j, then 1 ≤ k_0 <k_1 … < k_m;
(b) setting h_j = k_j - k_j-1 then there is a corresponding isometric and G-invariant splitting of X as Y×ℝ^h_0×ℝ^h_1×⋯×ℝ^h_m. Moreover, there exist 0<L_j < 1 such that the natural action of G on the space
X' := Y×(1/L_0·ℝ^h_0) ×(1/L_1·ℝ^h_1)×⋯×( 1/L_m·ℝ^h_m)
is Δ_0-cocompact with free-systole at least s_0.
§ CONVERGENCE AND COLLAPSING
In this last section we will finally attack Theorem <ref> using the tools developed until now.
First of all, for the reader's convenience, we briefly recall the notion of equivariant Gromov-Hausdorff convergence and ultralimits.
§.§ Equivariant Gromov-Hausdorff convergence and ultralimits
An isometric action on a pointed space is a triple (X,x,G) where X is a proper metric space, x ∈ X is a basepoint and G < Isom(X) is a closed subgroup. An equivariant isometry between isometric actions of pointed spaces (X,x,G) and (Y,y,H) is an isometry F X → Y such that
– F(x)=y;
– F_*(X) →(Y) defined by F_*(g) = F∘ g ∘ F^-1 is an isomorphism between G and H.
The best known notion of convergence for isometric actions of pointed spaces is the equivariant pointed Gromov-Hausdorff convergence, as defined by K. Fukaya in <cit.>:
we will write (X_j,x_j,G_j) ⟶ (X_∞,x_∞,G_∞)
for a sequence (X_j,x_j,G_j) of isometric actions converging in the equivariant pointed Gromov-Hausdorff sense to an isometric action (X_∞,x_∞,G_∞).
Forgetting about the group actions and considering just pointed metric spaces (X_j,x_j), this convergence reduces to the pointed Gromov-Hausdorff convergence: we will write (X_j,x_j) ⟶ (X_∞,x_∞) for a sequence of pointed metric spaces (X_j,x_j) converging to the pointed metric space (X_∞, x_∞) in this sense. If moreover all the spaces X_j under consideration are compact we can even drop the basepoints, and we will simply write X_j ⟶ X_∞ when X_j converges to the compact metric space X_∞.
We do not recall here the definition of equivariant, pointed Gromov-Hausdorff convergence.
An equivalent approach uses ultralimits. We present it in this section, and then we recall the relations with the usual Gromov-Hausdorff convergence, referring to <cit.> and <cit.>.
A non-principal ultrafilter ω is a finitely additive measure on ℕ such that ω(A) ∈{ 0,1 } for every A⊆ℕ and ω(A)=0 for every finite subset of ℕ. Accordingly, we will write ω-a.s. or for ω-a.e.(j) in the usual measure theoretic sense.
Given a bounded sequence (a_j) of real numbers and a non-principal ultrafilter ω there exists a unique a∈ℝ such that for every ε > 0 the set { j ∈ℕ s.t. | a_j - a | < ε} has ω-measure 1 (cp. <cit.>). The real number a is then called the ultralimit of the sequence a_j and it is denoted by ω-lim a_j.
A non-principal ultrafilter ω being given, one can define the ultralimit pointed metric space
(X_ω, x_ω)= ω-lim (X_j, x_j) of any sequence of pointed metric spaces (X_j, x_j):
– first, one says that a sequence (y_j), where y_j∈ X_j for every j, is admissible if there exists M such that d(x_j,y_j)≤ M for ω-a.e.(j);
– then, one
defines (X_ω, x_ω) as set of admissible sequences (y_j) modulo the relation (y_j)∼ (y_j') if and only if ω-lim d(y_j,y_j') = 0.
The point of X_ω defined by the class of the sequence (y_j) is denoted by y_ω = ω-lim y_j.
Finally, the formula d(ω-lim y_j, ω-lim y_j') = ω-lim d(y_j,y_j') defines a metric on X_ω which is called the ultralimit distance on X_ω.
Using a non-principal ultrafilter ω, one can also talk of limits of isometries and of isometry groups of pointed metric spaces. A sequence of isometries g_j of pointed metric spaces (X_j, x_j) is admissible if there exists M≥ 0 such that d(x_j, g_jx_j) ≤ M ω-a.s. Any such sequence defines a limit isometry g_ω = ω-lim g_j of X_ω=ω-lim (X_j, x_j) by the formula: g_ω y_ω = ω-lim g_jy_j (<cit.>).
Given a sequence of groups G_j < Isom(X_j) we set
G_ω = {ω-lim g_j s.t. g_j ∈ G_j for ω-a.e.(j)}.
In particular the elements of G_ω are ultralimits of admissible sequences.
One has a well-defined composition law on G_ω (<cit.>): if g_ω = ω-lim g_j and h_ω = ω-lim h_j we set g_ω ∘ h_ω := ω-lim(g_j ∘ h_j).
With this operation G_ω becomes a group of isometries of X_ω, which we call the ultralimit group of the sequence of groups G_j.
Notice that if X_ω is proper then G_ω is always a closed subgroup of isometries of X_ω <cit.>.
In conclusion, a non-principal ultrafilter ω being given, for any sequence of isometric actions on pointed spaces (X_j,x_j, G_j) there exists an ultralimit isometric action on a pointed space
(X_ω, x_ω, G_ω) = ω-lim (X_j,x_j, G_j).
The ultralimit approach and the Gromov-Hausdorff convergence are essentially equivalent.
Let (X_j, x_j, G_j) be a sequence of isometric actions of pointed spaces:
(i) if (X_j,x_j,G_j) ⟶ (X_∞,x_∞,G_∞), then (X_ω, x_ω, G_ω) ≅ (X_∞,x_∞,G_∞) for every non-principal ultrafilter ω;
(ii) reciprocally, if ω is a non-principal ultrafilter and (X_ω, x_ω) is proper, then (X_j_k,x_j_k,G_j_k) ⟶ (X_ω,x_ω,G_ω) for some subsequence {j_k}.
Moreover, if for every non-principal ultrafilter ω the ultralimit (X_ω,x_ω,G_ω) is equivariantly isometric to the same isometric action (X,x,G), with X proper, then (X_j,x_j,G_j) ⟶ (X,x,G).
The sequence (X_j,x_j, G_j) is called D-cocompact if each G_j is D-cocompact.
The ultralimit of a sequence of isometric actions on pointed spaces does not depend on the choice of the basepoints,
provided that the actions have uniformly bounded codiameter.
Let (X_j,x_j,G_j) be a sequence of isometric actions on pointed spaces, which are all D-cocompact and let x_j'∈ X_j be different basepoints. Then, for any non-principal ultrafilter ω, the ultralimit of (X_j,x_j,G_j) is equivariantly isometric to the ultralimit of (X_j,x_j',G_j).
Therefore, when considering the convergence of uniformily cocompact isometric actions,
we will often omit the basepoint x, if unnecessary for our arguments.
Finally, we remark that the equivariant pointed Gromov-Hausdorff convergence of a sequence of isometric actions with uniformly bounded codiameter implies the pointed Gromov-Hausdorff convergence of the quotients.
Let (X_j,G_j) be a given sequence of D-cocompact isometric actions.
If (X_j, G_j) ⟶ (X_∞,G_∞) then G_j\ X_j =: M_j ⟶ M_∞ := G_∞\ X_∞.
In general the group G_∞ is unknown and the structure of the quotient G_∞\ X_∞ is not clear at all. We want to clarify what happens in our specific setting. With this purpose, we now precisely define the following classes of isometric actions of CAT(0)-groups we are interested in:
- _0^(D_0): this is the set
of isometric actions (X, G)
where X is a proper, geodesically complete, CAT(0)-space, and G is a D_0-cocompact, totally disconnected and unimodular subgroup of Isom(X);
- _0^(P_0,r_0,D_0): the subset of _0^(D_0) made of the
actions (X, G)
such that X is, moreover, (P_0,r_0)-packed.
We will denote by 𝒪-CAT_0^ (D_0), 𝒪-CAT_0^ (P_0, r_0, D_0) the respective classes of quotients M=G \ X: these are compact generalized CAT(0)-orbispaces, with
(X, G) respectively in CAT_0^ (D_0) and CAT_0^ (P_0, r_0, D_0).
We say that a sequence of spaces X_j
(or a sequence of actions on X_j) is uniformly packed if there exists (P_0, r_0) such that every X_j is (P_0, r_0)-packed. By <cit.>, the class of proper, geodesically complete, pointed (0)-spaces (X_j,x_j) which are (P_0, r_0)-packed is closed under ultralimits and compact with respect to the pointed Gromov-Hausdorff convergence. Moreover, the proof of <cit.> implies that
_0^(D_0) = ⋃_P_0, r_0_0^(P_0,r_0,D_0).
More precisely, we have:
A subset K ⊆_0^(D_0) is precompact (with respect to the equivariant pointed Gromov-Hausdorff convergence) if and only if there exist P_0,r_0 > 0 such that K⊆_0^(P_0,r_0,D_0).
Let us consider a sequence of isometric actions
(X_j, G_j) in _0^(D_0),
(X_j, G_j) ⟶ (X_∞, G_∞).
Our goal is to describe the limit group G_∞ acting on X_∞ and the quotient space G_∞\ X_∞.
[Standard setting of convergence]
We say that we are in the standard setting of convergence when we have a sequence
(X_j,G_j)
in _0^(P_0,r_0,D_0) such that
(X_j,G_j) ⟶ (X_∞,G_∞).
We will denote by M_j = G_j\ X_j and M_∞ = G_∞\ X_∞ the quotient spaces.
The standard setting of convergence (or, equivalently, the convergence of the isometric actions
(X_j,G_j)
and of the generalized orbispaces M_j) will be called:
- without collapsing if lim sup_j→+∞^♢(G_j,X_j) > 0;
- with collapsing if lim inf_j→+∞^♢(G_j,X_j) = 0.
These cases are mutually exclusive.
Actually, the two cases are respectively equivalent to the the conditions that the dimension of the limit M_∞ = G_∞\ X_∞ equals the dimension of the quotients M_j or decreases, as we will prove in Theorem <ref>.
§.§ Convergence of Haar measures
One of the main issues in the proof of Theorem <ref> is to find some kind of stability of unimodularity under equivariant pointed Gromov-Hausdorff convergence. In this section we study this problem finding a sufficient condition for such stability.
We start by improving <cit.> to general groups, using the distances defined in (<ref>).
Suppose that (X_j,x_j,G_j) ⟶ (X_∞,x_∞,G_∞), with X_j geodesic spaces. Endow G_j with the metric d_x_j^ℓ (resp. d_x_j^) and G_∞ with the metric d_x_∞^ℓ (resp. d_x_∞^). Then (G_j,, d_x_j^ℓ) ⟶ (G_∞,, d_x_∞^ℓ) and (G_j,, d_x_j^) ⟶ (G_∞,, d_x_∞^).
We do the proof for the left invariant metric, the other case is similar.
By definition X_∞ is proper, so (X_∞) is locally compact. Moreover G_∞ < (X_∞) is closed, so locally compact as well. As recalled in Section <ref> the metric d_x_∞^ℓ is therefore proper. This means that we are in position to apply Proposition <ref>. In particular the thesis is equivalent to prove that (G_ω, id, d_x_ω^ℓ) is isometric as pointed metric space to ω-lim (G_j, id, d_x_j^ℓ) for every non-principal ultrafilter ω. Let us fix a non-principal ultrafilter ω. Let (g_j) be an admissible sequence defining a point of ω-lim (G_j, id, d_x_j^ℓ). This means that ω-lim d_x_j^ℓ(g_j,𝕀) < +∞. This implies ω-lim d(g_jx_j,x_j) < +∞ by the very definition (<ref>), i.e. the sequence g_j∈ G_j defines a limit isometry g_ω∈ G_ω. We set Φω-lim (G_j, id, d_x_j^ℓ) → G_ω defined by (g_j) ↦ g_ω. The distance function on ω-lim (G_j, id, d_x_j^ℓ) will be denoted by ω-lim d_x_j^ℓ.
Good definition. Let us take two admissible sequences (g_j), (h_j) defining the same point of ω-lim (G_j, id, d_x_j^ℓ), i.e. ω-lim d_x_j^ℓ(g_j,h_j)=0. By definition, for every ε > 0 we have d_x_j^ℓ(g_j,h_j) < ε for ω-a.e.(j). By (<ref>) there exists R> 1/ε such that d_x_j,R^ℓ(g_j,h_j) < ε for ω-a.e.(j). If we now fix some R > 0 and we let ε go to zero we get that g_ω coincides with h_ω on B(x_ω, R). Since this is true for every R>0 we conclude that g_ω = h_ω, so Φ is well defined.
Isometric embedding. We now prove that Φ is an isometry. By definition
d_x_ω^ℓ(Φ((g_j)), Φ((h_j))) = d_x_ω^ℓ(g_ω, h_ω) = inf_R > 0{1/R + d_x_ω,R^ℓ(g_ω, h_ω) }.
Recalling (<ref>) we have
d_x_ω,R^ℓ(g_ω, h_ω) = max_y_ω∈B(x_ω,R) d(g_ω y_ω, h_ω y_ω) = max_y_ω∈B(x_ω,R)ω-lim d(g_j y_j, h_j y_j).
Using the fact that B(x_ω, R) = ω-limB(x_j,R) (see <cit.>) it is straightforward to conclude that
d_x_ω,R^ℓ(g_ω, h_ω) = ω-limmax_y_j∈B(x_j,R) d(g_j y_j, h_j y_j) = ω-lim d_x_j,R^ℓ(g_j,h_j).
Therefore d_x_ω^ℓ(g_ω, h_ω) = inf_R > 0{ω-lim(1/R + d_x_j,R^ℓ(g_j, h_j) ) }.
On the other side we know that d_x_j^ℓ(g_j, h_j) = inf_R > 0{1/R + d_x_j,R^ℓ(g_j, h_j) }. We obtain
ω-lim d_x_j^ℓ((g_j), (h_j)) := ω-lim d_x_j^ℓ(g_j,h_j)≤ d_x_ω^ℓ(g_ω, h_ω).
To prove the other inequality we fix ε > 0 and for every j we take R_j >0 such that 1/R_j + d_x_j,R_j^ℓ(g_j, h_j) ≤ d_x_j^ℓ(g_j,h_j) + ε. Since we know that ω-lim d_x_j^ℓ(g_j,h_j) is finite we deduce that ω-lim R_j > 0. If ω-lim R_j =: R_ω is finite then we conclude, as before, that ω-lim1/R_j + d_x_j,R_j^ℓ(g_j, h_j) = 1/R_ω + d_x_ω,R_ω^ℓ(g_ω, h_ω). So
d_x_ω^ℓ(g_ω, h_ω) ≤1/R_ω + d_x_ω,R_ω^ℓ(g_ω, h_ω) ≤ω-lim d_x_j^ℓ(g_j,h_j) + ε.
If R_ω = +∞ we proceed as follows. We fix R > 1/ε. For ω-a.e.(j) we have R_j≥ R. Therefore
1/R + d_x_ω, R^ℓ(g_ω, h_ω) ≤ d_x_ω,R^ℓ(g_ω, h_ω) + ε = ω-lim d_x_j,R^ℓ(g_j,h_j) + ε
≤ω-lim d_x_j,R_j^ℓ(g_j,h_j) + ε
≤ω-lim d_x_j^ℓ(g_j,h_j) + 2ε.
This implies d_x_ω^ℓ(g_ω, h_ω) ≤ω-lim d_x_j^ℓ(g_j,h_j) + 2ε.
In any case, by arbitrariness of ε, we conclude that d_x_ω^ℓ(g_ω, h_ω) ≤ω-lim d_x_j^ℓ(g_j,h_j), i.e. Φ is an isometric embedding.
Surjectivity. In the last step we just need to show that Φ is surjective. Indeed if it is the case then it is a surjective isometric embedding between ω-lim (G_j, id, d_x_j^ℓ) and (G_ω, 𝕀, d_x_ω^ℓ) with Φ(𝕀) = 𝕀, i.e. an isometry of pointed metric spaces. Let g_ω∈ G_ω. By definition it is the ultralimit of a sequence of isometries g_j ∈ G_j with ω-lim d(g_j x_j, x_j) =: M < ∞. Fix R = 1 and y_j ∈B(x_j,1). Then d(g_jy_j, y_j) ≤ 2 + 2M for ω-a.e.(j), by triangular inequality. Then d_x_j^ℓ(g_j, 𝕀) ≤ 3 + 2M for ω-a.e.(j). In other words the sequence (g_j) defines a point in ω-lim (G_j,𝕀,d_x_j^ℓ). It is clear that the image through Φ of this point is g_ω, so Φ is surjective.
In the situation above the groups G_j are also equipped with Haar measures μ_j, so we can ask when the sequence of pointed metric measure spaces (G_j,𝕀,d_x_j^ℓ,μ_j) converges to a pointed metric measure spaces (G_∞, 𝕀,d_x_j^ℓ, μ_∞), and if it is the case when μ_∞ is a Haar measure of G_∞.
We recall that we denote by B_x_j^ℓ(𝕀,r) (resp. B_x_j^(𝕀,r)) the closed ball of center 𝕀 and radius r in G_j with respect to the metric d_x_j^ℓ (resp. d_x_j^), for j∈ℕ∪{∞}.
For reader's convenience we state the definition of pointed measured Gromov-Hausdorff convergence in this special situation, only for the left-invariant metrics. For the right invariant ones the definition is similar.
We say that that the sequence (G_j,𝕀,d_x_j^ℓ,μ_j) converges in the pointed measured Gromov-Hausdorff sense to (G_∞, 𝕀, d_x_∞^ℓ, μ_∞), and if it is the case we write (G_j, 𝕀, d_x_j^ℓ, μ_j) ⟶ (G_∞, 𝕀, d_x_∞^ℓ, μ_∞), if there exist sequences R_j → +∞, ε_j → 0 and ψ_j G_j → G_∞ satisfying:
(a) ψ_j(𝕀)=𝕀;
(b) sup_g_j,h_j ∈B_x_j^ℓ(𝕀,R_j)| d_x_j^ℓ(g_j,h_j) - d_x_∞^ℓ(ψ_j(g_j), ψ_j(h_j)) | < ε_j;
(c) for every g_∞∈B_x_∞^ℓ(𝕀,R_j - ε_j) there exists g_j ∈B_x_j^ℓ(𝕀,R_j) such that d_x_∞^ℓ(g_∞, ψ_j(g_j)) < ε_j;
(d) for all f G_∞→ℝ continuous with compact support it holds
∫_G_∞ f d(ψ_j)_* μ_j = ∫_G_∞ f ∘ψ_j dμ_j j→ +∞⟶∫_G_∞ f dμ_∞.
We recall that if we remove condition (d) we have exactly the definition of pointed Gromov-Hausdorff convergence. An equivalent notion of convergence can be stated with ultrafilters (cp. <cit.>).
Before stating the main result of the section we need to improve Proposition <ref>.
Same assumptions of Proposition <ref>. Let ψ_j (G_j, 𝕀, d_x_j^ℓ) → (G_∞, 𝕀, d_x_∞^ℓ) be maps realizing the pointed Gromov-Hausdorff convergence as above.
Let g_j ∈ G_j such that ψ_j(g_j) converges to g_∞∈ G_∞. Consider the multiplications on the left L_g_j G_j → G_j for j∈ℕ∪{∞}. Then the sequence of maps L_g_j converges to L_g_∞ in the following sense. For all R≥ 0 it holds
lim_j→ +∞sup_h_j ∈B_x_j^ℓ(𝕀,R) d_x_∞^ℓ(ψ_j (L_g_j(h_j)), L_g_∞(ψ_j(h_j))) = 0.
A similar statement holds for the right invariant metrics and multiplications on the right.
We prove the statement for the left case, the right one is similar. Since the metric d_x_j^ℓ is left-invariant each L_g_j is an isometry and L_g_j(𝕀) = 𝕀. We fix a non-principal ultrafilter ω and we consider ω-lim (G_j,𝕀,d_x_j^ℓ) which is isometric to (G_ω, 𝕀, d_x_ω^ℓ) by Proposition <ref> and so to (G_∞, 𝕀, d_x_∞^ℓ) by Proposition <ref>. The sequence of isometries L_g_j is admissible, so it defines a limit isometry ω-lim L_g_j in the usual way (cp. <cit.>). It is clear that this limit isometry is sent via the map Φ of the proof of Proposition <ref> to L_g_ω, where L_g_ω is the multiplication on the left by g_ω of G_ω. Using again Proposition <ref> we get that the sequence of isometries L_g_j defines a limit isometry on G_∞ and it coincides with L_g_∞. Now (<ref>) is exactly the definition of convergence of maps.
Suppose that (X_j,x_j,G_j) ⟶ (X_∞,x_∞,G_∞), X_j geodesic spaces, and let μ_j be a left invariant Haar measure on G_j. Then:
(i) (G_j,𝕀, d_x_j^ℓ, μ_j) ⟶ (G_∞,𝕀,d_x_∞^ℓ,μ_∞) up to a subsequence if and only if sup_j μ_j(B_x_j^ℓ(𝕀,R)) < +∞ for every R ≥ 0 . If it is the case then μ_∞ is left invariant.
(ii) The measure μ_∞ is a left invariant Haar measure of G_∞ if and only if inf_jμ_j(B_x_j^ℓ(𝕀,r)) > 0 for every r>0.
A similar statement holds for right invariant measures and metrics.
We provide the proof only for the left invariant case, the other one being similar.
The first part of (i) follows by <cit.> together with <cit.> and Proposition <ref>. We need to check that μ_∞ is left invariant. Fix maps ψ_j G_j → G_∞ realizing the pointed measured Gromov-Hausdorff convergence. Fix an element g_∞∈ G_∞: it is limit of elements g_j ∈ G_j. Call L_g_j and L_g_∞ the corresponding left multiplication maps. Fix a continuous function f G_∞→ℝ with compact support, say contained in B_x_∞^ℓ(𝕀,R) for some R> 0.
The continuous function f is uniformly continuous.
Using the compact support of f, its uniform continuity and (<ref>) we obtain the following estimate. For every ε > 0 we have
|∫_G_j f ∘ (ψ_j ∘ L_g_j)dμ_j - ∫_G_j f∘ L_g_∞∘ψ_j dμ_j | < εμ_j(B_x_j^ℓ(𝕀,2R))
for j big enough. Since μ_j(B_x_j^ℓ(𝕀,2R)) is uniformly bounded in j and ε is arbitrary, we get
lim_j → + ∞∫_G_j f ∘ (ψ_j ∘ L_g_j)dμ_j = lim_j→ + ∞∫_G_j f∘ L_g_∞∘ψ_j dμ_j.
The left hand side equals ∫_G_∞ f dμ_∞ because (L_g_j)_*μ_j = μ_j and (<ref>). The function f∘ L_g_∞ is still continuous and with bounded support, so again by (<ref>) the right hand side coincides with ∫_G_∞ f∘ L_g_∞ dμ_∞. By the arbitrariness of f we get (L_g_∞)_*μ_∞ = μ_∞, i.e. μ_∞ is L_g_∞-invariant. This shows that μ_∞ is left invariant.
Clearly μ_∞ is finite on compact sets. Moreover it is positive on open sets if and only lim inf_j→ +∞μ_j(B_x_j^ℓ(𝕀,r)) > 0 for every r>0. This follows from the left invariance and from, for instance, <cit.>. In this case μ_∞ is a left invariant Haar measure of G_∞ by Lemma <ref>.
Suppose that (X_j,x_j,G_j) ⟶ (X_∞,x_∞,G_∞), X_j geodesic spaces, and G_j unimodular. If there exist Haar measures μ_j of G_j such that:
(i) sup_jμ_j(S_R(x_j)) < + ∞ for every R≥ 0;
(ii) inf_jμ_j(S_r(x_j,1/r)) < + ∞ for every r>0;
then G_∞ is unimodular.
The conditions (i) and (ii) imply, and are essentially equivalent to, conditions in (i) and (ii) of Proposition <ref> by Lemma <ref>.
§.§ Almost stabilizers
In this part we generalize <cit.> for totally disconnected, unimodular groups. This is the part where unimodularity plays a role. Recall the function σ_P_0,r_0,D_0: (0,ε_0] → (0,ε_0] given by Theorem <ref>, where ε_0 is the Margulis constant.
Remark that Theorem <ref>.(iv) yields, for any x∈ X, a slice { y }×ℝ^k which is preserved by G_ε^∗(x) (with σ_P_0,r_0,D_0(ε)< ε^∗ < ε), provided that the free-systole of G is smaller than σ_P_0,r_0,D_0(ε), so that X splits as Y ×ℝ^k. It is useful to know that, for at least a specific point x, the slice can be chosen to be the one passing through x. This is proved, in the unimodular case, by the following:
Let X be a proper, geodesically complete, (P_0,r_0)-packed (0)-space, and let G < (X) be closed, totally disconnected, unimodular and D_0-cocompact.
Assume that ^♢(G,X)< σ_P_0,r_0,D_0(ε), so that X splits as Y×ℝ^k by Theorem <ref>.
Then there exists x_0 = (y_0, v)∈ X such that G_ε^∗(x_0) preserves { y_0 }×ℝ^k.
The proof is based on a maximality argument, similar to the proof of <cit.>.
We just need the following additional fact.
Let X be a proper, geodesically complete, (0)-space and let G < (X) be closed, totally disconnected and cocompact.
For every x∈ X and r > 0 there exists an open set U ∋ x such that S_r(y)⊆S_r(x) for all y∈ U.
Suppose it is not true. We can find y_j converging to x and g_j ∈ G such that d(y_j,g_jy_j) ≤ r but d(x,g_jx) > r. We can suppose that g_j converges to g ∈ G. Clearly d(g^-1g_jx,x) tends to 0, so by Theorem <ref>.(a) we have g^-1g_j x = x for j big enough. Therefore d(x, g x) = d(x,g_jx) > r but on the other hand it holds d(x, g x) = lim_j→ + ∞d(y_j,g_jy_j) ≤ r, a contradiction.
Let μ be a bi-invariant Haar measure of G. Observe that 0<μ(S_ε^∗(x)) <+∞ for every x∈ X, because the set S_ε^∗(x) is open and compact by Theorem <ref>.(b). The map x↦μ(S_ε^∗(x)) is upper semicontinuous by Lemma <ref> and G-invariant by unimodularity. Therefore it admits a maximum by cocompactness of G. Let x be a point where the maximum is realized. Apply Theorem <ref>.(iv) to the point x: there exists y_0∈ Y such that G_ε^∗(x) preserves { y_0 }×ℝ^k. If x ∈{ y_0 }×ℝ^k, then we set x_0=x and there is nothing more to prove.
Otherwise call x_0 the projection of x on the closed, convex, G_ε^∗(x)-invariant subset { y_0 }×ℝ^k.
Let c [0,d(x,x_0)] → X be the geodesic [x,x_0] and set
T=sup{ t ∈ [0,d(x,x_0)] s.t. S_ε^∗(c(t)) = S_ε^∗(x)}.
By definition T≥ 0 and we claim that T=d(x,x_0) and it is a maximum.
In order to do so we need the following claim.
Claim: if S_ε^∗(y) ⊆S_ε^∗(y') then the equality holds if and only if μ(S_ε^∗(y)) = μ(S_ε^∗(y')). One direction is clear, so suppose μ(S_ε^∗(y)) = μ(S_ε^∗(y')). The two sets are both open and closed by Theorem <ref>.(b). Therefore if S_ε^∗(y) ⊊S_ε^∗(y') then the set S_ε^∗(y') ∖S_ε^∗(y) would be open and non-empty, so of positive μ-measure, contradicting the assumption.
We can now continue the proof. Assume S_ε^∗(c(t)) = S_ε^∗(x).
Then, S_ε^∗(c(t + t')) ⊆S_ε^∗(c(t)) = S_ε^∗(x) for all t'>0 small enough, by Lemma <ref>.
On the other hand, for every g ∈S_ε^∗(x) we have d(x_0, g x_0) ≤ d(x,gx) (since g acts on Y fixing y_0).
Therefore, by the convexity of the displacement functions we deduce that S_ε^∗(x) ⊆S_ε^∗(c(t+t')) too.
This shows that the supremum defining T is not realized, unless T = d(x,x_0). Let now 0≤ t_j < T such that t_j tends to T, so the points c(t_j) converge to c(T). By Lemma <ref> we get S_ε^∗(c(t_j)) ⊆S_ε^∗(c(T)) for all n big enough. But S_ε^∗(c(t_j)) = S_ε^∗(x) has maximal measure among the sets of this form, therefore μ(S_ε^∗(x)) = μ(S_ε^∗(c(t_j))) = μ(S_ε^∗(c(T))) for all j. The claim implies that S_ε^∗(c(T)) = S_ε^∗(x), i.e. T is actually a maximum and so T=d(x,x_0).
Hence, S_ε^∗(x_0) = S_ε^∗(x),
and x_0 is the point we are looking for.
The following corollary is new even for discrete groups. Observe that in general the closure of the projection on the Euclidean factor is not totally disconnected (i.e. not discrete), also for discrete groups (see examples in <cit.>, <cit.> and <cit.>).
Let X be a proper, geodesically complete, (P_0,r_0)-packed (0)-space, and let G be a closed, totally disconnected, unimodular, D_0-cocompact subgroup of (X).
Assume that ^♢(G,X)< σ_P_0,r_0,D_0(ε), so that X splits as Y×ℝ^k by Theorem <ref>.
Let x_0 = (y_0, v)∈ X be such that G_ε^∗(x_0) preserves { y_0 }×ℝ^k, as provided by Proposition <ref>. Then the closure G_Y of the projection G_Y of G on (Y) is totally disconnected and the orbit G_Yy_0 is made of ε^∗/2-separated points.
We will prove that the orbit G_Yy_0 is made of ε^∗/2-separated points. This implies that also G_Yy_0 is made of ε^∗/2-separated points and that G_Y is totally disconnected by Theorem <ref>.(iv). Let us suppose there exists g∈ G, g=(g',g”) ∈Isom(Y)×Isom(ℝ^k), such that 0<d(g'y_0,y_0) ≤ε^∗/2. By Theorem <ref>.(v) we know that the projection of the group G_ε^∗(x_0) on Isom(ℝ^k) is a crystallographic group which is ε^*/2-cocompact.
We can therefore compose g with an element of G_ε^∗(x_0) in order to obtain an isometry h = (h',h”) with d(h” v, v) ≤ε^∗/2. Each element of G_ε^∗(x_0) acts on Y by fixing y_0, hence d(h'y_0, y_0) = d(gy_0, y_0). Hence d(hx_0,x_0) ≤ε^∗, so h ∈G_ε^∗(x_0), implying h'y_0 = y_0, which is a contradiction.
Another consequence of the work that we developed in Section <ref> is the following control of the almost stabilizers, which will be useful in studying converging sequences.
Let X be a proper, geodesically complete, (P_0,r_0)-packed (0)-space, and let G < (X) be closed, totally disconnected, unimodular and D_0-cocompact.
Let σ (G, X) := σ_P_0,r_0,D_0 (ε^♢)
be the constant obtained for ε^♢= min{ε_0, ^♢(G,X) }.
Then
(i) the almost stabilizers G_σ (G, X)(x) are compact, for all x∈ X;
(ii) there exists x_0∈ X such that G_σ (G, X)(x_0) fixes x_0.
Observe that, since by (i) the group G_σ(G,X)(x) is compact, then it has a fixed point (cp. <cit.>). Here we are saying that for at least one specific point x_0 we have that G_σ(G,X)(x_0) fixes exactly x_0.
We first prove part (i), and set, for short, σ= σ (G, X).
Suppose that there exists x∈ X such that G_σ(x) is not compact. If rk(G_σ(x)) = 0 then G_σ(x) should fix all points of every set C(A) as in Proposition <ref>.(i), implying that G_σ(x) is compact. Therefore rk(G_σ(x)) ≥ 1.
Then, we can repeat all the arguments in the proof of Theorem <ref> (namely, the construction of the sequence ε_i and Proposition <ref>) for ε=ε^♢ to show that X splits isometrically and G-invariantly as Y×ℝ^k, with k≥ 1, and that there exists ε^∗∈(σ, ε^♢) such that G_ε^∗ (x) projects on Isom(ℝ^k) as a crystallographic group.
Moreover, we can find elements g=(g',g”) of G_ε^*(x) such that g” is a translation of length at most ε^*/2√(n_0) and g' fixes a point.
But these elements are hyperbolic isometries of G with translation length at most ε^∗/2√(n_0) < ^♢(G,X), a contradiction.
Assertion (ii) follows from an argument similar to that of Proposition <ref>:
we consider the same function x↦S_σ(x), which admits a maximum by unimodularity and cocompactness of G. We fix a point x where the maximum is realized. The group G_σ(x) is compact, so the closed, convex set Fix(G_σ(x)) is not empty. If x∈Fix(G_σ(x)), then x_0=x and there is nothing more to prove.
Otherwise we call x_0 the projection of x on Fix(G_σ(x)), we consider again the geodesic c [0,d(x,x_0)] → [x,x_0] ⊂ X and we show as in Proposition <ref> that
T :=sup{ t ∈ [0,d(x,x_0)] s.t. S_σ(c(t)) = S_σ(x)} = d(x,x_0)
by Lemma <ref> and the convexity of the displacement function. Moreover, again by Lemma <ref> and by the maximality of x it follows that T=d(x,x_0) is a maximum,
hence G_σ(x_0) = G_σ(x), and x_0 is the announced fixed point.
§.§ Controlled convergence, without unimodularity
In this part we develop the main technical tools we need to deal with both the collapsed and the non-collapsed case. We introduce now the setup. Let P_0,r_0,D_0 be fixed constants. An isometric action (X,x,G) is said to be σ-controlled if
(a) X is a proper, geodesically complete, (P_0,r_0)-packed, (0)-space;
(b) G is a closed, totally disconnected and D_0-cocompact group of isometries, not necessarily unimodular;
(c) there exists σ > 0 such that for every g ∈ G either d(x,gx)=0 or d(x,gx) ≥σ.
We present the two main cases of σ-controlled isometric actions.
- Let (X,G) ∈_0^(P_0,r_0,D_0). Corollary <ref>.(ii) shows that (X,x,G) is σ-controlled with σ = σ(G,X), for a suitable choice of the basepoint x ∈ X.
- Let (X,G) ∈_0^(P_0,r_0,D_0) and suppose ^♢(G,X) ≤ε≤ε_0. Let X = Y×ℝ^k be the splitting and x = (y, v) be the point provided by Proposition <ref>. Then the isometric action (Y, y, G_Y) satisfies the assumptions (a), (b) and (c) with σ = σ_P_0,r_0,D_0(ε), as shown by Corollary <ref>, where G_Y is the closure of the projection of G on (Y). Indeed σ_P_0,r_0,D_0(ε) ≤ε^*/2.
We write (X_j, x_j, G_j) σ⟶ (X_∞, x_∞, G_∞) if each (X_j,x_j,G_j) is σ-controlled and if (X_j, x_j, G_j) ⟶ (X_∞, x_∞, G_∞) .
If
(X_j, x_j, G_j) σ⟶ (X_∞, x_∞, G_∞) then (X_∞, x_∞, G_∞) is σ-controlled. So G_∞ is closed, totally disconnected and D_0-cocompact.
We fix a non-principal ultrafilter ω.
By Proposition <ref>, it is enough to show the thesis for the ultralimit group G_ω.
By property (c) of the definition of σ-controlled isometric action the following holds: every isometry g_j ∈ G_j either fixes x_j or it moves it by at least σ. This implies that every isometry g_ω∈ G_ω either fixes x_ω or moves it by at least σ. In particular the orbit G_ω x_ω is discrete. It is clear that G_ω is D_0-cocompact and closed (<cit.>), so G_ω is totally disconnected by Theorem <ref>.(iv).
This fact has many important consequences. We begin with a quantified version of <cit.>.
Let P_0,r_0,D_0, σ,R >0. Then there exists N_0(σ, R) = N_0(P_0,r_0,D_0, σ, R) > 0 such that the following holds true.
Let (X,x,G) be σ-controlled. Then for all x∈ X the cardinality of the image of the map _G(x) →(B(x,R)) is at most N_0(σ, R).
Let us suppose the thesis is not true, so we can find σ-controlled isometric actions (X_j, x_j, G_j) and points y_j ∈ X_j such that the image of the map _G_j(y_j) →(B(y_j,R)) has at least j elements.
Fix a non-principal ultrafilter ω. By Proposition <ref> and Proposition <ref> we know that G_ω is closed, totally disconnected and D_0-cocompact.
By <cit.> the image of the map ΦStab_G_ω(y_ω) →Isom(B(y_ω, R + 1) has a finite number of elements, say Φ(g_ω,1),…,Φ(g_ω, N), with g_ω,1 = id. So there exists 0<ε <1 such that d_y_ω,R+1^ℓ(g_ω, i, g_ω,m) > 4ε for all 1≤ i < m ≤ N.
By definition of ultralimit group we can write these isometries as g_ω,i = ω-lim g_j,i, with g_j,1 = id. We claim that the following statement holds for ω-a.e.(j):
(i) for all h_j ∈Stab_G_j(y_j) there exists i∈{ 1,…, N} such that d_y_j,R+1^ℓ(h_j, g_j,i) ≤ε.
Indeed if for ω-a.e.(j) we can find isometries h_j contradicting this statement we get that the isometry h_ω, which is well defined since each h_j fixes y_j, satisfies h_ω∈Stab_G_ω(y_ω) and d_y_ω,R+1^ℓ(h_ω, g_ω,i) ≥ε for all i=1,…,N which is absurd.
Moreover we have
(ii) d_y_j,R+1^ℓ(g_j,i, g_j,m) ≥ 4ε for all 1≤ i < m ≤ N and for ω-a.e.(j).
This implies that the ball B_d_y_j,R+1^ℓ(id, ε) is a subgroup of Stab_G_j(y_j) for ω-a.e.(j). Indeed if h_j,h_j' ∈B_d_y_j,R+1^ℓ(id, ε) then h_jh_j' ∈B_d_y_j,R+1^ℓ(id, 2ε). By (i) the element h_jh_j' is ε-close to some g_j,i, so d_y_j,R+1^ℓ(id, g_j,i) = d_y_j,R+1^ℓ(g_j,1, g_j,i) ≤ 3ε. Now (ii) implies that g_j,i = g_j,1 = id.
The last thing we need to recall is
(iii) the multiplication on the left by g_j,i defines a d_y_j,R+1^ℓ-isometry between B_d_y_j,R+1^ℓ(id, ε) and B_d_y_j,R+1^ℓ(g_j,i, ε).
Recall that we are supposing that the image of Stab_G_j(y_j) →Isom(B(y_j, R)) has cardinality bigger than N+1 for ω-a.e.(j). This means, together with (iii), that the following holds true ω-a.s.: there exists an isometry h_j ∈Stab_G_j(y_j) such that d_y_j,R+1^ℓ(h_j, id) ≤ε and d_y_j,R^ℓ(h_j, id) > 0.
We claim that d_y_j,R+1^ℓ(h_j^k,id) > 1 for some k∈ℤ, which is a contradiction to the fact that B_d_y_j,R+1^ℓ(id, ε) is a subgroup. We proceed as follows. Since d_y_j,R^ℓ(h_j, id) > 0 we can find a point p_j ∈B(y_j,R) which is not fixed by h_j. Call z_j the projection of p_j to Fix(h_j) and extend the geodesic [z_j,p_j] beyond p_j up to find a point w_j at distance R+1 from x_j.
Three things hold: z_j is the projection of w_j to Fix(h_j), d(w_j,z_j) > 1 and w_j∈B(y_j,R+1). If the whole orbit ⟨ h_j ⟩ w_j would be contained in B(w_j,1) then the center of this orbit (cp. <cit.>) would be a fixed point of h_j at distance at most 1 from w_j, which is a contradiction since d(w_j,Fix(h_j)) > 1.
Therefore there must be some k∈ℤ such that d(h_j^kw_j, w_j) > 1, i.e. d_y_j,R+1^ℓ(h_j^k,id) > 1 for some k∈ℤ.
This gives a similar uniform estimate for all unimodular groups.
Let P_0,r_0,D_0,R >0. There exists a constant N_0^(R) = N_0^(P_0,r_0,D_0, R) > 0 such that the following holds true.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space and let G < (X) be closed, totally disconnected, unimodular and D_0-cocompact. Then for all x∈ X the cardinality of the image of the map _G(x) →(B(x,R)) is at most N_0^(R).
If ^♢(G,X) > σ_0:=σ_P_0,r_0,D_0(ε_0) then it is enough to take N_0^(R) = N_0(σ_0, R). Otherwise we apply Theorem <ref> with ε = ε_0. We have a splitting X= Y ×ℝ^k, k≥ 1 and we write x = (y, v). Every element of _G(x) is of the form (g',g”), where g' fixes y and g” is a finite order isometry of a crystallographic group fixing v. The order of the stabilizer of v inside this crystallographic group is at most J_0 by Proposition <ref>. The isometric action (Y,y,G_Y) is σ_0-controlled by Corollary <ref>, see also the beginning of this section. Therefore the cardinality of the image of the map Stab_G_Y(y) →Isom(B(y,R)) is at most N_0(σ_0, R). The thesis follows taking N_0^(R) = N_0(σ_0, R) · J_0.
As a consequence of Corollary <ref> we have a quantified version of <cit.> for unimodular groups. A similar statement for σ-controlled groups with N_0(σ,R) in place of N_0^u(R) holds, but we do not write it.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space and let G < (X) be closed, totally disconnected, unimodular and D_0-cocompact. Let g be an elliptic isometry of G, x∉(g) and y be the projection of x on (g). Then
(i) ∠_y(x,gx) ≥ 2arcsin(1/2N_0^(1));
(ii) d(x,gx) ≥1/N_0^(1)· d(x,(g)).
We need to recall the formula <cit.>:
α:=∠_y(x,gx) = lim_t→ 0 2arcsin(d(c(t), gc(t))/2t),
where c(t) is the geodesic [y,x], and where the right hand side is monotone decreasing as t goes to 0. By Corollary <ref> we know that the order of g, when seen as an isometry of B(x,1), is at most N_0^u(1). Suppose α < 2arcsin(1/2N_0^(1)) and choose ε > 0 so that α+ε < 2arcsin(1/2N_0^(1)). We can find t> 0 such that
α +ε≥ 2arcsin(d(c(t), gc(t))/2t),
i.e. 2tsin(α+ ε/2) ≥ d(c(t),gc(t)). If k∈{ 1,…,N_0^(1)} we get
d(c(t),g^kc(t)) ≤ 2ktsin(α+ ε/2) < k/N_0^(1)t ≤ t.
Therefore the orbit ⟨ g ⟩ c(t) is contained in a ball of center c(t) and radius smaller than t. The center of this orbit (cp. <cit.>) is a point which is fixed by g and which is at distance <t from c(t), contradicting the fact that y is the projection of c(t) on Fix(g).
A direct application of (<ref>) gives (ii). Indeed, recalling that the quantity inside the limit in (<ref>) is decreasing as t goes to 0, we get
2arcsin( d(x,gx)/2d(x,y)) ≥α.
The proof of (i) is conceptually equivalent to the proof of <cit.>, just notice that they wrote the wrong inequality, essentially opposite to (<ref>). However their proof still works using the limit procedure as we did.
We can also prove Theorem <ref>, actually a bit stronger result.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space. Let G<(X) be closed, totally disconnected, unimodular and D_0-cocompact. Let x∈ X, let g ∈_G(x) and let R≥ 0. Let m = p_1^α_1⋯ p_k^α_k be the prime decomposition of the order of g restricted to (B(x,R)). Then max{ p_1,…, p_k}≤ N_0^(1).
In particular the maximal prime number appearing in the prime decomposition of the order of any finite order isometry is at most N_0^(1).
Clearly the α_k's are not bounded in general. A similar statement holds in the σ-controlled case with N_0(σ,1) in place of N_0^u(1), but we do not report it here.
The thesis is equivalent to bound the order of g|_B(x,R) when it has prime order p > 1. For all 1≤ j < p we can find k∈ℤ such that kj ≡ 1 modulo p, so
Fix(g) ⊆Fix(g^j) ⊆Fix(g^kj) = Fix(g),
forcing the containments to be equalities. Let y be a point of B(x,R) which is not fixed by g and let x' be the closest point to y along [x,y] which is fixed by g. Finally let y' be a point of [x',y] at distance at most 1 from x' and different from x'. By (<ref>) we conclude that the points g^jy' are all distinct for 1≤ j < p and contained in B(x',1). Therefore p ≤ N_0^u(1).
The last important consequence is the following uniform bound, that we explicit in both the unimodular and the σ-controlled situation.
Let X be a proper, geodesically complete, (P_0,r_0)-packed, (0)-space and x∈ X. Let G < (X) be closed, totally disconnected, unimodular (resp. σ-controlled) and D_0-cocompact. Then for every R≥ 0 the group _G(x,R) has index at most N_0^(R) (resp. N_0(σ,R)) in _G(x).
Corollary <ref> (resp. Proposition <ref>) implies that the cardinality of the group Stab_G(x)/Stab_G(x,R) is at most N_0^(R) (resp. N_0(σ,R)), which is the thesis.
§.§ Convergence without collapsing
We are ready to describe the limit group in the non-collapsed standard setting of convergence.
Assume that we are in the standard setting of convergence of
(X_j, G_j) ⟶ (X_∞, G_∞),
without collapsing. Then the limit group G_∞ is closed, totally disconnected, unimodular and D_0-cocompact.
Up to pass to a subsequence, which converges to the same limit, we can suppose that there exists 0<ε≤ε_0 such that
sys^♢(G_j,X_j) ≥ε > 0 for every j. Therefore each (X_j,G_j) is σ := σ_P_0,r_0,D_0(ε)-controlled, as explained at the beginning of Section <ref>. Proposition <ref> implies that G_∞ is closed, D_0-cocompact and totally disconnected.
Let x_j be points of X_j such that G_σ(x_j) = Stab_G_j(x_j).
Let μ_j be bi-invariant Haar measures on G_j normalized in such a way that μ_j(Stab_G_j(x_j)) = 1.
We claim that conditions (i) and (ii) of Corollary <ref> are satisfied, and so that G_∞ is unimodular. We first show that sup_jμ_j(S_R(x_j)) < + ∞ for every R≥ 0. The condition G_σ(x_j) = Stab_G_j(x_j) implies that the set G_jx_j is made of σ/4-separated points. In particular the distinct points of the set S_R(x_j) x_j are at most Pack(R,σ/8) =: P, a number that does not depend on j by Proposition <ref>. Let us take elements g_1,…,g_m ∈S_R(x_j) such that { g_i x_j }_i=1^m = S_R(x_j) x_j. By the discussion above we know that m ≤ P. Moreover if g ∈S_R(x_j) is arbitrary then there exists i ∈{ 1,…,m} such that gx_j = g_ix_j, so g_i^-1g ∈Stab_G_j(x_j). This implies
S_R(x_j) = ⋃_i=1^m g_i Stab_G_j(x_j).
Therefore the claim is proved since
μ_j(S_R(x_j)) ≤∑_i=1^m μ_j (g_i Stab_G_j(x_j)) = ∑_i=1^m μ_j (Stab_G_j(x_j)) = m ≤ P.
On the other hand Corollary <ref> implies that for every r>0 we have
inf_jμ_j(S_r(x_j,1/r)) ≥inf_jμ_j(Stab_G_j(x_j, 1/r)) ≥1/N_0^u(1/r) > 0.
§.§ Convergence with collapsing
We now deal with the collapsing case.
Assume that we are in the standard setting of convergence of
(X_j, G_j) ⟶ (X_∞, G_∞)
with collapsing. Then:
(i) X_∞ splits isometrically and G_∞-invariantly as X_∞' ×ℝ^ℓ, for some ℓ≥ 1. In particular X_∞' is a proper, geodesically complete, (P_0,r_0)-packed, (0)-space.
(ii) G_∞^∘ = {}×(ℝ^ℓ), and
X_∞' = G_∞^∘\ X_∞.
(iii) ℓ is characterized as follows: there exists ρ^* > 0 such that for every 0<ρ≤ρ^* there exists j_ρ such that (G_j,ρ(z_j)) = ℓ for all j≥ j_ρ and all z_j∈ X_j. Here G_j,ρ(z_j) := (G_j)_ρ(z_j) as defined in (<ref>).
(iv) the group G_∞ is unimodular.
(v) the projection G_∞' of G_∞ on (X_∞') is closed, totally disconnected, unimodular and D_0-cocompact.
(vi) the quotient spaces M_j=G_j \ X_j converge to M_∞ = G_∞\ X_∞, which is isometric to the quotient of X_∞' by G_∞'. In particular M_∞∈𝒪_0^(P_0,r_0,D_0).
This theorem is stronger than and refines <cit.>, which is stated for discrete groups. The main, and technically more important, improvement is the characterization of ℓ given in (iii).
We recall a couple of facts we need.
Let (X_j, x_j, G_j), (X_j', x_j', G_j') be two sequences of isometric actions and ω be a non-principal ultrafilter. Then the ultralimit ω-lim(X_j× X_j', (x_j,x_j'), G_j × G_j') is equivariantly isometric to (X_ω× X_ω', (x_ω, x_ω'), G_ω× G_ω').
Let A be a group of translations of ℝ^k.
Then A ≅ℝ^ℓ×ℤ^d with ℓ + d ≤ k. Moreover, there is a corresponding A-invariant metric factorization of ℝ^k as ℝ^ℓ×ℝ^k-ℓ, such that the connected component A^∘≅ℝ^ℓ can be identified with (ℝ^ℓ).
Recall again the function σ_P_0,r_0,D_0(·) provided by Theorem <ref>, and the value σ_0 = σ_P_0,r_0,D_0(ε_0).
The sets defined in (<ref>) will be denoted by S_j,r(z_j, R), S_j,r(z_j, R), G_j,r(z_j, R), G_j,r(z_j, R) if they refer to the group G_j.
Proof of (i) and (ii). We fix a non-principal ultrafilter ω such that ω-lim^♢(G_j,X_j) = 0. By Proposition <ref> it is enough to show the thesis for the ultralimit ( X_ω, G_ω). By our choice of the ultrafilter ω, we have ^♢(G_j,X_j) ≤σ_0 for ω-a.e.(j). Thus X_j splits as Y_j×ℝ^k_j with k_j ≥ 1 for ω-a.e.(j), by Theorem <ref>. If k = ω-lim k_j, we have k≤ n_0 and k_j = k for ω-a.e.(j).
By Lemma <ref> the limit does not depend on the choice of the basepoints, so we can assume that the basepoints are the points x_j=(y_j, v_j) provided by Proposition <ref> applied to ε = ε_0. In particular G_j,σ_0(x_j) preserves the slice { y_j }×ℝ^k.
Choose a positive
η≤1/2min{σ_0 ,
√(2 sin( π/J(k)) )}. By Lemma <ref>, every element of a crystallographic group of ℝ^k moving every point of B_ℝ^k ( v_j, 1/2η) less than 2η is a translation. The group G_η(x_ω, 1/η) is open, so G_η(x_ω, 1/η) ∩ G_ω^∘ = G_ω^∘.
Remark that if g_ω = ω-lim g_j belongs to S_η(x_ω, 1/η), then
for ω-a.e.(j) the isometry g_j belongs to S_j,2η(x_j, 1/2η).
Every element g_j ∈ S_j,2η(x_j, 1/2η) belongs to G_j,σ_0(x_j) since 2η≤σ_0, therefore it acts on X_j= Y_j ×ℝ^k as g_j = (g_j',g_j”), where g'_j fixes y_j (because of our choice of basepoints); moreover, g_j” is a global translation of ℝ^k, since it moves the points of B_ℝ^k ( v_j, 1/2η) less than 2η.
An application of Lemma <ref> says that also X_ω splits isometrically as Y_ω×ℝ^k, where Y_ω is the ultralimit of the (Y_j,y_j)'s. Moreover G_ω preserves the product decomposition.
Every element of g_ω∈ G_ω^∘ can be written as a product u_ω(1) ⋯ u_ω(n), with each u_ω(i) in S_η(x_ω, 1/η).
As u_ω(i) = ω-lim u_j(i) with u_j(i)= (u_j(i)', u_j(i)”) ∈ S_j,2η(x_j, 1/2η), where u_j(i)' fixes y_j and u_j(i)” is a translation, it follows that also
g_ω can be written as (g_ω',g_ω”) ∈Isom(Y_ω) ×Isom(ℝ^k) where g_ω' y_ω = y_ω and g_ω” is a global translation (being the ultralimit of Euclidean translations). Let us call π G_ω→Isom(Y_ω) the projection map. The group π(G_ω^∘) is normal in π(G_ω). The set of fixed points Fix(π(G_ω^∘)) is closed, convex, non-empty and π(G_ω)-invariant (since π(G_ω^∘) is normal in π(G_ω )). Since G_ω is clearly D_0-cocompact, so it is π(G_ω). Then the action of π(G_ω) on Y_ω is minimal (cp. <cit.>) which implies that Fix(π(G_ω^∘)) = Y_ω, that is π(G_ω^∘) = {id}.
We conclude that G_ω^∘ is a connected subgroup of
{id}×Transl(ℝ^k). By Lemma <ref>, ℝ^k splits isometrically as ℝ^ℓ×ℝ^k-ℓ and G_ω^∘ can be identified with the subgroup of translations of the factor ℝ^ℓ,
for some ℓ≤ k.
Setting X_ω':=(Y_ω×ℝ^k_ω-ℓ), this is still a proper, geodesically complete, (P_0,r_0)-packed, CAT(0)-space, and clearly X_ω' =G_ω^∘\ X_ω.
Notice that, since G_ω^∘ is normal in G_ω, then the splitting X_ω =X_ω' ×ℝ^ℓ is G_ω-invariant. Let us now show that G_ω^∘ is non-trivial, hence ℓ≥ 1.
Actually, if G_ω^∘ was trivial then G_ω would be totally disconnected, implying sys^♢(G_ω, X_ω) > 0 by Theorem <ref>.(ii). However, we are able to exhibit hyperbolic isometries of G_ω with arbitrarily small translation length, which will prove that G_ω^∘ is non-trivial. Indeed, fix any λ > 0 and an error ξ > 0.
By the collapsing assumption we can find hyperbolic isometries g_j ∈ G_j with ℓ(g_j) ≤ξ, for ω-a.e.(j). By D_0-cocompactness, up to conjugating g_j we can suppose g_j has an axis at distance at most D_0 from x_j. Take a power m_j of g_j such that λ < ℓ(g_j^m_j) ≤λ + ξ. Then, the sequence (g_j^m_j) is admissible and defines a hyperbolic element of G_ω whose translation length is between λ and λ + ξ. By the arbitrariness of λ and ξ we conclude.
At this point we notice the following. Suppose to be in the standard setting of convergence (X_j,G_j)eq-pGH⟶ (X_∞, G_∞). Theorem <ref> shows that if along a subsequence { j_h} we have lim_h → +∞^♢(G_j_h, X_j_h) > 0 then G_∞ is totally disconnected. The proof above shows that if along a subsequence { j_h} we have lim_h → +∞^♢(G_j_h, X_j_h) = 0 then G_∞ has a non trivial connected component of the identity. Therefore there cannot be a mixed behaviour: either along any subsequence the above limit is positive or along any subsequence it is zero. In particular the fact that ω-lim^♢(G_j, X_j) is zero or not does not depend on the non-principal ultrafilter ω. As a consequence all what we proved and all we are going to prove is true for every possible non-principal ultrafilter ω and not only for the one we fixed at the beginning of the proof. So from now on we fix an arbitrary non-principal ultrafilter ω.
Proof of (iii). We proceed by steps. We first show
lim_ρ→ 0ω-limrk(G_j,ρ(x_j)) = ℓ
for a generic non-principal ultrafilter ω.
For 0 < ρ≤ε_0 the group G_j,ρ(x_j) is almost abelian, so it is meaningful to speak about is rank. For simplicity we set ℓ_ρ := ω-limrk(G_j,ρ(x_j)). Observe that ℓ_ρ is a non-decreasing sequence of integers, depending on ρ.
We start proving ℓ≤lim_ρ→ 0ℓ_ρ. For every 0 < ρ < 2η, where η is as in the proof of (i) and (ii), we choose π/2-linearly independent translations g_ω,1,…,g_ω,ℓ∈ G_ω^o = {𝕀}×Transl(ℝ^ℓ) of length ρ/2. Each isometry can be written as g_ω,i = ω-lim g_j,i. For ω-a.e.(j) the following conditions are true: d(g_j,ix_j,x_j)≤ρ for every i and the projections of the g_j,i's on ℝ^k are π/4-linearly independent translations, in particular rk(G_j,ρ(x_j)) ≥ℓ. The second condition holds because the projection of the g_j,i's on ℝ^k are isometries moving every point of B_ℝ^k( v_j, 1/2η) less than 2η, for ω-a.e.(j), and they belong to a crystallographic group. Therefore by Lemma <ref> they must be translations. The condition on the uniform linear independence is clear. Since this happens for ω-a.e.(j) we deduce that ℓ_ρ≥ℓ for every ρ small enough, then lim_ρ→ 0ℓ_ρ≥ℓ.
We now move to the other inequality. Let ℓ' = lim_ρ→ 0ℓ_ρ. Then there exists ρ' > 0 such that ℓ_ρ = ℓ' for all 0<ρ≤ρ' because the sequence ℓ_ρ takes integer values.
We claim that for every R, λ > 0 we can construct isometries g_ω,1^R,λ, … g_ω,ℓ'^R,λ∈ G_ω such that their projections on Y_ω fix pointwise B(y_ω,R) and whose projections on ℝ^k are translations of length λ that are ϑ_0-linearly independent, where ϑ_0 = min_k≤ n_0ϑ_k, and ϑ_k is provided by Lemma <ref>. Let us first observe why this concludes the proof of (<ref>). Once we have isometries g_ω,1^R,λ, … g_ω,ℓ'^R,λ∈ G_ω as above we let R go to +∞. We get limit isometries g_ω,1^λ, … g_ω,ℓ'^λ∈ G_ω with the property that their projections on Y_ω is the identity and their projections on ℝ^k are still translations of length λ that are ϑ_0-linearly independent. We now consider the group H_ω = { g_ω∈ G_ω whose projection on Y_ω is trivial}. H_ω is a closed subgroup of G_ω that is topologically isomorphic to a closed subgroup of Isom(ℝ^k). Therefore it is a Lie group. In particular its connected component of the identity is open. Observe that each sequence g_ω,i^λ belongs to H_ω and by construction it converges to 𝕀 as λ goes to 0, so there must be λ small enough such that g_ω,i^λ∈ H_ω^o for every i=1,…,ℓ'. But H_ω^o ⊆ G_ω^o, because H_ω^o is a connected set containing 𝕀. Then g_ω,i^λ∈ G_ω^o for every i=1,…,ℓ', and these translations of ℝ^k are ϑ_0-linearly independent. It follows that ℓ' ≤ℓ.
In order to prove (<ref>) it remains to show the construction of the isometries g_ω,1^R,λ, … g_ω,ℓ'^R,λ. We fix R,λ > 0 and we choose ρ < min{ρ', λ/2^n_0 - 1· J_0 · N_0(σ_0,R)}, where N_0(σ_0,R) is the constant provided by Proposition <ref>. By Corollary <ref> the closure of the projection G_Y_j of G_j on Y_j is σ_0-controlled (see also the beginning of Section <ref>), while the projection of G_j,ρ(x_j) on ℝ^k is contained in a crystallographic group of rank k≤ n_0. Moreover another application of Theorem <ref> with ε = ρ provides another splitting X_j = Y_j ×ℝ^k - ℓ'×ℝ^ℓ', where the splitted factor has dimensions ℓ' because this is the rank of G_j,τ(x_j) for every τ < ρ. This splitting is also compatible with the original splitting because of Proposition <ref>. In particular by Theorem <ref>.(v) we can find isometries g_j,1, …, g_j,ℓ'∈G_j,ρ(x_j) whose projections on ℝ^ℓ' are translations of length <ρ and generate a lattice ℒ_j. Hence λ(ℒ_j) < ρ. By Lemma <ref> we can replace the g_j,i's with other isometries, that we still denote g_j,i, whose projections on ℝ^ℓ' still generate ℒ_j, have length at most 2^n_0-1ρ and are ϑ_0-linearly independent. The projection of each g_j,i on the factor ℝ^k - ℓ' is elliptic by Theorem <ref>.(iv). Remember that each g_j,i lives in a crystallographic group of ℝ^k. Proposition <ref> implies that a power not greater than J_0 of each g_j,i acts as a translation on ℝ^k. We replace each g_j,i with this power, without changing the notation. From what we said above we deduce that the projection of g_j,i on ℝ^k - ℓ' is the identity. On the other hand they project as translations of length at most 2^n_0 - 1· J_0 ·ρ that are ϑ_0-linearly independent on ℝ^ℓ'.
Since the projection G_Y_j is σ_0-controlled, then the projection of g_j,i^N_0(σ_0,R) on Y_j is the identity on B_Y_j(y_j,R), by Proposition <ref>. We set g_j,i^N_0(σ_0,R) =: g_j,i^R,λ and we claim they do the job. By construction their projections on Y_j are the identity on B_Y_j(y_j,R). Moreover their projections on ℝ^k - ℓ' are the identity. Finally their projections on ℝ^ℓ' are translations of length at most 2^n_0 - 1· J_0 · N_0(σ_0,R) ·ρ = λ that are ϑ_0-linearly independent. This concludes the proof of (<ref>).
By the arbitrariness of the non-principal ultrafilter ω and using <cit.> we can promote (<ref>) to
lim_ρ→ 0lim inf_j→ +∞(G_j,ρ(x_j)) = lim_ρ→ 0lim sup_j→ +∞(G_j,ρ(x_j)) = ℓ.
Since all the quantities involved are integer-valued we conclude that there exists ρ' > 0 such that for all 0<ρ≤ρ' it holds
lim inf_j→ +∞(G_j,ρ(x_j)) = lim sup_j→ +∞(G_j,ρ(x_j)) = ℓ.
In particular for all 0<ρ≤ρ' we have: ∃lim_j→ + ∞(G_j,ρ(x_j)) = ℓ.
As a consequence for all 0<ρ≤ρ' there exists j_ρ such that if j≥ j_ρ then (G_j,ρ(x_j)) = ℓ. In order to conclude the proof of (iii) we need to extend this result to arbitrary z_j ∈ X_j. We choose ρ^* = σ_P_0,r_0,D_0(ρ') and we claim it satisfies the thesis of (iii).
Fix an arbitrary 0<ρ≤ρ^*. We apply Theorem <ref> with ε = ρ'. Theorem <ref>.(ii) applied to ε = ρ' says that there exists ε^* ∈ (ρ^*, ρ') such that
rk(G_j,ρ(z_j)) ≤rk(G_j,ε^*(z_j)) = rk(G_j,ε^*(x_j)) ≤rk(G_j,ρ'(x_j)) = ℓ
for every z_j ∈ X_j, where the last equality is true for all j≥ j_ρ'. On the other hand, if we apply Theorem <ref>.(ii) to ε = ρ we find ε^* ∈ (σ, ρ), where σ = σ_P_0,r_0,D_0(ρ) such that
rk(G_j,ρ(z_j)) ≥rk(G_j,ε^*(z_j)) = rk(G_j,ε^*(x_j)) ≥rk(G_j,σ(x_j)) = ℓ,
where the last equality is true for all j≥ j_σ. Therefore for all j≥max{ j_ρ', j_σ} we have that rk(G_j,ρ(z_j)) = ℓ for all z_j ∈ X_j. This ends the proof of (iii).
Let ρ^* be the quantity provided by (iii) and set σ^*=σ_P_0,r_0,D_0(ρ^*). Theorem <ref> applied to ε = ρ^* says that, for j big enough, each X_j splits as X_j' ×ℝ^ℓ, where X_j' = Y_j ×ℝ^k-ℓ by Proposition <ref>. This is because the rank of the group G_j,ε^*(x_j) with ε^* ∈ (σ^*, ρ^*) is exactly ℓ for all j≥max{ j_σ^*, j_ρ^*}. It follows that X_j' converges to X_∞'. Corollary <ref> applied to ε = ρ^* shows that we can suppose, up to change the basepoint x_j = (x_j', v_j) ∈ X_j' ×ℝ^ℓ, that G_j, ρ^*(x_j) preserves the slice { x_j' }×ℝ^ℓ and that the closure G_X_j' of each G_j on X_j' has σ^*-separated orbit.
Moreover, by Theorem <ref>.(v), the maximal lattice of the projection of G_j,ρ^*(x_j) on ℝ^ℓ is generated by elements of length at most ρ^*/2√(n_0).
Proof of (iv). We want to show that G_∞ is unimodular. Let ρ^* be the positive number provided by (iii). Take bi-invariant Haar measures μ_j of G_j normalized in such a way that μ_j(S_j,2ρ^*(x_j)) = 1. It is enough to verify that conditions (i) and (ii) of Corollary <ref> are satisfied.
Every isometry g ∈ G_j will be written as (g', g”) ∈Isom(X_j') ×Isom(ℝ^ℓ).
As we noticed above, the orbit G_X_j'x_j' is σ^*-separated. In particular there are at most Pack(R,σ^*/2) =: P points of this orbit inside B_X_j'(x_j',R). Notice that P does not depend on j by Proposition <ref>. Let us take g_1,…,g_m ∈S_j,R(x_j) such that for every g∈S_j,R(x_j) there exists i ∈{ 1,…,m} such that g'(x_j')=g_i'(x_j'). Therefore for every g∈S_j,R(x_j) there exists i∈{ 1,…,m} such that (g_i^-1g)'x_j'=x_j' and d_ℝ^ℓ((g_i^-1g)” v_j, v_j) ≤ 2R. Let us denote by S_j' the set of isometries of G_j whose projection on X_j' fixes x_j' and whose projection on ℝ^ℓ moves v_j by at most 2R. We just showed that
S_j,R(x_j) ⊆⋃_i=1^m g_i S_j',
so μ_j(S_j,R(x_j)) ≤ P ·μ_j(S_j'). To conclude the thesis we just need to bound μ_j(S_j') from above by a quantity that does not depend on j. The projection of
G_j,ρ^*(x_j) on ℝ^ℓ is a crystallographic group whose maximal lattices ℒ_j,ρ^* ia generated by elements of length at most ρ^*/2√(n_0). We first apply Lemma <ref> to find a sublattice ℒ'_j,ρ^* < ℒ_j,ρ^* with λ(ℒ_j,ρ^*) = λ(ℒ'_j,ρ^*) ≤ 2τ(ℒ'_j,ρ^*) ≤ρ^*/√(n_0). Secondly, by replacing each generator of ℒ'_j,ρ^* by a fixed power, depending on j, we can suppose that
ρ^*/2√(n_0)≤τ(ℒ'_j,ρ^*) ≤λ(ℒ'_j,ρ^*) ≤ρ^*/√(n_0).
Let g_j = (g'_j,g”_j) ∈ S_j'. First we find an element h_j = (h_j',h_j”) ∈G_j,ρ^*(x_j) such that h_j”∈ℒ'_j,ρ^* and d_ℝ^ℓ(g_j” v_j, h_j” v_j) ≤ 2ρ^*, because ℒ'_j,ρ^* is 2ρ^*-cocompact by (<ref>). In particular h_j^-1g_j ∈S_j,2ρ^*(x_j) because h_j'^-1g_j' x_j' = x_j'.
Moreover ‖ h_j”‖≤ 2R + 1, so by Lemma <ref> applied to ℒ'_j,ρ^* we can find a word w_j = (w_j',w_j”) of length at most (2R + 1)2√(n_0)/ρ^* + 1 in S_j,2ρ^*(x_j) such that w_j'^-1h_j' fixes x_j' and w_j” = h_j”.
In particular w_j^-1h_j^-1∈S_j,2ρ^*(x_j). Combining all together we conclude that g_j can be written as a word of length at most (2R + 1)2√(n_0)/ρ^* + 3 =: p in the alphabet S_j,2ρ^*(x_j). Observe that p does not depend on j. In other words
S_j' ⊆ (S_j,2ρ^*(x_j))^p.
Corollary <ref> ensures that
μ_j(S_j') ≤ M_0^p-1μ_j(S_j,2ρ^*(x_j)) = M_0^p-1,
which is independent of j.
We now move to the proof of condition (ii) of Corollary <ref>.
First we show
inf_j μ_j(S_j, ρ(x_j)) > 0
for every 0<ρ≤ρ^*.
For fixed ρ we restrict the attention to the indices j≥ j_ρ, so that rk(S_j, ρ(x_j)) = rk(S_j, ρ^*(x_j)) = ℓ. We can apply Theorem <ref>.(iv) with ε = ρ to find a lattice ℒ_j,ρ of the projection of G_j,ρ(x_j) on ℝ^ℓ generated by elements of length at most ρ/2√(n_0).
Using Lemma <ref> as above we can find a sublattice ℒ'_j,ρ of ℒ_j,ρ satisfying
ρ/4√(n_0)≤τ(ℒ'_j,ρ) ≤λ(ℒ'_j,ρ) ≤ρ/2√(n_0).
The lattice ℒ'_j,ρ is ρ-cocompact because of (<ref>). Repeating word by word the same argument as before
we can write each element of S_j,2ρ^*(x_j) as a word of length at most 8√(n_0)(ρ^*/ρ + 3) =: p in the alphabet S_j,ρ(x_j).
In other words
S_j, 2ρ^*(x_j) ⊆S_j,ρ(x_j)^p.
Again by Corollary <ref> we have
1 = μ_j(S_j, 2ρ^*(x_j)) ≤ M_0^p-1μ_j(S_j, ρ(x_j)),
where p is independent of j. This shows (<ref>).
We now fix r >0. We choose ρ = r/J_0· N_0(σ^*,1/r). Let w be any word in the alphabet S_j, ρ(x_j) of length J_0 · N_0(σ^*,1/r), that we write as w=(w', w”). Since the projection of G_j on X_j' is σ^*-controlled we deduce that w' acts as the identity on B_X_j'(x_j,1/r) by Proposition <ref>. On the other hand w” is a translation of ℝ^ℓ by Proposition <ref>. So w acts on X_j' fixing the ball B_X_j'(x_j, 1/ε) and on ℝ^ℓ as a translation of length at most ρ· J_0· N_0(σ^*, 1/r) ≤ r. In particular w belongs to S_r(x_j, 1/r). This shows that S_j,ρ(x_j) can be covered with at most J_0· N_0(σ^*,1/r) translated of S_r(x_j, 1/r).
Therefore μ_j(S_r(x_j, 1/r)) ≥1/J_0· N_0(σ^*,1/r)·μ_j(S_j,ρ(x_j)) for every j big enough. This implies the thesis by (<ref>). In conclusion G_∞ is unimodular.
Proof of (v). We first show that G_∞' is closed, where G_∞' is the projection of G_∞ on X_∞'. Let g_i' be a sequence of elements of G_∞' and suppose they converge to g'. By applying elements of G_∞^∘ = {𝕀}×Transl(ℝ^ℓ) we can suppose to have elements g_i = (g_i', g_i”) ∈ G_∞, with g_i” v_∞ = v_∞. In particular g_i” converges, up to subsequence, to some g”. Therefore g_i converges to g = (g',g”) ∈ G_∞, since G_∞ is closed. Therefore g' ∈ G_∞'.
Let us show now that G_∞' is unimodular. Let μ_∞ be a bi-invariant Haar measure of G_∞, which exists by (iv). Let us denote by S_1( v_∞) the subset of the projection of G_∞ on Isom(ℝ^ℓ) made of the isometries that move v_∞ by at most 1. Let A⊆ G_∞' be any set and consider the set
A^* := A ×S_1( v_∞).
We claim that the formula ν(A):=μ_∞(A^*) defines a bi-invariant Haar measure of G_∞'. Clearly ν(∅)=0, while if A_i ⊆ G_∞' are pairwise disjoint then also A_i^* ⊆ G_∞ are pairwise disjoint. Therefore
ν(⋃_i A_i) = μ((⋃_i A_i)^*) = μ(⋃_i A_i^*) = ∑_i μ(A_i^*) = ∑_i ν(A_i).
This shows that ν is actually a measure on the Borel subsets of G_∞', since if A is Borel then A^* is Borel too. The next step is to show that it is finite on compact sets and positive on open sets. Observe that if A is compact then A^* is compact, so ν(A) = μ(A^*) < +∞. On the other hand if A is open then A^* has non-empty interior, so ν(A) = μ(A^*) > 0. It remains to show it is left and right invariant. Every element g' ∈ G_∞' comes from some g=(g',g”) ∈ G_∞. After a composition with an element of G_∞^∘ = {𝕀}×Transl(ℝ^ℓ) we can suppose that g” fixes v_∞. In particular g”S_1( v_∞) = S_1( v_∞) = S_1( v_∞) g”.
Therefore we have
(g' A)^* = g'A ×S_1( v_∞) = g'A × g”S_1( v_∞) = (g',g”) (A×S_1( v_∞)) = (g',g”) A^*.
This implies
ν(g'A)=μ((g'A)^*) = μ ((g',g”) A^*) = μ (A^*) = ν (A).
In the same way we can prove the right invariance of ν, so G_∞' is unimodular.
The last step is to show that it is totally disconnected. This can be done observing that, with the notation of the previous steps, each G_X_j' is σ^*-controlled and using Proposition <ref>. There is also another way to get the conclusion. Indeed let us consider the natural action of the totally disconnected group G_∞ / G_∞^o on X_∞'. By construction the image of such action is G_∞'. The kernel is
{ g = (g',g”) ∈ G_∞ / G_∞^o s.t. g' = 𝕀}.
Any class in the set above has a representative of the type (𝕀, g”), with g” v_∞ = v_∞. Therefore the kernel of the action is a compact, normal subgroup of G_∞ / G_∞^o. Then the thesis follows since the group G_∞' is the quotient of the locally compact, totally disconnected group G_∞ / G_∞^o by a closed, normal subgroup. Even if it is not necessary, observe that the fact that the kernel of the action is compact shows also that the group G_∞ / G_∞^o is unimodular. The fact that G_∞' is D_0-cocompact is obvious.
Proof of (vi). The quotients M_j=G_j \ X_j converge to M_∞ = G_∞\ X_∞, by Lemma <ref>. Moreover
G_∞\ X_∞ = (G_∞ /G_∞^∘) \ (G_∞^∘\ X_∞) = G_∞' \ X_∞'.
§.§ Characterization of collapsing
If X ∈_0^(P_0,r_0,D_0) then dim(X) ≤ n_0=P_0/2, by Proposition <ref>(ii). Moreover, if (X_j,x_j) ⟶ (X_∞,x_∞) then, by <cit.>,
dim(X_∞) ≤lim inf_j → +∞dim(X_j).
The following theorem precisely relates the collapsing (as defined in <ref>) in the standard setting of convergence to the dimension of the limit quotients, and is a direct consequence of the convergence Theorems <ref> & <ref>. The proof is the same of <cit.>.
[Characterization of collapsing]
In the standard setting of convergence of
(X_j, G_j) ⟶ (X_∞, G_∞),
let M_j = G_j \ X_j and M_∞ = G_∞\ X_∞ be the quotient spaces. Then:
(i) the sequence is non-collapsing iff (M_∞) = lim_j→ +∞(M_j);
(ii) the sequence is collapsing iff (M_∞) < lim_j→ +∞(M_j).
Moreover, in the above characterization, the topological dimension can be replaced by the Hausdorff dimension .
Notice that since we proved that lim_j→ +∞TD(M_j) exists, Theorem <ref> excludes to have sequences (X_j,G_j) converging with mixed behaviour (that is, such that along some subsequence the convergence is collapsed, and along other subsequences it is non-collapsed). We have already noticed this fact during the proof of Theorem <ref>. Finally we conclude with the
Let M_j ∈𝒪-CAT_0^td,u(P_0,r_0,D_0) and suppose it converges to some M_∞ in the Gromov-Hausdorff sense. By definition there are isometric actions (X_j, G_j) ∈CAT_0^td,u(P_0,r_0,D_0) such that G_j\ X_j = M_j. Up to pass to a subsequence we can suppose that (X_j,G_j) eq-pGH⟶ (X_∞, G_∞) by Proposition <ref>. We are in the standard setting of convergence. Observe that by Lemma <ref> the space M_∞ is isometric to the quotient G_∞\ X_∞. If the sequence (X_j,G_j) is non-collapsed then we get M_∞∈𝒪-CAT_0^td,u(P_0,r_0,D_0) by Theorem <ref>. If it is collapsed then M_∞∈𝒪-CAT_0^td,u(P_0,r_0,D_0) by Theorem <ref>. By the remark above these two cases cover all possible cases, so we conclude the proof of the closure of 𝒪-CAT_0^td,u(P_0,r_0,D_0). The compactness is a consequence of Proposition <ref> and Lemma <ref>.
alpha
|
http://arxiv.org/abs/2307.04458v1 | 20230710101221 | Analyzing the Evolution of Inter-package Dependencies in Operating Systems: A Case Study of Ubuntu | [
"Victor Prokhorenko",
"Chadni Islam",
"Muhammad Ali Babar"
] | cs.SE | [
"cs.SE"
] |
V. Prokhorenko et al.
CREST - The Centre for Research on Engineering Software Technologies, the University of Adelaide, Australia
victor.prokhorenko, [email protected]
Cyber Security Cooperative Research Centre (CSCRC), Australia
Queensland University of Technology, Brisbane, Australia
[email protected]
Analyzing the Evolution of Inter-package Dependencies in Operating Systems: A Case Study of Ubuntu
Victor Prokhorenko1,2 Chadni Islam3 Muhammad Ali Babar1,2
August 12, 2023
==================================================================================================
An Operating System (OS) combines multiple interdependent software packages, which usually have their own independently developed architectures. When a multitude of independent packages are placed together in an OS, an implicit inter-package architecture is formed. For an evolutionary effort, designers/developers of OS can greatly benefit from fully understanding the system-wide dependency focused on individual files, specifically executable files, and dynamically loadable libraries. We propose a framework, DepEx, aimed at discovering the detailed package relations at the level of individual binary files and their associated evolutionary changes. We demonstrate the utility of DepEx by systematically investigating the evolution of a large-scale Open Source OS, Ubuntu. DepEx enabled us to systematically acquire and analyze the dependencies in different versions of Ubuntu released between 2005 (5.04) to 2023 (23.04). Our analysis revealed various evolutionary trends in package management and their implications based on the analysis of the 84 consecutive versions available for download (these include beta versions). This study has enabled us to assert that DepEx can provide researchers and practitioners with a better understanding of the implicit software dependencies in order to improve the stability, performance, and functionality of their software as well as to reduce the risk of issues arising during maintenance, updating, or migration.
This work is accepted for publication in The 17th European Conference on Software Architecture (ECSA 2023),
Istanbul, Turkey.
§ INTRODUCTION
Combining multiple independent software packages together is commonly used to form complex inter-connected ecosystems. A typical example of such large software ecosystems is various Linux distributions. Such ecosystems tend to consist of hundreds or thousands of packages, libraries, binaries, and configuration files with an order of magnitude more dependencies among them <cit.>, <cit.>.
Developers and researchers have expressed interest in software complexity measurement in an attempt to reason about characteristics of large code bases <cit.>. Software complexity is viewed as a result of different design decisions and implementation specifics and is a crucial component of long-term effects like the maintainability of software <cit.>. Although software complexity is a crucial consideration for package managers, Linux distributors, and maintainers, we currently have limited knowledge about the evolution of this complexity over the software lifespan. While the complexity of individual packages is tamed by their corresponding developers, combining thousands of packages materializes a new emergent layer of complexity. It is also uncertain whether different metrics for measuring software complexity exhibit similar or varying patterns of evolution.
A significant amount of research has extensively explored source-level software complexity <cit.>. As a result, various complexity metrics have been defined, such as cyclomatic, branching, or data flow complexity <cit.>. These metrics are primarily used for software design, debugging, and optimization purposes <cit.>.
These metrics are, however, not applicable when analyzing closed-source software distributed only in binary form without access to the source code. In such cases, binary dependency analysis is required to understand the interactions and dependencies between compiled binary executables. Additionally, even when source code is available, there may be situations where the compiled binary may behave differently from what is expected due to specific environment configurations. Thus, binary dependency analysis can provide a more accurate and complete understanding of run-time software behavior, which can be crucial for identifying potential issues or vulnerabilities.
This work considers an OS as a whole rather than focusing on analyzing individual software binaries. Considering an OS enables the identification of cross-application relations, which make up an emergent inter-package relation architecture instead of just the intra-package software complexity. We propose a framework that enables the extraction of binary-to-library dependencies and constructs a full OS dependency graph to obtain insights on overall OS complexity which we determine through inter-package dependency coupling. By coupling we mean any type of dependency of one code fragment on another (library inclusion, function call, etc).
Our study focused on Ubuntu as a case study to examine the evolution of large software ecosystems over almost two decades. Through empirical research and evidence-based findings, we aimed to assess the current state of package, library, and binary dependencies and identify areas for improvement in management tools, policies, and ecosystem analysis platforms.
We believe that a deep understanding of emergent inter-package architecture resulting from combining a multitude of independently developed software subsystems would benefit software developers and OS maintainers. The proposed techniques and tools are expected to minimize manual labor associated with multi-package maintenance.
Following are the key contributions of our work
* We have introduced a framework for dependency coupling analysis for multi-package software to extract the inter-package relations architecture that is applicable to a broader range of OS due to the binary-level analysis.
* We have defined four techniques to quantitatively measure software coupling in terms of executable and dynamically loadable library dependencies at different granularities.
* We have investigated the evolution of Ubuntu OS in terms of the proposed library presence dependency type, which revealed the changes in OS-wide inter-package relations over time.
§ BACKGROUND AND MOTIVATION
§.§ Software Complexity
Throughout the lifetime of any software system, various code modifications must be implemented in order to adapt to ever-changing user requirements and environmental conditions. An intuitive expectation is that large and complex software systems may be more difficult to update and maintain. Thus, in efforts to gain a stricter definition of complexity, multiple code complexity measurement techniques, such as straightforward line count or cyclomatic complexity, have been proposed so far <cit.>. However, analyzing multiple diverse software systems as a whole is not trivial due to (i) lack of access to the source code of all third-party components,
(ii) lack of formal interoperability specification and
(iii) highly dynamic state of execution environment at run time.
Several techniques are typically employed to handle the growing complexity of large software systems (such as a full OS). For instance, the system package manager may track package dependency information at the OS level. This tracking enables detecting incompatibilities between separate software subsystems and repairing them if possible. Unfortunately, manual labor is commonly used in determining and maintaining information on such version-level incompatibilities <cit.>. Due to the large number of files in a typical OS, manual efforts typically target only high-level dependency definitions, such as package level only <cit.>. As each package may consist of multiple files containing executable code (i.e., executable binaries and libraries), such package dependency understanding may not represent the dependencies precisely.
Further challenges arise due to modern complex software systems commonly developed in various programming languages. For instance, purely-binary compiled languages are intertwined with interpreted script languages leading to execution flow frequently being transferred between them. The dependency chains within such complex systems may propagate through a significant portion of files in the file system through the indirect reliance of different code fragments on each other. A typical example includes PHP web pages relying on the PHP interpreter, web server, and third-party PHP libraries. Such immediately obvious (direct) dependencies, in their turn, recursively rely on other system-provided and third-party libraries. Therefore we argue that automated and precise dependency tracking would benefit software system maintainers and administrators and may provide useful insight to software developers.
§.§ Code dependency types
One piece of code can depend on another in numerous ways. For instance, within the source code analysis context, a function may call other functions. Similarly, class methods may work by invoking other class methods. These types of dependencies present in the same code base are well understood and routinely used in modern IDEs (Integrated Development Environments) to aid software developers. In contrast, cross-language code dependencies spanning across multiple independently developed software systems are less formal and challenging to identify. For instance, a PHP-oriented IDE would not detect incompatible changes in the library which is required by the PHP interpreter itself.
Focusing solely on software running within the same OS while not taking network-based dependencies into consideration, we propose the following four conceptual types of dependencies suitable in the executable code analysis context. These four types include (i) the presence of third-party libraries, (ii) the extent of library coverage, (iii) library function call occurrences, and (iv) the run-time usage of functions (Figure <ref>).
The third-party library presence dependency relates to file-level granularity. This type of dependency indicates a requirement for a dynamically loadable library to be present in the system for an executable binary to be able to load and start. In Windows-based systems, libraries and executables are denoted by .dll and .exe file extensions, while on Linux-based these are .so and typically extension-less ELF (Executable and Linkable File) correspondingly. While high-level, this file granularity is crucial as a missing library file typically causes the executable file loader to indicate an error and prevents any further file execution.
Coverage dependency focuses on the library fragments (e.g., functions or class methods) that a developer explicitly uses or relies on. This type of dependency refers to specific function existence requirements. Thus, the library coverage aspect reflects the degree of reliance on a given library by the executable. Depending on the OS, programming language, and execution environment, individual function-level requirements can be implemented in various ways. For instance, in the context of the Windows PE executable, the list of required functions is tied to a specific library. In contrast, the lists of required libraries and functions are independent in the Linux ELF executable <cit.>. These implementation specific differences complicate coverage analysis in the general case.
Function occurrence dependency type attempts to provide further insight into the code dependency by observing that a single external function can be referred to multiple times in the original code. For instance, some heavily used functions can be mentioned all over the code, while some rarely used functions may only appear once. Extracting this type of dependency is extremely complicated and involves computationally-heavy disassembling of compiled code or parsing of interpreted languages. Initial unoptimized attempts revealed a significant time overhead for extracting such occurrence-level dependencies. While certain optimizations can be taken for production-ready usage, it can be concluded that this type of analysis is currently unsuitable for real-time applications.
Lastly, dependency usage refers to the actual run-time external code flow control transfers (i.e., the actual function calls). This level of detail may, for example, reveal that one function call is contained within a high-count loop while other function calls may be a part of a condition rarely satisfied at run time. Run-time observation would reveal a deeper understanding of the level of reliance on third-party libraries in both cases. Despite seemingly most accurate and closest to reality, relying on this type of dependency suffers from a major drawback. Different executions or instances of the same executable may exhibit different behavior due to different run-time conditions. In other words, observing a single execution does not guarantee to reveal all external code usage cases.
Note that a purposefully crafted executable may incorporate external dependencies that would not be reflected using the proposed dependency measurement techniques. For instance, if an executable downloads code over the network and executes it in place, no third-party library references, function names, or function calls related to the downloaded code may be present in the original executable. Moreover, the downloaded code downloaded can be different on each program invocation, making any dependency analysis futile in such a context. Based on the identified dependency types, we propose an extensible plugin-based framework suitable to extract code dependencies for various types of executable code.
§ OUR APPROACH AND IMPLEMENTATION
Analyzing the full file system enables a more complete and consistent understanding of the dependencies. Software developers only express a requirement for dynamically loadable library presence, but do not have actual guarantees of the library's existence in a given system. We implement a Python-based proof of concept solution to analyze system-wide dependencies.
On a conceptual level, our proposed approach for Dependency Extraction (DepEx consists of a file system scanner, a plugin dispatcher, multiple user-definable file-type-specific plugins, and the resulting database. The following steps provide an overview of the DepEx operation:
* The existing dependency extraction plugins (also Python-based) are queried to prepare the list of all supported file types
* The specified file system is iterated over and each file of a supported type is passed to a corresponding plugin for dependency extraction
* The dependencies extracted by the plugin are stored in an SQLite database
Having the knowledge of individual file type structures, each plugin is responsible for external dependency detection and extraction. Note that while the current implementation assumes one-to-one relation between file types and plugins, it is possible for multiple plugins to process the same files to extract different types of dependencies.
While we have implemented a proof of concept plugins for PHP, Bash, and, to a lesser degree, Python scripts, in this research we primarily focus on ELF executables and .so libraries with the library presence dependency.
Once the unattended phase of the dependency extraction is complete, several interactive analysis and usage scenarios become accessible. These include visualization, statistical reporting, and forward and reverse update impact estimation. For instance, various system health characteristics, such as "number of missing libraries" or "number of executables with unfulfilled dependencies" can be queried and plotted if necessary. Similarly, update impact calculation enables obtaining the list of executables and libraries that would be potentially affected in case a given library is updated.
In order to aid comprehension of the large amounts of data collected, we developed a visualization subsystem. Using DOT language for graph representation enables rendering the resulting graphs using existing tools as well (such as GraphViz or Gephi). While the individual executable file graphs were readable, the full-system dependency graph was too cluttered for human comprehension. At this stage, interactive filtering was implemented to allow the hiding of popular libraries responsible for most of the visual noise (as shown in Figure <ref>). We are also planning to implement automated filtering based on various features, such as node type, sub-string matching, and popularity.
Other auxiliary scripts for dependency graphs exploration include querying all binaries and libraries that depend on a given library () and individual binary/library dependency graph generation ( and ). Individual library dependencies can also be visualized in a more detailed view.
§ STUDYING THE ARCHITECTURAL ASPECTS OF UBUNTU
We focus on the following Research Questions (RQs) to investigate the file-level package relation architecture in Ubuntu systems using DepEx. We considered the presence dependency in this case study. We collected and analyzed the dependencies of 84 consecutive live Ubuntu Linux images that span over 18 years of development and evolution. The research questions we primarily focus on revolve around the emergent inter-package OS-wide architecture implicitly forming as a result of combining multiple independent software packages as well as the related architectural changes observed throughout longer time periods. In addition, we investigate the complexity perception from the perspectives of individual software package developers and whole system maintainers.
* RQ1. How do binary-to-library dependencies manifest in the Ubuntu OS in terms of a system-wide dependency graph?
* RQ2. What is the difference between individual library complexity directly exposed to developers vs. overall internal system complexity that emerges as a result of combining multiple subsystems together (direct vs. recursive dependencies)?
* RQ3. How does the whole Ubuntu OS binary-to-library dependency graph evolve over a longer period?
Having high popularity, rich history, and open-source nature, Ubuntu serves as a comprehensive data source. Despite other Linux distributions, such as Alpine, gaining popularity, we were unable to find another dataset comparable in size and quality. Specifically, older Alpine versions were unavailable for download and Debian produced fewer live images.
Throughout the development of our DepEx framework, we relied on well-established existing open-source software, such as
squashfs-tools[<https://github.com/plougher/squashfs-tools>], binutils[<https://www.gnu.org/software/binutils/>] and ldd[<https://man7.org/linux/man-pages/man1/ldd.1.html>]. SquashFS-related tools were used to expose compressed live Ubuntu images for analysis. Note that different versions of had to be used depending on the age of the Ubuntu image. Binutils package, particularly the GNU tool, was used to extract ELF-specific data such as imported library names. Lastly, was used to extract library search locations. Special precautions had to be taken to lookup for the library paths inside the mounted image rather than resolving paths within the host system that conducted the analysis. For this purpose, we relied on standard Linux functionality.
Solely mounting the Ubuntu ISO files directly does not provide access to the live file system, as another layer of compression is typically present for disk space optimization purposes. Thus, we implemented a two-step unpacking process to gain visibility of the inner live file system.
Interestingly, extracting the images generated over 18 years revealed how live image preparation changed over time. We noticed different compression techniques used throughout the time period analyzed that ranged from compressed loop files (cloop) to SquashFS versions 2.1-4.0. We also observed that modern SquashFS kernel modules could not transparently mount images compressed by older versions. Thus, we developed a supporting script to provide access to all of the downloaded images in a uniform manner.
Using our DepEx framework, we recursively built the full library dependency graph for each identified executable using , and tools. Extracting library dependencies requires analyzing and variables, system library cache as well as the binary executable file path. Finally, we used an SQLite database to store the collected dependency data for all the scanned Ubuntu images. This data can be queried for further analysis and visualization.
§ FINDINGS AND RESULTS
The dependency data extracted from a typical OS is a rich source of information on the high-level system architecture. In contrast to planned layer of architecture, this layer refers to the unwritten architectural aspects that emerge as a result of combining a multitude of independently-developed software packages. Coupled with temporal updates, this data can serve as a basis for a deeper system evolution trends analysis. For instance, long-term trends such as libraries gaining or losing popularity or executable complexity inflation may be detected. Predicting potential OS library or executable removal may help developers adjust the development plans. In addition, determining and removing unused libraries could be useful in optimizing disk space usage and reducing the attack surface.
Throughout the data collection conducted, we focused on three key aspects. Firstly, we investigated the OS-level dependency graph as a whole (RQ1). Secondly, we examined various aspects of complexity in binary dependencies determined through coupling analysis (RQ2). Lastly, we analyzed evolutionary trends in the OS dependency graph (RQ3).
§.§ OS-wide Dependency Graph
Analyzing the resulting SQLite database, which covers 84 Ubuntu images, revealed the following number of binaries, libraries and dependencies per image. We found that from Ubuntu 5.04 to 23.04 the number of binary executables ranged from 1519 to 2753 and the number of libraries ranged from 1683 to 3673. In terms of dependencies detected, the numbers ranged from 18165 to []37641 in the images scanned. A total of 408364 binary and library files were processed to extract the dependencies, which returned almost 2 million dependencies. The total SQLite database size generated is over 83MB of raw dependency data.
We noticed that highly popular libraries such as () make the graphs unreadable. Thus we implemented filtering out libraries from the sorted (by popularity) list of all the involved libraries. We observe that hiding the top 10-15 libraries increases the readability of the whole system graph. Notably, loosely coupled subsystems, such as the networking subsystem, become apparent. The libraries presented alongside the diagram also provide insight into the relative popularity of individual libraries within a system.
We have observed that number of libraries imported but not present in the system varied from 20 (v5.04) to 8 (v23.04) with the highest number being 92 (v21.10b). As a consequence, the number of other libraries directly impacted by the missing dependencies varied from 4 (v17.10 and v17.10.1) to 27 (v13.04 and v9.04). Similarly, we see that the number of unused libraries (i.e., not imported by any other library or executable) ranged from 1301 (v5.04) to 1666 (v23.04). These numbers constitute a significant proportion of the total number of libraries included (around 77% and 62% respectively). Potential explanations for such a high number of unused libraries could be a) plugin-based applications that do not import libraries directly, b) "forgotten" legacy libraries and c) libraries shipped "just in case" for use by applications commonly installed at a later stage.
§.§ Dependencies Coupling Aspects
Software dependencies represent the reliance of a given piece of code on external code. In practice, software developers only deal with a subset of the code required for an application to run. A graphics-oriented library may expose a simpler set of functions to developers, while relying on a multitude of other complex hardware-specific libraries to implement the advertised functionality. Thus, a complex and large code base is made to look simple from the developer's perspective.
This perception difference opens the possibility of measuring code coupling in direct and recursive ways. The direct coupling of an application reflects how many specific libraries a developer deals with explicitly. In contrast, recursive coupling takes all the underlying dependencies into consideration as well.
In addition, there is an inherent asymmetry in dependency tracking. Forward tracking from a given binary to all the required libraries is trivial, as this information is contained within the binary. Reverse tracking from a given library to determine all the binaries and libraries that require the specified library is complicated, as this information is not stored explicitly. Reverse tracking essentially reflects the popularity of a given library and requires scanning the whole file system to be calculated. Thus we developed functionality to measure the (i) direct coupling, (ii) total (recursive) coupling, and (iii) library popularity.
Figures <ref> and <ref> illustrate the changes in the average and maximum number of dependencies correspondingly. As can be seen from Figure <ref>, whereas the average total number of dependencies largely stays the same, developer-facing complexity tends to decrease over time. This indicates that developers tend to re-arrange code within libraries to minimize the coupling they face directly. The large spike in Figure <ref> is caused by the introduction of Gnome Shell in Ubuntu 17.10. We, therefore can conclude that while maintaining roughly the same external coupling, GNOME Shell has a complicated internal structure. Particularly, we found that binary has the largest amount of dependencies. This is explained by the fact that the configuration tool needs to interact with most of the GNOME Shell subsystems.
A complementary aspect of dependency coupling is popularity. We define library popularity through the number of other libraries or executables that depend on it. In other words, damaging or removing more popular libraries would impact a larger number of executables in a system. In terms of popularity, the top 10 most used libraries (i.e. imported from other libraries and executables) in Ubuntu are: . The numbers alongside the libraries refer to the number of uses (i.e., library importing) averaged across all Ubuntu versions the library was present in.
We notice that 7 out of the top 10 directly-coupled libraries relate to various GNOME subsystems while the other 3 relate to the Evolution mail client. Interestingly, the most complex executable with 100 direct dependencies was only present in two Ubuntu versions. This likely indicates that such high coupling was not tolerated, leading to the application removal.
Lastly, analyzing total coupling by taking recursive dependencies into account, we found the top 10 complex libraries and binaries:(154),
(156), (273), (155), (154), (155), (158),
(169), (158),
(164).
§.§ Dependency Graphs Evolutionary Trends
Running a large-scale analysis on a set of Linux distributions developed and released over 18 years revealed a number of shifts occurring in the domain. In constant efforts to attract users, Ubuntu is known for conducting experiments, such as introducing new large software packages as a replacement for existing ones. For instance, the significant dip in the number of dependencies on Figure <ref> is explained by the replacement of GNOME 2 with Unity. On a longer scale it is also visible that despite limited local successes of such experiments, the overall trend indicates a slow growth of the number of files and dependencies.
Interestingly, we also observed a significant amount of not explicitly required files are present in the system (Figure <ref>). In other words, up to 37% of libraries physically located in the file systems were not mentioned in the import tables of any of the binaries or libraries. This likely indicates that such libraries are primarily used as plugins and could be loaded at run-time through dynamic directory scanning if necessary. Note that these conditional dependencies may be impossible to detect in advance due to the unpredictable nature of external factors. For instance, a user controlled application configuration can determine whether a given plugin library should be loaded at run time. The overall trend also hints that such a dynamic plugin-based approach gains popularity as the proportion of libraries not imported keeps steadily growing.
Another observation discovered throughout our analysis relate to the longevity of the libraries and binaries in Ubuntu. Namely, while complex binaries are periodically removed in search of better alternatives, highly popular libraries tend to stay around. Once a popular library is introduced in a particular Ubuntu version, it is unlikely to be removed as such removal would impact all libraries and executables that rely on the library's existence. Even internal code reorganizations affecting highly popular libraries require extra care to maintain compatibility[https://developers.redhat.com/articles/2021/12/17/why-glibc-234-removed-libpthread].
§ DISCUSSION
§.§ Threats to Validity
While we primarily focused on dependency-centric package management in Linux OS, other factors may explain some of the observations. Despite high popularity, packages might get removed from the system due to licensing, compatibility, security, or maintainability issues. Dependency analysis should, therefore, be coupled with change log analysis to verify and confirm the findings.
To enhance the external validity of our dependency analysis, we selected a highly popular Linux distribution. By including all of the available versions we expect our approach to be generalizable and applicable to a broader range of OSs. Widening the input data set on the time axis enabled the discovery of uncommon cases and long-term trends.
Being well-maintained, Ubuntu served as a high-quality dataset. Legacy Ubuntu versions and their corresponding change logs were still available for download[Ubuntu wiki: Releases - https://wiki.ubuntu.com/Releases]. In contrast, Alpine (another popular Linux distribution) archives did not go far back in time. Moreover, the Alpine archives contained broken links for older versions, preventing image downloading. Similarly, while considering Debian systems, we discovered different and incompatible system image layouts which would complicate the analysis.
Primary threats to external validity are abrupt changes causing significant paradigm shifts, lower granularities skewing the results, and implicit dependencies.
Abrupt changes may be introduced throughout evolution. Such changes introduce incompatibilities, forcing to amend the scanning process accordingly. Notable examples we observed include compression algorithm changes, folder hierarchy alterations, and transition from to . We noticed a different layout of binary files in the file system that required consideration due to the changes introduced in Ubuntu 19.04. Specifically, and directories were converted to symbolic links to and correspondingly[<https://lists.ubuntu.com/archives/ubuntu-devel-announce/2018-November/001253.html>]. Depending on whether 19.04 is being installed from scratch or on top of the previously installed version, the number of binaries may look like being suddenly doubled in version 19.04. We alleviated this problem by resolving symbolic links.
In addition to library dependencies stored in executable binary file import tables, other types of coupling occur in practice. For instance, network communication, special files like Unix domain sockets, Inter-Process Communication (IPC) calls, message-oriented buses, and pipes provide various means of code interactions. Discovering such code coupling instances may not be possible in practice (e.g., new code fragments might be downloaded over a network). Taking into account these code coupling types may significantly skew our findings.
§.§ Challenges and Limitations
The two primary technical challenges we encountered throughout our data collection and analysis are the large data set sizes and performance issues related to extracting dependencies at lower granularities.
As the distributed Ubuntu images are growing in size, so do the number of executable files and their individual sizes. This steady growth is observed over all Ubuntu versions analyzed. For example, within 18 years analyzed, the live Ubuntu image size grew from 600MB (version 5.04) to 3.7GB (version 23.04). Likewise, the number of executable files experienced a 70% increase in size (1605 in 5.04, 2753 in 23.04).
Through practical experiments, we established that restricting the dependency granularity is crucial to achieving acceptable processing speed as lower granularity dependency extraction incurs large overheads. Disassembling executable binaries to identify individual third-party library function calls slows the dependency extraction and incurs significant memory overheads. For instance, we have observed cases of over-disassembly and analysis of a single executable taking 40 minutes on an average laptop-class CPU. Thus, while technically possible and potentially interesting to gain further insights, lower-level granularity analysis is out of reach for real-time applications we initially aimed for. At this stage, we restricted the analysis to the file level only.
§ RELATED WORK
The prior work primarily revolves around two aspects, (i) diverse conceptual complexity metrics definitions and (ii) dependency extraction and analysis.
Various types of software complexity metrics have been widely studied in the literature <cit.>. Some studies have focused on metrics that are useful in source code analysis but are not easily applicable in binary code analysis <cit.> <cit.> <cit.>. Others have discussed the deficiency of methods to obtain global dependency knowledge and the difficulty in visualizing the resulting graphs <cit.>. The use of software complexity metrics to detect vulnerabilities has also been investigated, with some studies proposing dependency-oriented and execution-time complexities <cit.>. Dependency extraction aspects and challenges have also been explored, with some studies focusing on specific languages or ecosystems <cit.> <cit.>.
Package management and dependency validation have been popular research topics, with a set of studies proposing methods to address issues arising from package evolution (e.g., splitting into multiple different packages) <cit.> <cit.> <cit.>. User questions related to package management, such as calculating the consequences of removing or modifying a package, have also been explored <cit.> <cit.>.
Efficient package management tools and query languages have been proposed, including tools for efficient package management and relations lookup <cit.>. However, similar to software complexity metrics research efforts, multiple studies have focused only on source-level rather than binary dependencies <cit.> <cit.>.
In efforts to resolve binary compatibility issues, some works have investigated relying on version ranges rather than minimum version requirements <cit.>. Unfortunately, the large downside of the proposed approach is the requirement of debug symbols availability, which is rare in commercial software. An interesting use of dependency extraction has been proposed for Windows executables for malware detection <cit.>. Taking the notion of the extent of a dependency into account enables detecting and eliminating insignificant dependencies <cit.>.
Overall, it should be noted that dependency related studies primarily focus on source code dependency analysis and package-level relations<cit.> <cit.> and do not typically examine software package evolution over time. We, therefore, conclude that a more precise file-based dependency extraction is an under researched area that might benefit from providing better structural visibility for large-scale systems comprising multiple independently developed packages. We also see that understanding software evolution is essential for maintaining software, ensuring compatibility, and improving security. Having this understanding aids developers in making informed decisions about updates and maintenance, ensures software remains compatible with other systems, and reduces the risk of security issues. Additionally, understanding software evolution can lead to new innovations and improvements in software design and development.
§ CONCLUSION AND FUTURE WORK
In this study, we introduce automated extraction of dependency graphs for a whole system at the executable files level (as opposed to manually maintained traditional package-level dependency graphs). The resulting system-wide dependency graph provides a high-level view of the OS architecture emerging from interactions between the different subsystems and user packages. In addition, this study enabled the discovery of general high-level trends/common patterns in Ubuntu Linux architecture evolution over time.
We also differentiate between developer-facing complexity (defined through direct dependency coupling) and overall system complexity (defined through recursive dependency coupling). The motivation behind such a separation is that developers typically deal with third-party libraries without having full visibility of the back-end side of the libraries. In other words, a developer may include one library, while the library itself can have a complicated graph of dependencies not directly visible to the developer. These invisible dependencies may cause software bloating and increase the attack surface.
We believe the findings of this study will provide useful insights for software developers and OS maintainers in terms of gaining a holistic quantitative understanding of inter-package architecture management that would be useful, for example, in optimizing disk space and improving system maintainability.
We have identified two main directions for future research lines. Specifically, expanding the dependency extraction approach to a wider set of platforms to support and more types of dependencies to extract.
For future research, we aim to perform Windows-based analysis and implement support for other levels of granularity, such as individual function dependencies. Also, in contrast to the convenient, holistic file system structure used in live editions, non-live distribution variants are composed of multiple compressed packages, complicating the dependency extraction and analysis. Implementing analysis for such non-live distributions could be a potential future research line.
As opposed to fixed library imports, code fragments interacting through various communication channels are loosely coupled. Such non-obvious dependencies are not trivial to detect. For instance, changing code on one side of a UNIX pipe may negatively affect the results of the next program in the pipeline. Furthermore, such dependencies may not be predefined in advance and are only required intermittently while being completely unnoticeable most of the time. We believe that comprehensive and accurate detection of such concealed dependencies would greatly enhance the overall system architecture, evolution, and run-time operation understanding and visibility and enable early detection of potential compatibility breaks caused by code modifications.
§ ACKNOWLEDGMENT
The work has been partially supported by the Cyber Security Research Centre Limited whose activities are partially funded by the Australian Government’s Cooperative Research Centres Programme.
§ DATA AVAILABILITY
As the current project is funded by industry partners, we are unable to publish the source code at this stage. However, aiming to increase transparency and reproducibility in research, we have made the obtained dataset available for public access <cit.>. Researchers and interested parties can access the dataset and utilize it to replicate or build upon our findings.
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|
http://arxiv.org/abs/2307.04621v2 | 20230710150816 | Recipes for Jet Feedback and Spin Evolution of Black Holes with Strongly-Magnetized Super-Eddington Accretion Disks | [
"Angelo Ricarte",
"Ramesh Narayan",
"Brandon Curd"
] | astro-ph.HE | [
"astro-ph.HE",
"astro-ph.GA"
] |
Super-Eddington Spin Evolution
Ricarte, Narayan, & Curd
Angelo Ricarte
[email protected]
0000-0001-5287-0452]Angelo Ricarte
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
0000-0002-1919-2730]Ramesh Narayan
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
0000-0002-8650-0879]Brandon Curd
Department of Physics & Astronomy, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA
A spinning black hole accreting from a disk of strongly magnetized plasma via a magnetically arrested disk is known to produce an efficient electromagnetic jet powered by the black hole's spin energy. We present general relativistic radiative magnetohydrodynamic simulations of magnetically arrested systems covering a range of sub- to super-Eddington accretion rates. Using the numerical results from these simulations, we develop formulae to describe the magnetization, jet efficiency, and spin evolution of an accreting black hole as a function of its spin and accretion rate. A black hole with near-Eddington accretion experiences a mild degree of spin-down because of angular momentum loss through the jet, leading to an equilibrium spin of 0.8 rather than 1.0 at the Eddington limit. As the accretion rate increases above Eddington, the spin-down effect becomes progressively stronger, ultimately converging on previous predictions based on non-radiative simulations. In particular, spin evolution drives highly super-Eddington systems toward a black hole spin near zero. The formulae developed in this letter may be applied to galaxy and cosmological scale simulations that include black holes. If magnetically arrested disk accretion is common among supermassive black holes, the present results have broad implications for active galactic nucleus feedback and cosmological spin evolution.
§ INTRODUCTION
Astrophysical black holes (BHs) accreting from disks of plasma are known to launch relativistic jets and outflows <cit.>. Such energy injection from supermassive BHs (SMBHs) at the centers of galaxies, a process referred to as active galactic nucleus (AGN) feedback, is believed to be essential for stopping runaway gas cooling and star formation in massive galaxies and dark matter halos <cit.>. In this paradigm, accretion and feedback processes are critical for a complete picture of SMBH growth and galaxy co-evolution. However, the details remain poorly understood.
For magnetized accretion disks, an electromagnetic analogue of the <cit.> process known as the <cit.> (BZ) mechanism provides the most widely accepted model for jet launching. The power of a jet launched by the BZ mechanism scales approximately proportional to both the square of the BH spin and the square of the magnetic flux threading the horizon. In systems with high enough spin and with maximal magnetic field strength, corresponding to a so-called magnetically arrested disk (MAD) <cit.>, more jet power can be launched than the entire rest mass energy of the material flowing into the BH <cit.>. The extra energy is supplied by the spin kinetic energy of the BH, which thereby may cause the BH to spin down with time. In this way, jets that travel through dark matter halos for hundreds of kiloparsecs are ultimately linked to the evolution of BH spin and the transport of magnetic fields on event horizon scales.
Since the BZ mechanism powers a jet by extracting BH spin energy, if the process continues long enough a BH could continuously spin down and equilibrate near a spin value a_* ≈ 0. This has been explicitly demonstrated via general relativistic magnetohydrodynamic (GRMHD) simulations of radiatively inefficient, geometrically thick, MAD models <cit.>. Several recent publications have begun to study the implications of this spin-down effect for BH populations over cosmic time. The systems simulated so far largely belong to the regime of advection-dominated accretion <cit.>, or hot accretion <cit.>, which corresponds to highly sub-Eddington accretion. Spin-down is relatively slow for such low Eddington-ratio systems simply because the mass accretion rate is very small; nevertheless, continuous jet feedback from such BHs is implicated for maintaining low star formation for Gyrs in some galaxies <cit.>, which can lead to cosmologically significant BH spin evolution <cit.>.
Super-Eddington accretion disks are geometrically thick and advection-dominated, just like low-Eddington ratio hot accretion flows, and can also reach the MAD state <cit.>. Such systems can produce extremely powerful jets <cit.>, and because of the very large accretion rate their BHs could spin-down very rapidly. <cit.> developed a physical semi-analytic model for this spin-down phenomenon. Using this model, <cit.> predicted rapidly decreasing collapsar BH spins to a_* ≲ 0.2 near birth.
Self-consistent BH spin evolution is now being implemented in some galaxy and cosmological-scale simulations, which may then be used to model radiative efficiency and jet power <cit.>. Although galaxy-scale simulations cannot possibly resolve accretion disk scales, such an approach still represents a substantial improvement over most contemporary work to link SMBH spin evolution to the angular momentum of resolved gas on scales of parsecs. <cit.> and <cit.> implement spin-down during periods of thick disk accretion, employing fitting functions for the magnetic flux as a function of spin from GRMHD simulations. Again assuming the same results that have been demonstrated for very low Eddington ratio disks also hold for super-Eddington disks, <cit.> consider super-Eddington growth in high-redshift galaxies. While spin-down is noticeable in this simulation, it is counteracted by periods of thin disk accretion.
All such calculations require some a priori knowledge or assumptions about the magnetic field strength. For magnetized geometrically thick disks in the low-Eddington rate limit, the MAD model offers one well-studied solution. In contrast to the weak-field “Standard and Normal Evolution” (SANE) model <cit.>, a MAD system is characterized by such strong magnetic fields that magnetic pressure and tension is comparable to the gas pressure near the horizon <cit.>. MAD models are characterized by a dimensionless magnetic flux parameter ϕ (defined in <ref>) saturating at a spin-dependent maximum value <cit.>, as well as “flux eruption events” that occur when the BH expels magnetic flux <cit.>. The saturated fields that characterize the MAD state lead to highly efficient jets powered by the BZ mechanism.
Spatially resolved and polarimetric observations of the nearby low-luminosity AGN, M87* and Sgr A*, currently favor MAD models over their SANE counterparts <cit.>, suggesting that the saturated values of ϕ characteristic of MAD models are easily achieved in low Eddington-ratio geometrically thick hot accretion disks. However, it remains to be confirmed that the same saturation values found for hot accretion flows at low Eddington ratios also hold for super-Eddington accretion flows where radiation plays an important role. It is also unknown whether the BZ mechanism operates efficiently in such systems and how efficiently BH spin-down proceeds. We explore these questions here.
In this letter, we introduce and analyze a suite of super-Eddington general relativistic radiative magnetohydrodynamic (GRRMHD) simulations in the MAD regime to explicitly calculate the magnetization ϕ, jet power P_jet, and spinup parameter s (defined in equation <ref>), as a function of the dimensionless BH spin parameter a_* and the Eddington ratio f_Edd (defined in <ref>) of the accretion flow. As we shall show, highly super-Eddington accretion disks (f_Edd≫ 1) behave similarly to their very low Eddington-ratio (f_Edd≪ 1) counterparts. However, we find reduced magnetization and spin-down for Eddington ratios f_Edd≲ 10. Based on this behavior, we devise fitting functions for jet power and spin evolution that can be adapted into cosmological and galaxy-scale simulations.
§ GRRMHD SIMULATIONS
Radiation plays a critical role in the dynamics of BH accretion disks for Eddington-ratios f_ Edd≳ 0.01. In these systems, radiative cooling acts to thin the disk at lower Eddington ratios, while radiative pressure puffs up the disk vertically as the mass accretion rate approaches or exceeds Eddington <cit.>. In super-Eddington systems, winds and jets driven purely by radiation can also occur <cit.>
The numerical treatment of radiation in BH accretion problems is quite difficult as the algorithm must treat both optically thin and thick regions in a curved spacetime. <cit.> pioneered global, non-relativistic, radiation hydrodynamics (RHD) simulations of super-Eddington accretion disks using flux-limited diffusion. Following this work, radiation was first included in the fully general relativistic radiation magnetohydrodynamics (GRRMHD) code, koral, by <cit.> using the M1 closure scheme and a semi-implicit method to handle the radiation terms. Since then, the M1 closure scheme has been applied in other GRRMHD codes <cit.> as well as a GPU accelerated GRRMHD code <cit.>. Alternative methods of treating radiation in GRRMHD include directly solving the radiative transfer equations to obtain the Eddington tensor <cit.>, Monte Carlo methods <cit.>, or using a discretized radiation tensor <cit.>. The M1 closure scheme allows limited treatment of anisotropic radiation fields. It is superior to the Eddington approximation in optically thin regions, and is well suited for global GRRMHD simulations of super-Eddington disks. However, for complicated radiation fields, it cannot match methods based on the full Eddington tensor.
<cit.> explored the role of BH spin in super-Eddington accretion by running a suite of 2D GRRMHD simulations for different spin values. They considered the SANE regime of accretion for which 2D simulations are sufficient. The MAD accretion regime, however, requires 3D simulations and this is the focus of our work. We present a suite of 38 3D numerical simulations of near-Eddington to super-Eddington MAD simulations carried out with the GRRMHD code, koral <cit.>. We include 2 BH masses, M = 10, 10^4 M_⊙, 6 BH spin values, a_*= -0.9, -0.68, 0, 0.68, 0.9, and 0.97 (where a minus sign denotes retrograde accretion), and a range of Eddington ratios, 0.4 ≲ f_Edd≲ 40. Since prolonged super-Eddington accretion is often invoked for the growth of BH seeds in the early universe, as we will later explore in <ref>, these two masses are loosely motivated by exploring both “light” and “heavy” seeding scenarios <cit.>. We define f_Edd as follows,
f_Edd = Ṁ/Ṁ_Edd,
where Ṁ is the mass accretion rate through the BH horizon (<ref>) and Ṁ_Edd is the Eddington mass accretion rate corresponding to the radiative efficiency of a thin disk (see <ref> and <ref>). Thin disks below and near the Eddington limit are notoriously difficult to simulate, due to difficulties resolving the disk scale height. However, the additional magnetic pressure of the MAD state helps to inflate even moderately sub-Eddington disks (see <ref>), making this problem computationally tractable <cit.>.
Using a mesh-based, finite-difference method in a stationary Kerr space-time, koral solves the conservation equations of GRMHD, with the addition of radiative heating, cooling, and plasma coupling. Modeled radiative processes include synchrotron radiation, opacities from electron scattering, free-free and bound-free emission/absorption from the <cit.> model, and Compton scattering. While ideal GRMHD simulations without radiation are rescalable to different masses and accretion rates, the inclusion of radiative processes sets absolute physical scales and necessitates individual simulations for each combination of M, a_*, and f_Edd.
Each simulation is initialized as a torus of gas in hydrostatic equilibrium threaded by a large-scale poloidal magnetic field, either perfectly aligned or anti-aligned with the BH spin axis. To limit computational expense, but still allow non-axisymetric structures that commonly arise in MAD disks, we simulate a periodic π/2 wedge in azimuth. From the torus initial conditions, the magnetorotational instability naturally develops to allow the plasma to lose angular momentum and accrete onto the BH, advecting along with it magnetic field which saturates at the MAD state. One example is shown in <ref>, where in the upper panels we visualize the density and magnetic field lines of the M=10^4 M_⊙, a_*=0.9, f_Edd=9.3 model in the plane and in a perpendicular slice respectively. The BH has accumulated a significant poloidal magnetic field, and turbulent eddies are evident in the disk. A flux eruption event characteristic of the MAD state, the low-density bubble near the horizon, is visible during this snapshot.
Throughout this work, we use gravitational units to describe physical parameters. For distance we use the gravitational radius r_g ≡ GM/c^2 and for time we use the gravitational time t_g ≡ GM/c^3. We set G = c = 1, so the above relations would be equivalent to r_g = t_g = M. We restore G and c in cases where it helps to keep track of units. Each of the 38 models was run for a total time of 30000 t_g. Summary statistics are given in <ref> and correspond to averages over the final 5000 t_g of the run when we expect each simulation to be most nearly in steady state.
§ RESULTS
§.§ Magnetization
The dimensionless magnetization parameter ϕ(t) at time t is defined by <cit.>,
ϕ(t) = √(4π)/2√(Ṁ(t))∫_ϑ∫_φ|B^r|_r=r_ H √(-g) dϑ dφ,
where B^r is the radial component of the magnetic field, g is the metric determinant, Ṁ(t) is the BH accretion rate, and the integral is evaluated at the BH horizon. MAD systems are characterized by a value of ϕ that has saturated at a spin-dependent value of ∼ 30-50 <cit.>, as is the case for the example plotted in Figure <ref>. The value of ϕ tends to decrease during a flux eruption event; note that our example snapshot visualized in <ref> coincides with a local minimum in ϕ. Although both Ṁ and ϕ are time variable, we assign a single value to each simulation by averaging each quantity over the time period t=25000t_g - 30000t_g. These are the values listed in <ref>.
In the left panel of <ref>, we show the values of ϕ obtained from our 38 simulations, both as a function of the Eddington ratio f_Edd and the BH spin a_*. Different spins are encoded in different colors, and different masses are encoded by symbol size. At large Eddington ratios, the simulations approach spin-dependent values similar to those found in pure GRMHD simulations of MADs <cit.>. However, ϕ decreases as f_Edd decreases. Interestingly, simulations with f_Edd=1 remain substantially magnetized, with ϕ values typically about a third of the limiting value for f_ Edd≫1. As we explore in <ref>, this trend can be explained by increased pressure scale height as Eddington ratio increases, allowing the disk to confine stronger magnetic fields.
We model the behavior shown in the simulation data by fitting the following function:
ϕ(a_*,f_Edd) = ϕ_MAD(a_*)(f_Edd/f_c)^α/1+(f_Edd/f_c)^α,
where f_c is a critical Eddington ratio determining the mid-point of the transition, and α is a free parameter determining the rapidity of the evolution around f_c. The function ϕ_MAD(a_*) is the saturated value of ϕ found in non-radiative MAD simulations. We use the approximation given in <cit.>,
ϕ_MAD(a_*) = 52.6 + 34a_* - 14.9a^2_* - 20.2a^3_*.
By construction, in <ref>, ϕ→ 0 as f_Edd→ 0 and ϕ→ϕ_MAD(a_*) as f_Edd→∞. Via least-squares fitting, we arrive at α=1.29 and f_c=1.88. The spin-dependent ϕ(a_*,f_Edd) curves are plotted in the background of <ref>, and describe the main trends fairly well. We intentionally transition ϕ→ 0 as f_Edd→ 0 to connect to the thin disk solution, but we caution that the shape and rapidity of this transition may be sensitive to our poor sampling of the f_Edd≲ 1 regime. We note that the GRRMHD simulations of both <cit.> and <cit.> produced ϕ∼ 30 for f_Edd∼ 0.3, which our fitting function would underestimate.
§.§ Jet Efficiency
The electromagnetic jet efficiency η_EM = P_jet / Ṁc^2 can be calculated analytically given a_* and ϕ. For small to moderate values of spin, η_ EM∝ a_*^2ϕ^2 <cit.>, but for spin values up to and including a_*=1, the following expression including higher order correction factors is more accurate <cit.>:
η_EM = κ/4πϕ^2Ω^2_ H[1 + 1.38Ω^2_ H - 9.2Ω^4_ H],
where
Ω_ H≡|a_*|/2r_ H=|a_*|/2(1+√(1-a_*^2))
is the angular velocity of the horizon and κ is a constant dependent on the initial field geometry, for which we adopt κ = 0.05.
In the right panel of <ref>, we plot the MHD energy outflow efficiency η_MHD as a function of magnetization, with spin once again encoded in color and mass encoded in symbol size. Note that unlike η_EM predicted by <ref> this quantity also includes the hydrodynamic energy flux. The colored curves correspond to the fitting function <ref> for each spin sampled by our simulation suite. The data points are from the simulations, where we have computed the mass and energy fluxes at a radius of 5 r_g since numerical floors cause inaccuracies closer to the horizon <cit.>. Radiative flux is neglected (which is again affected by floors, particularly in the jet region), but this introduces only a small error since the radiation contribution near the BH tends to be small.
Despite the wide range of mass, spin and accretion rate considered in the right panel of <ref>, we find that the fitting function <ref> performs remarkably well, implying that the BZ mechanism dominates the jet physics in MAD super-Eddington accretion flows. Note that at a_*=0, the BZ prediction is identically 0 because the BH has no spin energy. However, the simulations still give η_MHD>0. In these models, the outflowing energy is from the accretion disk, presumably in a hydrodynamic wind. As a point of reference, we plot the radiative efficiencies of thin disks with a_* ∈{0,0.68,0.9,0.97 } as colored horizontal lines. The MHD outflow from the a_*=0 simulation is similar in energetic output to an equivalent thin disk's radiative output. Meanwhile, the radiative efficiency of a thin disk around a maximally spinning black hole can be easily be exceeded with enough spin and magnetic flux.
§.§ Spin Evolution
Since the BZ mechanism extracts spin energy from the BH, this can result in astrophysically significant spin evolution of an accreting BH, which we study here. We describe the evolution in terms of a dimensionless spin-up parameter <cit.>,
s = da_*/dtM/Ṁ = l - 2 a_* e,
where l is the inward specific angular momentum flux and e is the inward specific energy flux, each of which we measure at a radius of 5 r_g. Spinup as a function of a_* computed from our GRRMHD simulations is shown in the upper panel of <ref>, where the color encodes different Eddington ratios and the symbol size encodes different masses. The thin disk solution, which always pushes the BH towards maximal prograde spin (a_*→ 1), is shown as a dotted line <cit.>. A fitting function which we presented in previous work for MAD GRMHD (f_ Edd≪ 1) models <cit.> is shown as a dashed line and is given by
s_MAD(a_*) = 0.45 - 12.53 a_* -7.80 a_*^2 + 9.44 a_*^3
+5.71 a_*^4 - 4.03 a_*^5.
The simulated GRRMHD models generally transition from the thin disk solution to the MAD GRMHD solution as the Eddington ratio increases (blue to red colors in <ref>). This is not unexpected, since highly super-Eddington disks are geometrically very thick and are highly advection-dominated <cit.> and therefore closely resemble the low-f_ Edd hot accretion flows studied in <cit.>. Retrograde models do not follow this trend, however, in fact spinning up more rapidly than the thin disk solution. These models overshoot the thin disk curve because both the BZ mechanism and accretion of oppositely rotating material torque the BH towards a_*=0.[As Eddington ratio increases, the disk dynamics evolve from the thin disk solution and the hydrodynamic torques become weaker (see <ref>). At the same time, the magnetization increases, so the electromagnetic torque becomes stronger. Whether or not a retrograde disk spins up faster or slower than a thin disk depends on the balance between these effects.]
<cit.> built a semi-analytic model to understand spin evolution in non-radiative MAD systems based on the spin evolution equations appropriate for a disk-plus-jet system introduced in <cit.>. In this model, the spinup parameter is explicitly split up into hydrodynamic spinup by the accretion disk gas and spindown via a jet powered by the BZ mechanism. The spinup parameter is then expressed as
s = s_HD + s_EM,
where
s_HD = l_HD - 2 a_* e_HD,
and
s_EM =sign(a_*) η_EM( 1/k Ω_H - 2 a_* ).
We detail the calculation and modeling of s_HD from l_HD (the hydrodynamic specific angular momentum flux) and e_HD (the hydrodynamic specific energy flux) in <ref>. As explained there, we develop a fitting function for s_HD given by <ref> that smoothly interpolates between the thin disk solution as f_Edd→ 0 and non-radiative GRMHD results as f_Edd→∞. Meanwhile, the electromagnetic component s_EM depends on η_EM and the parameter k, which is the ratio of the angular frequency of field lines relative to that of the BH. We estimate η_EM as a function of a_* and f_Edd by combining <ref> and <ref>. For k, we adopt the following fit from the non-radiative GRMHD simulations of <cit.>:
k(a_*) =
0.23, a_* < 0
min(0.1+0.5a_*,0.35), a_* > 0
this gives k slightly less than the <cit.> monopole value of 0.5, which broadly agrees with other simulations in the literature <cit.>.
As one final modification to allow our model to support hot accretion flows, we make the following adjustment:
s=
s_HD + s_EM f_Edd > f_c
s_MAD f_Edd≤ f_c
where f_c is a critical Eddington ratio below which the accretion flow should transition to the radiatively inefficient hot accretion mode <cit.>. Following previous efforts to model the evolution of black hole populations, we adopt f_c=3× 10^-2 <cit.>. The exact Eddington ratio at which this transition occurs is poorly constrained and unlikely to be a sharp transition <cit.>. Different values of f_c may be adopted without qualitatively changing our formulae.
Our final result for the spinup parameter s (<ref>) can thus be obtained from just two parameters (a_* and f_Edd) by inserting our fitting functions for ϕ(a_*,f_Edd) (<ref>), s_HD(a_*,f_Edd) (<ref>), and η_EM(a_*,ϕ) (<ref>). As constructed, <ref> can be applied to all physical values of a_* ∈ [-1,1] and f_Edd∈ (0,∞).
The model predictions from <ref> are shown in the bottom panel of <ref>. The model captures the behavior seen in the simulations (upper panel) exceptionally well, especially for spinning BHs. For a_*=0, it underestimates the evolution of s with f_Edd. We speculate that this may be due to the exclusion of angular momentum loss due to hydrodynamic wind, evident in <ref>. In light blue, we plot the model's prediction for s when f_Edd=1. It is quite similar to the thin disk solution, but has a root, which corresponds to an equilibrium value of a_* for fixed f_Edd, at a_*, eq≈ 0.8 instead of 1. In red, we plot the limit as f_Edd→∞. It follows the non-radiative GRMHD fitting function well, with minor deviations in the retrograde regime. This curve exhibits two kinks originating from the piece-wise nature of <ref>. As f_Edd→ f_c, s is well-approximated by the thin disk solution (dotted black line) by construction. In any case, the key result from the red line is that, as f_ Edd→∞, the equilibrium spin (where s=0) approaches a_*, eq≈0.
In <ref>, we plot the equilibrium spin a_*, eq as a function of Eddington ratio, found by taking <ref> and solving the condition s=0 at fixed f_Edd. We demarcate three different physical regimes: (i) hot accretion for f_Edd < f_c, (ii) what is classically modeled as a thin disk for f_c < f_Edd < 1, and (iii) super-Eddington accretion for f_Edd >1. In reality, s and a_*,eq should evolve more gradually around f_ Edd≈ f_c, but we lack a detailed understanding of this transition and are unable to model it more realistically in this work.
Our model permits the existence of BHs with a stable a_*, eq≈ 1 for Eddington ratios in the range f_ Edd∼ 0.03 - 0.3, but a_*,eq begins to decline above f_Edd≈ 0.3 and approaches 0 as the accretion rate becomes highly super-Eddington. The limiting equilibrium spin for extremely large values of f_Edd is a_*=0.035, as in the hot accretion regime <cit.>, but note that this exact value is not very accurate and depends on the details of how spin-down is modeled. On the upper x-axis, we plot the evolutionary timescale of both mass and spin for a given f_Edd, given by t_Sal/f_Edd where
t_Sal = ϵσ_T c/4 π G m_p = ϵ× 450 Myr
is called the Salpeter timescale, where σ_T is the Thomson cross-section and m_p is the proton mass. For the convenience of defining a spin-independent t_Sal, we adopt a fiducial value of ϵ=0.1 for its definition, such that t_Sal = 45 Myr. Since mass and spin evolve on the same time-scale, a BH must accrete a significant fraction of its own mass to reach equilibrium spin[However, note that s measures the ratio of the spin evolution rate to the mass evolution rate. Hence for values of |s| approaching 10, spin evolves 10 times faster than mass.]. In the hot accretion regime, this would occur on timescales easily exceeding the age of the universe, and thus such BHs will not naturally reach the equilibrium spin value through the BZ process <cit.>. However, BHs which accrete continuously near or above the Eddington limit can reach their equilibrium spins in less than (sometimes very much less than) a Hubble time. Interestingly, such continuous and rapid assembly is invoked to explain the existence of massive quasars at z ≳ 6 <cit.>, which have accumulated masses up to 10^10 M_⊙ when the Universe was approximately 1 Gyr old.
§ DISCUSSION AND CONCLUSIONS
In this letter we presented a suite of GRRMHD simulations of radiative MAD accretion disks around BHs. The simulations cover a range of BH spins a_* from +0.97 to -0.9, and Eddington ratios f_ Edd from 0.4 to 40. We find two key qualitative results.
First, radiative disks in the MAD state around spinning BHs produce powerful jets as efficiently as the better-studied non-radiative disks (which are found in systems with f_ Edd≪ 1), and the power in the jet comes similarly from the BZ mechanism (see the right panel of <ref>).
Second, the saturated magnetic flux ϕ depends not only on the BH spin (as already known for non-radiative MAD models) but also on the Eddington ratio (see the left panel of <ref>). As a result, radiative disks with f_ Edd≲ 0.3 behave roughly like the standard thin accretion disk model, but systems with f_ Edd≫ 1 are very different and closely resemble non-radiative models (see <ref>). In particular, when f_ Edd≫1, the accreting BHs spin-down rapidly toward an equilibrium a_*≈ 0.
At a quantitative level, using the above suite of MAD GRRMHD simulations we have devised fitting functions which can be used to estimate magnetization ϕ (<ref>), jet feedback efficiency η (<ref>), and spin evolution s (<ref>), as a function of spin and Eddington ratio. Spindown via the BZ mechanism grows more efficient as Eddington ratio increases, but is already noticeable at f_Edd≈ 1, where the equilibrium spin is a_*=0.8. This has important implications for feedback and spin-evolution of BHs in the near-Eddington to super-Eddington regime, such as flux-limited samples of AGN, rapidly assembling seeds in the early universe, and collapsar BHs.
In <ref>, we plot evolutionary tracks for a selection of cosmologically motivated scenarios, each of which results in a BH with M ≈ 10^9 M_⊙. In each case, we have integrated <ref> using a standard Runge-Kutta-Fehlberg 4(5) integrator with adaptive step-sizing. For these examples, we make an important assumption that the accretion disk and BH angular momentum axes are always perfectly aligned, which need not generally be the case. Variations in disk tilt over cosmic time are an uncertainty that can lead to substantial differences in spin evolution, leading to lower spins if the angular momenta of material is more randomized <cit.>. In the left column of <ref>, we plot evolutionary scenarios with different fixed f_Edd values shown as different colors. For f_Edd=20, 1, 0.1, 0.01, we initialize our BHs with M=10, 10^7, 3×10^8, 10^9 M_⊙ and a_*=0, 0, 0, 0.998, respectively. In all cases, 1 Gyr is enough for each of the BHs to approach their equilibrium spin (see <ref>). These scenarios result in very different spin evolution and feedback as a function of time.
Both the f_Edd=20 and the f_Edd=1 scenarios result in the accretion of 10^9 M_⊙ of material, but the f_Edd=20 scenario releases a total of 7.8 × 10^53 erg worth of feedback compared to 5.3 × 10^54 erg in the f_Edd=1 scenario, a factor of 7 difference. The reason is that the f_ Edd=20 model reaches a lower equilibrium spin, which results in less efficient jet feedback. A consequence of this interesting result is that a BH could potentially grow more efficiently in a super-Eddington state before having its mass supply cut off by excessive jet feedback. We have assumed a sharp transition between thin and thick accretion flows at an Eddington ratio of f_c = 3×10^-2. Evolving in the thin disk regime, the f_Edd=0.1 model spins up to maximal spin and cannot power a very efficient jet, since lower Eddington ratio sources maintain weaker magnetization. On the other hand, the f_Edd=0.01 model evolves in the hot accretion flow regime and spins down to near zero spin.
In the right column of <ref>, we plot two different fueling-limited scenarios. In the “Constant Ṁ” model, we envision that a galaxy provides constant Ṁ that the BH can consume, regardless of the f_Edd implied. In this model, we suggestively tune our parameters to match the formation of the <cit.> quasar, which is observed with f_Edd=0.67 and M=1.6× 10^9 M_⊙ at z=7.642, when the Universe was only 670 Myr old. After being initialized at 10^4 M_⊙ and a_*=0, the BH accumulates mass in the super-Eddington regime as spindown from the BZ mechanism keeps its spin low. Its spin increases only as f_Edd→ 1, and it reaches an equilibrium spin of 0.9. Qualitatively consistent with our predictions for a powerful jet, <cit.> report a relativistic outflow while also suggesting greater incidence of such powerful outflows at high redshift.
In the second “Power-Law Ṁ” model, a 10^5 M_⊙ a_*=0 seed initially accretes at f_Edd=15,000, then the accretion rate declines as Ṁ∝ (1+(t/10^7 yr)^2)^-1, motivated by <cit.>. Over the age of the Universe, this BH traverses all three accretion regimes, starting with a_* ≈ 0 while it is super-Eddington, rising to a_* ≈ 0.9 in the thin disk regime, then finally declining to a_* ≈ 0.5 in the hot accretion regime. It runs out of fuel before it can achieve the equilibrium spin ≈ 0 for its final f_ Edd. Ending with f_Edd∼ 10^-6 and M ∼ 10^9 M_⊙, this evolutionary track could represent the history of the most massive BHs resolvable on the sky, such as Event Horizon Telescope target Messier 87.
<ref> illustrates how a BH's assembly history is imprinted on its final spin value, motivating observational spin constraints of supermassive BHs. For 0.01 ≲ f_Edd≲ 0.3, X-ray reflection spectroscopy has been most successful in accumulating large spin samples. The measured spin values tend to be highly skewed towards a_* ≈ 1 <cit.>, in agreement with the equilibrium spin of a thin accretion disk, as well as the equilibrium spin value suggested by the present work for that range of f_Edd. To complement these thin disk spin constraints, the next-generation Event Horizon Telescope aims to measure spins of dozens of supermassive BHs in the hot accretion (f_Edd≪ 1) regime <cit.>. Taking the “Power-Law Ṁ” model in <ref> as an example, we would predict typical spin values roughly half-way between 1 and 0 <cit.>. It would be interesting to see what future observations show. Unfortunately, there is no known direct probe of spin in the super-Eddington regime, where we predict equilibrium spins close to 0. Current probes of spin rely on the existence of a sharp transition in the dynamics of the accreting disk at the innermost stable circular orbit. Such a feature is expected to be present in geometrically thin disks (and is the basis of the X-ray reflection method), but it is washed out in geometrically thick disks such as are found for f_ Edd≫1 (e.g., this work).
It is worth mentioning that in the present radiative MAD models, as well as others in the literature, roughly ∼ 60% of the jet power can be transformed into radiation at large radius <cit.>. This can occur because inverse Compton scattering can transform much of the kinetic energy of the jet fluid into highly beamed radiation. However, we refrain from providing radiative efficiencies from our simulations, because we find that numerical floors in the jet region can artificially inflate the total energy in the jet at large radii. Fortunately, this artificially injected energy simply outflows from the simulation box and does not affect the region of interest.
The analytic formulae devised in this work can be applied to galactic or cosmological scale simulations, conveniently bridging the sub-Eddington and super-Eddington regimes. When placing these models in an astrophysical context, the most important caveat is the assumption that these systems are magnetically saturated in the MAD state. Event horizon scale polarimetric imaging the largest black holes on the sky do currently favor MAD models over their SANE counterparts <cit.>, and ab-initio simulations of gas and magnetic field transport onto Sgr A* can indeed naturally produce MAD states <cit.>, but this evidence pertains only to low-Eddington ratio BHs. Super-Eddington MAD disks can explain jetted tidal disruption events <cit.>, but these objects are only ∼1% of known TDEs and may not be representative of the typical super-Eddington disk. Future observational and theoretical developments to test the robustness of the MAD state would help validate the modeling performed here. Furthermore, our simulations are limited to M=10 M_⊙ and M=10^4 M_⊙, and <ref> hints at a possible trend with mass. We do not expect our results to be very sensitive to BH mass on physical grounds, but this should be verified in future work in the context of varying the metallicity as well.
§ ACKNOWLEDGMENTS
This work was supported in part by NSF grants AST1816420 and OISE-1743747, and by the Black Hole Initiative at Harvard University, made possible through the support of grants from the Gordon and Betty Moore Foundation and the John Templeton Foundation. The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the Moore or Templeton Foundations.
koral <cit.>, Matplotlib <cit.>, SciPy <cit.>, NumPy <cit.>
§ DATA AVAILABILITY
Most plotted values can be downloaded from data files that accompany this publication. In addition, we provide a Python script including the equations presented in this work, as well as the integrator that was used to produce <ref> and <ref>.
§ ADDITIONAL GRRMHD DETAILS
Using the finite-difference method in a fixed, Kerr spacetime, koral solves the conservation equations:
(ρ u^μ)_;μ = 0,
(T^μ_ ν)_;μ = G_ν,
(R^μ_ ν)_;μ = -G_ν,
(nu^μ_R)_;μ = ṅ,
where ρ is the gas density in the comoving fluid frame, u^μ are the components of the gas four-velocity as measured in the “lab frame”, T^μ_ ν is the MHD stress-energy tensor in the “lab frame”:
T^μ_ ν = (ρ + u_g+ p_g + b^2)u^μ u_ν + (p_g + 12b^2)δ^μ_ ν - b^μ b_ν,
R^μ_ ν is the stress-energy tensor of radiation, G_ν is the radiative four-force which describes the interaction between gas and radiation <cit.>, and n is the photon number density. Here u_g and p_g=(γ_g - 1)u_g are the internal energy and pressure of the gas in the comoving frame, and b^μ is the magnetic field four-vector which is evolved following the ideal MHD induction equation <cit.>. For fitting purposes, it is useful to write the MHD stress-energy tensor in terms of hydrodynamic (HD) and electromagnetic (EM) components
T^μ_ ν, HD = (ρ + u_g+ p_g)u^μ u_ν + p_gδ^μ_ ν
and
T^μ_ ν, EM = b^2 u^μ u_ν + 12b^2δ^μ_ ν - b^μ b_ν.
The radiative stress-energy tensor is obtained via the M1 closure scheme. We include a radiative viscosity term to better approximate the radiation field in the funnel region as in <cit.>. We include the effects of absorption, emission, and scattering via the electron scattering opacity (κ_es), free-free absorption opacity (κ_a), thermal synchrotron, and thermal Comptonization <cit.>. For the M=10 M_⊙ models, we also account for the bound-free absorption opacity (κ_bf) using the Sutherland Dopita model <cit.> assuming a solar metal abundance for the gas[The 10M_⊙ models are quite hot, with temperatures >10^7K, and so the precise details of the atomic opacity prescription or the choice of metallicity are unimportant.]. We exclude the bound-free absorption opacity for the M=10^4 M_⊙ simulations, because these models are primarily meant to represent rapidly-growing “heavy” BH seeds in the early universe that are assumed to form in metal-free halos devoid even of star formation <cit.>.
We adapt modified Kerr-Schild coordinates with the inner radius of the simulation domain inside of the BH horizon. The uniformly spaced internal coordinates (x_1,x_2,x_3) are related to the Kerr-Schild spherical polar coordinates polar coordinates (r,ϑ,φ) by
r = e^x_1,
ϑ = [1 + (H_0π/2)tan(H_0π[-0.5 + (Y_1 + (-Y_1 + Y_2)(e^x_1/2)^P_0)(1 - 2x_2) + x_2])]π2,
φ = x_3.
The complicated form of the middle expression is designed such that (i) the minimum/maximum coordinate ϑ is radially dependent, and (ii) more cells are focused towards the midplane ϑ=π/2. We choose H_0=0.6 to add slightly more resolution in the midplane in order to better resolve the accretion disk. We also choose Y_1=0.0025, Y_2=0.025, and P_0=1.2 such that Y_2π<ϑ<(1-Y_2)π near the horizon but Y_1π<ϑ<(1-Y_1)π further away. This choice ultimately increases the minimum time step and decreases the computational cost of each simulation.
The radial grid cells are spaced logarithmically, and we choose inner and outer radial bounds R_min<r_H and R_max=10^4 r_g. We specify R_min for each model in Table <ref>. We also use a wedge of π/2 in azimuth instead of the full 2π in order to minimize computational costs and set φ_min=-π/4 and φ_max=π/4. We choose outflow boundary conditions at both the inner and outer radial bounds, reflective boundary conditions at the top and bottom polar boundaries, and periodic boundary conditions in φ. In each simulation, we employ a resolution of N_r× N_ϑ× N_φ=256×192×24.
The resolution in θ is especially important for GRRMHD (and also GRMHD) simulations. The θ resolution used in the present work is superior to most GRRMHD simulations in the literature. Our φ resolution is modest: 24 cells over a π/2 wedge, which corresponds to an effective resolution of 96 cells over 2π. This is a bit lower than 32 cells in the wedge, or 128 cells over 2π, used in <cit.>. However, it is superior to most other GRRMHD simulations reported in the literature, e.g., 64 cells over 2π in <cit.> and <cit.>, or even 32 cells over 2π used in other work.
We ensure that the fastest growing mode of the magnetorotational instability (MRI, ) is adequately resolved within each simulation. For this we compute the quantities <cit.>,
Q_ϑ = 2πΩ dx^ϑ|b^ϑ|√(4πρ),
Q_φ = 2πΩ dx^φ|b^φ|√(4πρ),
where dx^i (the grid cell size) and b^i (the magnetic field strength) are both evaluated in the orthonormal frame, Ω is the angular velocity, and ρ is the gas density. Q≥ 5 is sufficient to resolve the MRI. We weight Q by √(b^2ρ) and integrate over the disk (σ < 1). We then spatially average over r=10r_g-100r_g and temporally average over t=25000t_g-30000t_g. In our least resolved model, which has M=10M_⊙, a_*=0.97, and f_Edd=1.97 in <ref>, we find ⟨ Q_ϑ⟩=5 and ⟨ Q_φ⟩=47, which is sufficient to resolve MRI in the bulk of the disk. Q_ϑ and Q_φ increase with f_Edd since the disk becomes thicker; therefore, all of our models sufficiently resolve the MRI.
We initialize each simulation with a torus of gas in hydrodynamic equilibrium following <cit.>. The density was fixed by the entropy constant 𝒦=63 and assuming Γ=4/3. The angular velocity
at the equatorial plane was set to a constant fraction of ξ=0.975 of
the Keplerian angular velocity outside radius R_1=30 r_g, and followed
fixed angular momentum between R_in < r < R_1 with R_in=22 r_g being the inner edge of the torus. The angular momentum was kept constant along the von-Zeipel cylinders. We set the outer edge of the torus at r≈ 400 r_g. This method only gives the hydrodynamic quantities. To initialize the radiation, we split the total pressure given by the initial hydrodynamics solution into gas and radiation components by assuming local thermodynamic equilibrium (LTE). We assign the gas and radiation pressure by finding the LTE temperature given by
p_tot=p_gas+p_rad=k_Bρ T + 13a c T^4,
where p_tot is the sum of gas and radiation pressure given by the initial torus in pure hydrodynamics, p_gas is gas pressure, and p_rad is the radiation pressure.
We thread the torus with a large scale poloidal magnetic field defined by the vector potential A_ϕ. We adopt a definition of A_ϕ which is a function of r and ϑ given by
A_ϕ=q(r,ϑ)sin(F(r)-F(R_start)),
where we define
q(r,ϑ) =
(u_g(r,ϑ)-u_g(R_chop,π/2)) - 0.2(u_g(r,π/2)-u_g(R_chop,π/2))0.8(u_g(r,π/2)-u_g(R_chop,π/2))sin(ϑ)^3, R_start < r < R_chop
0 , r > R_chop
and
F(r)=1λ(53r^0.6 + 54r^-0.4).
We set each of the parameters R_start=1.25 R_in, R_chop=350 r_g, and λ=15. Note that q(r,ϑ) uses the midplane gas internal energy to scale the vector potential. Also note that the sin(F(r)-F(R_start)) term can vary the sign of A_ϕ across radius with a wavelength that varies with λ. Our parameter choices are designed to place a large poloidal field that does not vary in sign at all. We normalize the magnetic field strength by setting the pressure ratio β_max≡(2(p_gas+p_rad)/b^2)_max=20. From these initial conditions, the MAD state naturally develops as the magnetic field is advected towards the horizon in the accretion flow.
We artificially increase the gas density in high magnetization, σ≡ b^2/ρ, regions in order to ensure the simulation remains numerically stable by limiting σ≤60. Each simulation is carefully inspected to ensure that its accretion rate, magnetic flux parameter, and radial inflow profiles are in steady state for the window considered for further analysis. See <ref> for the full list of simulations described in this work.
§ FLUX CALCULATIONS
The mass accretion rate as a function of radius is computed as
Ṁ(r) = -∫_ϑ∫_φ√(-g)ρ u^r dφ dϑ.
As we discuss in <ref>, we model the hydrodynamic and electromagnetic parts of the spinup parameter separately, following the formalism of <cit.> and <cit.>. To that end, we compute the angular momentum flux normalized by the mass accretion rate in HD and EM components separately:
l_HD(r) = -1/Ṁ(r)∫_ϑ∫_φ T^r_ φ, HD√(-g) dφ dϑ,
l_EM(r) = -1/Ṁ(r)∫_ϑ∫_φ T^r_ φ, EM√(-g) dφ dϑ.
We similarly obtain the energy flux normalized by the mass accretion rate in HD and EM components:
e_HD(r) = -1/Ṁ(r)∫_ϑ∫_φ T^r_ t, HD√(-g) dφ dϑ,
e_EM(r) = -1/Ṁ(r)∫_ϑ∫_φ T^r_ t, EM√(-g) dφ dϑ.
Note that the choice of sign in each expression is such that we compute the flux of energy and angular momentum into the BH, both of which are positive. We are particularly interested in the total outflowing energy relative to the accreted rest mass energy. We characterize this numerically using the dimensionless MHD efficiency
η_MHD(r) = 1 -[e_HD(r)+e_EM(r)].
For the hydrodynamic spinup component, we first obtain the specific angular momentum fluxes l_HD (<ref>) and specific energy fluxes e_HD (<ref>) from the fluid simulations at a radius of 5 r_g. We plot the values calculated directly from the GRRMHD simulations in the leftmost panel of <ref>. The dotted line represents the analytic solution for a thin disk, which we refer to s_thin. As expected, the models approach s_thin as f_Edd→ 0. Meanwhile, the dashed line represents the fit found for non-radiative GRMHD simulations from <cit.>, which we refer to as s_min. They reported e_HD≈ 0.86 and l_HD≈ 0.97 independent of spin, and thus
s_min = 0.86 - 1.94a_*.
As f_Edd increases, our simulations appear to move from s_thin towards s_min. To build our model, we devise a fitting function that approaches s_thin as f_Edd→ 0, and s_min as f_Edd→∞. Thus, we fit for a single number to interpolate between these solutions, arriving at
s_HD = s_thin + s_minξ/1+ξ
with ξ = 0.017 f_Edd.
The results of this fitting function are shown in the central column of <ref>, and residuals are shown in the rightmost column. Without modeling an additional spin dependence, this fitting function underestimates the rapidity with which the a_*=0 models transition from s_thin to s_min. We speculate that this may be due to the lack of consideration of angular momentum loss due to a hydrodynamic wind, evident in <ref>.
For convenience, we reproduce the formulae to obtain s_thin here, following <cit.>. In units where G=c=M=1,
e_thin = ( 1 - 2/3 r_ms)^1/2,
and
l_thin = 2/3√(3)[ 1 + 2(3 r_ms-2)^1/2],
where r_ms is the radius of the marginally stable orbit, given by
r_ms = 3 + Z_2 - sign(a_*)[(3-Z_1)(3+Z_1+2Z_2)]^1/2,
for
Z_1 = 1 + (1-a_*^2)^1/3[(1+a_*)^1/3+(1-a_*)^1/3]
and
Z_2 = (3a_*^2+Z_1^2)^1/2.
Finally,
s_thin = l_thin - 2 a_* e_thin.
We also use the radiative efficiency of the thin disk model to define the Eddington ratio. The Eddington luminosity is the limiting luminosity above which radiation pressure exceeds gravitational pressure in a spherically symmetric system. It is given by
L_Edd = 4 π G M m_p c/σ_T,
where m_p is the proton mass and σ_T is the Thomson cross-section. Defining a radiative efficiency ϵ = L / Ṁ c^2 allows one to define the Eddington mass accretion rate,
Ṁ_Edd = 4 π G M m_p/ϵσ_T c,
Throughout this work, when defining the Eddington mass accretion rate, we assume the radiative efficiency of a thin disk, given by
ϵ = 1 - e_thin = 1 - ( 1 - 2/3 r_ms)^1/2.
Thus, our definition of Ṁ_Edd depends on both mass and spin.
§ PRESSURE SCALE HEIGHT
To gain greater insight into the link between magnetic flux and Eddington ratio presented in <ref>, we explore the pressure scale heights of our simulations. We define the pressure scale height to be
h/r = ∫∫ (P_gas + P_rad)|π/2-θ| √(-g) dθ dϕ/∫∫ (P_gas+P_rad) √(-g) dθ dϕ,
where P_gas is the gas pressure and P_rad is the radiation pressure (which dominates). Here, P_gas + P_rad has taken the place of ρ in the usual definition of the scale height. In <ref>, we plot the pressure scale height at a radius of 10 r_g as a function of Eddington ratio in our simulations, and color code by spin. In grey, we plot a linear regression to these data, from which we obtain h/r = 0.21 + 0.046log_10f_Edd. This increase in pressure scale height as a function of Eddington ratio suggests that a higher Eddington ratio results in more pressure, mostly due to radiation, that can drive the gas to confine stronger magnetic fields onto the horizon.
|
http://arxiv.org/abs/2307.05824v1 | 20230711221401 | Finite SSH chains coupled to a two-level emitter: Hybridization of edge and emitter states | [
"C. I. Kvande",
"D. B. Hill",
"D. Blume"
] | physics.atom-ph | [
"physics.atom-ph"
] |
Homer L. Dodge Department of Physics and Astronomy,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Center for Quantum Research and Technology,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Physics Department, Kalamazoo College, 1200 Academy Street, Kalamazoo,
Michigan 49006, USA
Homer L. Dodge Department of Physics and Astronomy,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Center for Quantum Research and Technology,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Homer L. Dodge Department of Physics and Astronomy,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Center for Quantum Research and Technology,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
The Hamiltonian for the one-dimensional SSH chain is one of the simplest Hamiltonians that supports topological states. This work considers between one and three finite SSH chains
with open boundary conditions
that
either
share a lattice site (or cavity), which—in turn—is coupled to a two-level emitter,
or are
coupled to the same two-level emitter.
We investigate the system properties as functions of the emitter-cavity coupling strength g and the detuning between the emitter energy and the center of the band gap.
It is found that the energy scale introduced by the edge states that are supported by the uncoupled finite SSH chains leads to a g-dependent hybridization of the emitter and edge states
that is unique to finite-chain systems.
A
highly accurate
analytical three-state model that captures the band gap physics
of k-chain (k ≥ 1) systems
is developed.
To quantify the robustness of the topological system characteristics, the
inverse participation ratio for the cavity-shared and emitter-shared systems consisting of k chains
is
analyzed as a function of the
onsite
disorder strength.
The g-dependent hybridization of the emitter and uncoupled edge states can be probed dynamically.
Finite SSH chains coupled to a two-level emitter:
Hybridization of edge and emitter states
D. Blume
August 12, 2023
===========================================================================================
§ INTRODUCTION
The study of individual photons confined in a reflective cavity interacting with matter, frequently a
few-level emitter (e.g., an atom), is at the heart of many quantum studies <cit.>. Chief accomplishments in the field of cavity quantum electrodynamics (QED), such as the manipulation of atoms through photons and, conversely, the manipulation of individual photons by atoms, were recognized by the 2012 Nobel Prize for Physics <cit.>. An important extension of cavity QED is wave guide QED, where the cavity is replaced by a one-dimensional radiation channel or wave guide <cit.>. The one-dimensional wave guide
confines the photons, which interact with one or more quantum emitters that are localized at specific positions along the wave guide.
Such systems can feature hybridized bound and propagating light-emitter states as well as super- and sub-radiance and play a central role in various quantum information processing protocols <cit.>. In dissipation engineering protocols, the wave guide serves as a highly-tunable non-trivial reservoir <cit.>.
The interplay between a topological wave guide QED bath and one or more localized two-level emitters
(2LEs)
was investigated in a seminal paper by Bello et al. in 2019 <cit.>. It was found that the coupling of a
2LE
to a
photonic bath
with periodic boundary conditions (BCs)
described by the Su-Schrieffer-Heeger (SSH) Hamiltonian gives rise to a chiral
zero-energy
bound state if the emitter's frequency is tuned to lie in the
middle of the
bandgap of the bath dispersion. The SSH model was originally introduced to describe solitons in polyacetylene <cit.> and has been used extensively as an analytically tractable
model for topological investigations <cit.>. The SSH bath consists of two sub-lattices (sub-lattice 1 and sub-lattice 2; see Fig. <ref>) with inter-unit hopping energy v and intra-unit hopping energy u.
For |v|>|u|,
the
zero-energy
chiral bound state supported by the emitter-chiral wave guide Hamiltonian
with the emitter tuned to be in resonance with the middle of the band gap,
was found to have the following characteristics
for all emitter-cavity coupling strengths g <cit.>:
(i) The photonic component of the bound state has only a finite amplitude in the sub-lattice that the two-level emitter does not couple to (in our case, the emitter couples to sub-lattice 1 of unit cell n^*, implying that the photonic component occupies sub-lattice 2).
(ii) The photonic component of the bound state occupies only the side of the chain where the cavity of sub-lattice 2 of the unit cell adjacent to unit cell n^* is connected to the cavity of sub-lattice 1 of the unit cell n^* via a strong bond (left arm in Fig. <ref>).
(iii) The bound state
inherits the properties of the topological edge state, e.g.,
is robust against disorder.
This work considers a finite SSH chain with open BCs coupled to a
2LE.
Without the 2LE,
the SSH Hamiltonian
with open BCs
supports two topologically protected edge states that live in the bandgap. Building on the work presented in the supplemental material of Bello et al. <cit.>,
the system properties are analyzed as functions of the emitter-bath coupling strength g and the emitter frequency, focusing on parameter combinations for which the
hybridization
between the emitter and the edge states plays a prominent role.
The finite wave guide with open BCs and zero detuning supports, as the corresponding system with periodic BCs,
a zero-energy state.
The photonic contribution of this state has its maximum at a cavity located at the edges of the chain as opposed to, as found for periodic BCs, at a cavity that sits next to the cavity that the emitter is coupled to.
For g larger than a value that depends on the edge energy of the finite SSH chain and the emitter location on the chain, the emitter contribution to the chiral zero-energy state is essentially zero. The emitter instead hybridizes with the two g=0 edge states, resulting in two finite-energy states that play the role of edge states in the arm of the chain that is not occupied by the chiral zero-energy state. A simple analytical three-state (3-state) model that captures the behaviors for both vanishing and non-vanishing detuning is presented.
Motivated by the possibility that cavities can be connected in non-trivial geometries,
we extend our studies to two and three “crossed chains"
that are connected either by sharing a cavity or by coupling to a shared
2LE.
Both scenarios can be thought of
as having a single site, either the shared cavity or the shared emitter, with a coordination number that is,
respectively,
two and three times larger for the
2-chain and 3-chains
scenarios
than the coordination number of the other cavities.
Even though the number of g=0 states in the middle of the gap increases with k, we find that the g-dependent characteristics of emitter- and cavity-shared k-chain systems can be described by the same analytical 3-state Hamiltonian as the 1-chain system, provided the effective coupling constant is chosen accordingly. Our results extend readily to k>3.
Our findings highlight that finite baths, as frequently realized experimentally, display—compared to infinite baths—distinct characteristics that are due to the new energy and length scales introduced by the finiteness of the system. In the case of the topological bath, there is not only an energy scale that emerges from the finite length of the chain (which sets, e.g., a time for photons to travel to the end of the chain and back
and which also exists for non-topological chains) but
also
an
energy scale that emerges from the splitting of the two edge states supported by the bath
(this energy scale does not exist in a non-topological bath and goes to zero for a topological bath as N approaches infinity).
The remainder of this paper is organized as follows.
Section <ref> introduces the system Hamiltonians and static properties, including a discussion of the eigen spectra of k-chain
systems (k=1-3) and the emergence of dark states; mathematical details are relegated to Appendices <ref>-<ref>.
Section <ref> investigates the robustness of the states that have topological characteristics to chiral-symmetry-breaking disorder while Sec. <ref> shows that the g-dependent hybridization can be probed dynamically.
Section <ref>
concludes and presents an outlook.
§ HAMILTONIAN,
EIGEN ENERGIES, AND EIGEN STATES
§.§ SSH Hamiltonian
For a chain that consists of N unit cells,
the SSH Hamiltonian
Ĥ_SSH, which is schematically illustrated in Fig. <ref>, reads
Ĥ_SSH =
∑_n=1^N-1[
(
u ĉ_n,1^†ĉ_n,2
+
v ĉ_n,2^†ĉ_n+1,1)
+ h.c.
] +
[ (
u ĉ_N,1^†ĉ_N,2
+
v_N ĉ_N,2^†ĉ_1,1)
+ h.c. ]
,
where ĉ_n,j
annihilates an excitation in the jth sub-lattice (j=1 or 2) of the nth unit cell.
The parameters u and v, which are taken to be real and positive throughout this paper, denote intra-unit cell and inter-unit cell hopping energies, respectively.
Periodic and open BCs are realized for
v_N=v and v_N=0, respectively.
Throughout, we
use
u and ħ/u as our
energy and time units.
The SSH model has elucidated phenomena in many sub-disciplines of physics including chemical physics <cit.>, condensed matter physics <cit.>, cold atom physics <cit.>, and relativistic field theories <cit.>.
Throughout this paper, we have a scenario in mind where each unit cell contains two cavities (one that belongs to sub-lattice 1 and one that belongs to sub-lattice 2; see Fig. <ref>) and where the operators
ĉ_n,j and ĉ_n,j^†
annihilate and create a photon in the cavity that belongs to sub-lattice j of the nth unit cell.
Since the SSH Hamiltonian
possesses a chiral symmetry, it is a paradigmatic model for studying topology.
Specifically, the chiral operator
Ĉ and Ĥ_SSH anti-commute,
ĈĤ_SSHĈ = -
Ĥ_SSH,
where Ĉ is defined in terms of the
projection operators
P̂_j (j=1 or 2),
Ĉ=P̂_1-P̂_2
and
P̂_j =
∑_n=1^N
ĉ_n,j^†ĉ_n,j.
For concreteness, we consider the set-up in Fig. <ref>. If v is smaller than u,
Ĥ_SSH is topologically trivial.
Since the inter-unit hopping strength is weaker than the intra-unit hopping strength,
the two cavities contained in a given unit cell are “binding together", i.e., the hopping strengths “respect" the chain's division into unit cells.
If, on the other hand,
v is larger than u, Ĥ_SSH is topologically non-trivial.
In this case, a cavity from the nth unit cell and a cavity from the
(n+1)th unit cell are binding together, leading to inter-unit cell bonds.
For open BCs, this leads to a single dangling or unpaired cavity
on each end of the chain and the emergence of two edge states
that are predominantly located at the first and Nth unit cells <cit.>.
For later reference, we introduce approximate expressions for the edge states | ψ_±^C1⟩:
|ψ_±^C1⟩
= 1/√(2)(
| ψ_edge,L^C1⟩± | ψ_edge,R^C1⟩),
where |ψ_edge,L^C1⟩ and |ψ_edge,R^C1⟩
are localized in sub-lattice 1 near the first unit cell and in sub-lattice 2 near the Nth unit cell, respectively:
|ψ_edge,L^C1⟩=
∑_n=1^N
c_n,1 |n,1 ⟩
and
| ψ_edge,R^C1⟩=
∑_n=1^N
c_n,2 |n,2 ⟩.
Here, the site basis states |n,j⟩, where n=1,⋯,N labels the unit cell and j=1,2 indicates the sub-lattice, are used.
The superscript “C1" (chain 1) is introduced with view toward the k>1 discussion below.
The expansion coefficients c_n,1 and c_n,2 read
c_n,1 = N (-1)^n+1( u/v )^n-1
and
c_n,2 = N (-1)^N-n(u/v)^N-n,
with N denoting a normalization constant,
N=
[1-
(u/v)^2
]^1/2[ 1-
( u/v )^2N]^-1/2
.
Equations (<ref>)-(<ref>) become exact in the N →∞ limit.
The states | ψ_±^C1⟩ have energy
± E_edge, where
E_edge= (-1)^N+1N^2
( u/v )^N-1
u.
The energy E_edge approaches zero exponentially with increasing N.
In the N →∞ limit, the edge states have vanishing energy
and are characterized by a localization length ζ_loc=2 a/ln(v/u), where 2a denotes the
separation between neighboring unit cells <cit.>.
For v=2u, as considered throughout this paper, ζ_loc evaluates to ≈ 2.89a.
Moreover, if a site from sub-lattice 1
(sub-lattice 2) sits at the end of the chain, the edge state has zero amplitude in sub-lattice 2 (sub-lattice 1) at that end.
Since
zero-energy eigen states are simultaneously eigen states of Ĉ, this
follows directly from the chiral symmetry.
Note that zero-energy eigen states, and correspondingly edge states, are not supported for periodic BCs. Yet, the systems with open and periodic BCs are intimately related through the bulk-edge correspondence <cit.>.
The blue circles in Fig. <ref> show
the energy spectrum, plotted as a function of the normalized eigen state index,
for a chain with open BCs, v=2u, and N=31.
In this case, the energy
± E_edge
of the states in the band gap is ± 6.98× 10^-10 u;
within the digits reported, this agrees with the approximate expression (<ref>). For comparison, the black solid line shows the eigen energies for the infinite chain with periodic BCs. The agreement between the finite N and infinite N energy bands is very good.
The
numerically determined
edge states are illustrated in the insets in the upper left and lower right of Fig. <ref> using the
site basis states
|n,j⟩.
The size of the filled circles in Fig. <ref>
is directly proportional to the square of the amplitude of the expansion coefficients in sub-lattice 1 (upper row) and sub-lattice 2 (lower row) in the nth unit cell; the color of the circles (blue and red) represents the sign (positive and negative) of the expansion coefficients.
Despite
the finite number of unit cells, the
eigen states inherit the key characteristics
of the thermodynamic system (N →∞
limit), i.e., the edge states have finite amplitude on just one sub-lattice. Moreover, the localization length of the finite-chain edge states is very close to the localization length ζ_loc for infinite N.
§.§
Single
SSH chain coupled to emitter
As alluded to in the introduction, we are interested in
k-chain
systems (k=1-3) coupled to a single
2LE
with ground state |g ⟩
(energy 0) and excited state |e⟩ (energy ħω_e).
Schematics of these systems are shown in Fig. <ref>. This section introduces the
1-chain
Hamiltonian Ĥ_C1-2LE,
which is written as a sum of the SSH chain, the
2LE
Hamiltonian
Ĥ_2LE,
and the coupling term
Ĥ_int (see, e.g., Ref. <cit.>),
Ĥ_C1-2LE=
Ĥ_SSH+
Ĥ_2LE+
Ĥ_int,
Ĥ_2LE=
ħω_e/2(σ̂^z
+1 ),
and
Ĥ_int=
g ( ĉ_n^*,1^†σ̂^-
+
ĉ_n^*,1σ̂^+
).
The operators
σ̂^z,
σ̂^+,
and
σ̂^-, which act in the Hilbert space of the
2LE,
read
σ̂^z=|e⟩⟨ e|-|g⟩⟨ g|,
σ̂^+=|e⟩⟨ g|,
and
σ̂^-=|g⟩⟨ e|.
In Eq. (<ref>),
the emitter couples,
with coupling energy g (g ≥ 0), to the
cavity
of sub-lattice 1 that belongs to the (n^*)th unit cell.
In the examples considered in this work,
N is odd and n^* is equal to (N+1)/2.
Coupling to a cavity that belongs to sub-lattice 2 can be treated in the same way and yields analogous results.
Since Eq. (<ref>) employs the rotating wave approximation <cit.>, we restrict ourselves to g/u ≤ 0.1 throughout this work.
The Hamiltonian Ĥ_C1-2LE
commutes with the excitation operator
N̂_exc,
N̂_exc=
P̂_1 + P̂_2 + |e ⟩⟨ e |,
and can thus be diagonalized separately for each
excitation manifold <cit.>.
Since the dynamics discussed in
Sec. <ref>
start with the emitter in state |e ⟩ and the SSH chain in the zero-photon vacuum state |vac⟩ (this state has ⟨N̂_exc⟩=1),
we are interested in the single-excitation manifold, which is spanned by the basis states
|n,j;g ⟩,
where the first two entries refer to the SSH chain (n=1,⋯,N and j=1,2) and the last entry refers to the emitter, and | vac;e⟩.
Since ĈĤ_C1-2LEĈ is not
equal to
-Ĥ_C1-2LE, the introduction of the emitter leads to a breaking of the chiral symmetry:
the emitter can be thought of as a
chiral symmetry-breaking perturbation.
Solid lines in the top row of Fig. <ref> show the near-zero eigen energies of Ĥ_C1-2LE
as a function of the emitter energy ħω_e for v/u=2, N=15, n^*=8, and four different g/u, namely g/u=10^-4-10^-1. The spectrum is calculated using open BCs.
The emitter energy can be interpreted as a detuning from the center of the band gap.
For g=0, the eigen states of the three eigen energies shown in Fig. <ref>
correspond to the two edge states
|ψ_-^C1;g⟩
and
|ψ_+^C1;g⟩
and the excited emitter
state |vac;e⟩.
As the coupling g is turned on, these three states mix and the corresponding eigen energies undergo avoided crossings.
The eigen energies that belong to the two nearly continuous bands (not shown) and their eigen states, in contrast, remain essentially unchanged.
For small g/u,
avoided crossings
between two states occur when the detuning (or emitter energy) ħω_e is equal to the energy ± E_edge of the edge states supported by Ĥ_SSH (for the N=15 system considered in Fig. <ref>, E_edge≈ 4.58 × 10^-5u). As expected,
both
avoided crossings
become broader with increasing g/u.
The two avoided crossings start to overlap (implying hybridization of three states)
for g/u ≳ 10^-2.
For g/u=0.1, Fig. <ref>(d) suggests that the green and red energy levels undergo an avoided crossing, with the energy level shown in blue being
decoupled and having, on the scale shown, zero energy.
Denoting the three eigen states whose energies lie in the gap by |ψ_l^gap⟩ (l=1-3), we find that the initial state
|vac;e⟩ considered in the dynamical studies discussed in
Sec. <ref>
can be decomposed with good accuracy as
|vac;e⟩≈∑_l=1^3
d_l^gap |ψ_l^gap⟩,
where
d_l^gap=
⟨ψ_l^gap |
vac;e ⟩.
Solid lines in the second row of Fig. <ref> show the overlap
square O_l,
O_l=|⟨vac;e | ψ_l^gap⟩|^2.
Since the quantity ∑_l=1^3 O_l is greater than
0.996
for all 1-chain systems considered in Fig. <ref>,
the results presented in the second row of Fig. <ref> allow us to forecast where population transfer is expected since population transfer occurs only if the initial state projects onto two or more eigen states of the coupled system.
Since the three gap states |ψ_l^gap⟩ can be written, with good accuracy, as a superposition of
the uncoupled approximate g=0 states
|ψ_-^C1;g⟩,
|ψ_+^C1;g⟩,
and |vac;e⟩ for all parameter combinations considered in Fig. <ref>, we use them
to construct
the 3-state Hamiltonian matrix
H_3-st.(G):
H_3-st.(G)=
(
[ -E_edge 0 G; 0 E_edge G; G G ħω_e; ]),
where the effective coupling energy G is defined through G=⟨ψ_±^C1;g|Ĥ_int|vac;e⟩; note, G is real.
Using the analytical expressions given in Eqs. (<ref>)-(<ref>), we find
G =
g c_n^*,1 / √(2).
While Eq. (<ref>) is characterized by the effective coupling constant G, the 3-state model introduced in the supplemental material of Ref. <cit.> contains both G and -G.
Our 3-state model
reproduces the
1-chain
gap energies and overlap square data shown
in Fig. <ref> to
0.5 %
or better.
Correspondingly, the model serves as
a highly reliable tool for understanding the gap physics for the parameter regime of interest in this paper.
Since Ĥ_3-st.(G) lives
in the space that is spanned by states that have non-vanishing amplitude on sub-lattice 1 only in the left arm of the SSH chain and non-vanishing amplitude on sub-lattice 2 only in the right arm of the SSH chain and since it describes the gap states |ψ_l^gap⟩ accurately,
the gap states inherit
chiral characteristics for all parameter combinations shown in Fig. <ref>.
The 3-state model predicts that the hybridization of the three uncoupled basis states occurs at |G/ E_edge| ≈ 1 (see Appendix <ref>).
For the parameters of Fig. <ref>, this corresponds to g/u≈ 10^-2. For fixed u/v, the transition moves to smaller g/u with increasing N [and, as before, n^*=(N+1)/2].
For fixed N and n^*=(N+1)/2, the transition moves to larger g/u with decreasing v/u (keeping v>u).
To compare the 1-chain systems with open and periodic BCs for v/u=2, N=15, and n^*=8 (same parameters as used in Fig. <ref>), we analyze the zero-energy state, which exists for ħω_e=0 for both open and periodic BCs. Red solid and green dashed lines in Fig. <ref> show the
probability |c_e|^2 of the zero-energy state to be in state |vac;e⟩
(approximate analytical expressions) as a function of g/u for open and periodic boundary conditions, respectively.
For comparison, the symbols are obtained by diagonalizing the Hamiltonian Ĥ_C1-2LE.
The agreement between the lines and symbols is excellent.
For open BCs,
|c_e|^2 is close to 1 for small g/u and drops to a value close to 0 around g/u values for which |G/E_edge| ≈ 1 (arrow in Fig. <ref>).
For periodic BCs, in contrast, |c_e|^2
does not decrease notably till g/u takes values of order one (or, more generally, when the coupling energy g becomes comparable to the width of the g=0 energy gap). The fact that the coupling strength where the contribution |c_e|^2 to the zero-energy state drops significantly differs for open and periodic BCs explicitly demonstrates the role played by E_edge.
We note that the behaviors of systems with g/u values larger than 0.1-0.5 need to be interpreted with the understanding that beyond-the-rotating-wave-approximation terms may play a non-negligible role.
For small g/u, the photonic contribution to the zero-energy state of the systems with open and periodic BCs is located on different arms and localized at different positions, namely, as far away from the emitter as possible for open BCs (see the inset in upper left corner) and on cavities close to the emitter for periodic BCs (see the inset in the middle left).
The blue dotted lines and circles show |c_e|^2 for the non-zero energy states of the finite-chain system: |c_e|^2
increases relatively sharply at |G/E_edge| ≈ 1. For g/u≳ 0.2, the agreement between the numerical results and the approximate analytical expression deteriorates. This is not surprising since this is the regime where g is strong enough to couple to states that are not part of Ĥ_3-st.. The insets in the lower right corner show the corresponding eigen states; both have finite photonic contributions on the left arm of the chain.
For finite detuning, direct comparisons between the systems with open and periodic BCs are less straightforward since the energy of the gap states (system with open BCs) and bound state (system with periodic BCs) changes differently with finite detuning.
§.§ 2- and 3-chain systems
The cavity- and emitter-shared
2- and 3-chain
systems are illustrated in Fig. <ref>. To make connections between the physics of the
2- and 3-chain
systems and that of the
single SSH chain system discussed in the previous section, we start with g=0 and then consider what happens for finite g values.
The emitter-shared
2- and 3-chain
systems
reduce, for g=0, to two and three independent copies of the single SSH
chain.
Correspondingly, the
2- and 3-chain
systems with open BCs support a total of four and six edge states, respectively. The energy degeneracy of the edge states is two (three) for the
2-chain (3-chain)
systems:
For the
2-chain (3-chain)
systems,
two (three) states have energy E_edge
and
two (three) states have energy -E_edge.
Forming appropriate linear combinations of the degenerate states (see Appendix <ref> for details), we find that four (six) of these states are, to a very good approximation, not affected by the coupling between the cavities and the emitter, i.e., their energies for finite g are approximately equal to ± E_edge.
The other three energies near zero
with eigen states |ψ_l^gap⟩
are, as in the case of the
1-chain
system, very well described by a 3-state model.
Specifically,
the 3-state model discussed in
the previous section applies also to the emitter-shared
k-chain
(k>1) system, provided g is not too large and provided the effective coupling constant is replaced by √(k)G
(see Appendix <ref> for details).
We note also that the
initial state |vac;e⟩ can, to a good approximation, be expanded in terms of
the three gap states |ψ_l^gap⟩.
The quantity ∑_l=1^3 O_l is greater than 0.993
for the emitter-shared 2- and 3-chain systems for the parameter combinations covered in Fig. <ref> (note, though, that the figure, is for the cavity-shared systems).
The cavity-shared
2- and 3-chain
systems are, even for g=0, distinct from the
1-chain
system.
Figure <ref> shows the energy of the
2-chain
system with open BCs as a function of the normalized state index for N=15 and v/u=2. Since the two chains share one cavity, the total number of sites of the
2-chain
system is 4N-1 (recall, N refers to the number of unit cells of one of the SSH chains).
As expected, the energy spectrum consists of two nearly continuous energy bands that are separated by an energy gap, which supports states with energy close to zero. The number of states in the gap is not four, as might be expected naively by doubling the number of edge states supported by the
1-chain
system, but five. Four states have finite energy and one state has vanishing energy.
The latter
is a delocalized dark state (see Appendix <ref>), which has non-vanishing amplitude in both sub-lattices (see the lower right inset of Fig. <ref>). An analogous non-topological dark state also exists for the
2-chain
system with periodic BCs but does not exist for the emitter-shared
2-chain
systems
with periodic and open BCs.
The other four states with energy close to zero can be divided into two pairs. The states belonging to the first pair, with energy
±ϵ,
are approximately unaffected when g is turned on.
The states belonging to the second pair with energy ± E_edge (see the lower left inset of Fig. <ref> for an example), couple to the emitter and form, together with the state | vac;e ⟩, the basis for a 3-state model
(same
3-state Hamiltonian matrix
as discussed above for the
1-chain
system, but with G replaced by G/√(2);
see Appendix <ref>).
For comparison, the cavity-shared
3-chain
system with open BCs (6N-2 sites with one cavity shared by all three
chains)
supports two dark states
for g=0
whose energy is exactly zero as well as another six states that also reside
in the energy gap. The existence of these six states might be expected based on the naive
argument that the number of edge states supported by the
1-chain
system triples for the
3-chain
system.
The six states can be divided into two groups, four states that have energy ±ϵ (two states with positive energy and two states with negative energy) and two states that have energy ± E_edge. The latter two states couple to the emitter when g is finite
and are described well by the 3-state
Hamiltonian matrix H_3-st.(G/√(3)) (see Appendix <ref>).
The energy spectrum for the cavity-shared
2-chain
system shown in Fig. <ref>
features one energy state below the bottom of the lower band and one energy state above the top of the upper band. These states have no analog in the
1-chain
system or the emitter-shared
2- and 3-chain
systems and can, since they reside outside the nearly continuous energy bands, be interpreted as bound states. This interpretation is consistent with the observation that the corresponding eigen states, which are shown in the upper left and upper right insets of Fig. <ref>, are localized in the vicinity of the cavity that is shared by the two SSH chains. The eigen state that sits below the bottom of the lower band is nodeless (upper left inset of Fig. <ref>)
while the eigen state that sits above the top of the upper band is highly oscillatory, i.e., the eigen state's expansion coefficients corresponding to neighboring cavities have opposite signs
(upper right inset of Fig. <ref>).
These localized bound states are non-topological
in nature since they have non-vanishing population in both sub-lattices and also exist for u/v=1
(for this ratio, the energy gap of Ĥ_SSH closes) as well as periodic BCs.
The binding energy, measured from the bottom/top of the energy band, increases—for fixed N—with increasing number of
chains.
As already alluded to, for finite
g/u and |ħω_e/u|≪ 1,
the emitter state |vac;e ⟩ has appreciable overlap
with only three of the eigen states that are located in the energy gap of the cavity-shared
2- and 3-chain
systems.
Dashed and dotted lines in Fig. <ref> show the energy of the three states in the gap that have finite overlap with |e;vac⟩ (top row) and the square of the overlap (bottom row) for,
respectively,
the cavity-shared
2- and 3-chain
systems
with open BCs
as a function of ħω_e for four different g/u values.
The behavior of the gap states and their energies for the
2- and 3-chain
systems is similar to that for the
1-chain
system, with the main feature that the avoided crossings are becoming somewhat narrower as the number of
chains
increases from 1 to 2 and again from 2 to 3. The observed behavior is consistent with the decrease of the effective coupling constant G by factors of 1/√(2) and
1/√(3) for the cavity-shared
2-chain and 3-chain
systems, respectively,
relative to the
1-chain
system
with effective coupling constant G (see Appendix <ref>).
Since the 3-state model applies, the hybridization discussed in the previous section carries over, with the basis states being those introduced in Appendix <ref>. The quantity ∑_l=1^3 O_l is greater than
0.998
for all cavity-shared 2- and 3-chain systems considered in Fig. <ref>.
This
is similar to what was discussed above for the corresponding emitter-shared
2- and 3-chain
systems.
§ RESPONSE TO DISORDER
To analyze the robustness of the topological characteristics, we introduce uniformly distributed onsite disorder of the photonic part of the Hamiltonian.
The disorder strengths ϵ_n are chosen from the disorder strength window
[ -Δ,Δ].
In the absence of the coupling to the emitter, the onsite disorder breaks the chiral symmetry of the
SSH part of the Hamiltonian; hopping disorder (not considered), in contrast, preserves the chiral symmetry of the
SSH part of the Hamiltonian <cit.>.
We diagonalize the full system Hamiltonian
for a large number of onsite disorder realizations and
analyze, for each disorder realization, the three eigen states |ψ_l'^gap,disorder⟩ that have the largest overlap with the
gap states |ψ_l^gap⟩
for the same g/u, ħω_e/u, v/u, and n^* in the absence of disorder (recall that the three gap states are defined as the states that have energy close to zero and depend,
for ħω_e 0,
on the value of g/u).
When the disorder strength Δ/u is small, the overlap criterion employed to identify the states |ψ_l'^gap,disorder⟩ is—since the largest overlap is pretty close to 1—“clean". For larger Δ/u, in contrast, the eigen states
|ψ_l'^gap,disorder⟩
are found to deviate notably from the disorder-free gap states |ψ_l^gap⟩;
despite of this, the largest overlap, while notably smaller than one, allows for an “unambiguous" identification of the
states |ψ_l'^gap,disorder⟩. s
The
decrease of the overlap with increasing Δ/u signals that the system characteristics are fundamentally altered when the onsite disorder strength is increased.
To quantify the degree of localization of the three eigen states |ψ_l'^gap,disorder⟩,
we calculate the inverse participation ratio IPR <cit.>,
IPR = ∑_m=1^M |c_m^(l')|^4 /| ∑_m=1^M |c_m^(l')|^2 |^2,
where the expansion coefficients
c_m^(l') are given by the overlap of the state |ψ_l'^gap,disorder⟩ and the mth site basis state, and M is equal to kN-k+2 and kN+1 for the cavity- and emitter-shared cases, respectively (in this context, the state |vac;e⟩ is counted as one of the site basis states).
The inverse participation ratio (IPR) is a measure of localization: IPRs
of 1 and 0 indicate maximal and minimal localization, respectively.
In addition, we analyze the polarization, i.e., we monitor if the states |ψ_l'^gap,disorder⟩ occupy just one sub-lattice or both sub-lattices in each of the 2k arms of the k-
chain
systems. The IPR (see Fig. <ref>) together with the polarization (not shown) quantify the robustness of the gap state characteristics against disorder.
Symbols in Fig. <ref>
show the IPR for the three states |ψ_l'^gap,disorder⟩ for
chain
systems
with open BCs for
v/u=2, ħω_e/u=-5 × 10^-5, N=15, n^*=8, and
g/u values ranging from
10^-4
[Fig. <ref>(ai)-<ref>(aiii)] to 10^-1
[Fig. <ref>(di)-<ref>(diii)] as a function of Δ/u. The scaled disorder strength Δ/u is shown on a logarithmic scale, which
covers 12 orders of magnitude; for each disorder strength, the IPR (symbols) and error bars are calculated by averaging 5 × 10^3 independent disorder realizations.
The colors employed in Fig. <ref> are “matched" with those in Fig. <ref>, i.e., the IPRs shown in red, blue, and green coincide—in the zero disorder limit—with those for the states |ψ_l'^gap,disorder⟩ whose energies and overlap squares are shown in red, blue, and green in Fig. <ref>. The left, middle, and right columns are for
the
1-chain
system, the cavity-shared
2-chain
system, and the emitter-shared
2-chain
system, respectively. It can be seen that the changes of the IPRs and the IPRs' error bars with disorder strength depend on both the
chain
geometry and the coupling strength g/u.
The IPR tends to change
in non-trivial ways with the disorder strength,
suggesting that the disorder modifies the states
|ψ_l'^gap,disorder⟩ in ways that depend intricately on the energy scales of the system.
Complementing the IPR, Fig. <ref> shows the distribution of the eigen energies, averaged
also
over 5 × 10^3 disorder realizations, as a function of the scaled disorder strength Δ/u.
The layout of Fig. <ref> is the same as that of Fig. <ref>, i.e., the two figures cover the same range of scaled disorder strengths,
chain
geometries, and coupling strengths g/u. For each g/u, the
chosen energy window (range of the y-axes) in Fig. <ref> is the same as in Fig. <ref>. Contrary to Fig. <ref>, Fig. <ref> includes not only the energy of the gap states but of all eigen energies that fall into the energy window. To connect the energy distribution in the band gap with the nearly continuous energy bands, Fig. <ref> shows the distribution of eigen energies of the cavity-shared
2-chain
system for a much larger energy window and somewhat smaller range of disorder strengths. The gap regime, which is the focus of Fig. <ref>, is not resolved on this scale. Plots (not shown) for the other
chain
geometries and coupling strengths g/u considered in this work look essentially identical to Fig. <ref>, with the exception of the bound states below and above the energy bands, which only exist for the cavity-shared
k-chain
systems (k≥ 2; see Fig. <ref>).
Combining Figs. <ref> and <ref>, three regimes can be identified.
(i) For small Δ/u (Δ/u ≲ 10^-4), the energies in the gap are robust to disorder, i.e., the energies in the gap are distinguishable from each other.
(ii) For intermediate Δ/u (10^-4≲Δ / u ≲ 1), the energies that used to lie in the gap form a band that is separated from the two nearly continuous energy bands; also, the bound states are separated from the two nearly continuous energy bands.
(iii) At large detunings (Δ/u ≳ 1), the bands are essentially “melted" entirely; we emphasize that there exists a state with energy ≈ħω_e for small g/u and large Δ/u that is only minimally impacted by the disorder.
Importantly, we find that the population in a given arm of the
k-chain
systems is, for all gap states |ψ_l'^gap,disorder⟩, to a very good approximation either located in sub-lattice 1 or in sub-lattice 2 up to disorder strength Δ/u ≈ 10^-3
[this includes the regime (i) as well as a portion of the regime (ii) introduced above], i.e., the topological characteristic
of population being localized on only one sub-lattice in a given arm
is preserved up to a critical disorder strength that depends relatively weakly on the coupling strength and
chain
geometry and is about 20 times larger than |E_edge|.
We now discuss selected limits.
We start with
the Δ/u → 0 limit (arbitrary g/u).
Appendices <ref> and <ref> show that the
IPRs for the
1-chain and emitter-shared 2-chain
system for Δ/u=0 are reproduced with high accuracy (at the percent level or better) by the analytical 3-state model expressions [see Eqs. (<ref>) and (<ref>)] for all g/u considered in Fig. <ref>. Notably, to approach the zero-disorder limit for the emitter-shared
2-chain
system with g/u=10^-4 [Fig. <ref>(aiii)], the scaled disorder strength must be smaller than 10^-8, i.e., more than three orders of magnitude smaller than ħω_e/u, g/u,
and |E_edge|/u.
For
the
1-chain and cavity-shared 2-chain
system, in contrast, Δ/u must be ≲ 10^-5 for the zero-disorder limit to be approached.
Next,
we consider
the small g/u limit [see Figs. <ref>(ai)-<ref>(aiii)
and Figs. <ref>(ai)-<ref>(aiii)].
For the smallest g/u considered
(namely, g/u=10^-4), the IPR for the
state |ψ_l'^gap,disorder⟩ that is dominated by the basis state |vac;e ⟩ is
very close to 1 for all disorder strengths [red triangles in Figs. <ref>(ai)-<ref>(aiii)].
The error bar is small for small Δ/u, then increases, and is small again for Δ/u larger than 10^-3. In the latter regime,
the state
has an energy close to ħω_e [yellow-ish stripe in Figs. <ref>(ai)-<ref>(aiii)] and is localized at the emitter. Scaled disorders around 10^-3 lead—for g/u=10^-4—to state localization and the reopening of an energy gap.
At very strong disorder, the IPRs of the other two states [shown in green and blue in Figs. <ref>(ai)-<ref>(aiii)] also approach 1, signaling Anderson localization <cit.>; this behavior is reminiscent of what was observed in Ref. <cit.>.
We note that most
of the energies of these states lie outside of the energy windows shown in Figs. <ref>(ai)-<ref>(aiii).
For the emitter-shared
2-chain
system,
a very weak disorder leads for
|G/E_edge| ≪ 1 to a change of one of the states with edge-like character [green symbols in Fig. <ref>(aiii)]: the state changes from having population in all four arms to having population in only two arms, which may belong to the same
chain or different chains.
For the cavity-shared
2-chain
system,
a weak disorder leads for
|G/E_edge |≫ 1 to a distinct change of the IPR shown by blue symbols in Fig. <ref>(dii): the state changes from being localized in two arms to being localized in one arm.
These examples illustrate that the response of the states that live in the gap to onsite disorder depends sensitively on how the SSH chains are
connected to each other (through a cavity or through an emitter).
In particular, very weak disorder can lead to distinct differences in the system response of the cavity-shared and emitter-shared
k-chain
systems in a regime where the disorder modifies the eigen energies extremely weakly.
This can be understood by realizing that the disorder breaks the discrete rotation symmetry associated with the
k-chain
systems (invariance of the Hamiltonian under exchange of any two
chain
indices for k ≥ 2), favoring—in some cases—states that localize on one
chain
as opposed to populating all k
chains
equally.
§ TIME-DEPENDENT SIGNATURE OF THE HYBRIDIZATION
This section shows that the transition from the excited emitter state contributing predominantly to one state to contributing predominantly to two states in k-chain systems with open BCs can be probed dynamically (see also Ref. <cit.>).
Figure <ref>
shows the population dynamics of the
1-chain
system
[Figs. <ref>(a) and <ref>(c)]
and cavity-shared
2-chain
system
[Figs. <ref>(b) and <ref>(d)]
as a function of the site basis
for ħω_e/u=-5 × 10^-5 and two different g/u values, namely
g/u=10^-3 (top row) and
g/u=10^-1 (bottom row).
The basis state
in which the emitter is excited (namely, basis state |vac;e⟩) is placed on the far right.
The other basis states are ordered to alternate between sub-lattice 1 and sub-lattice 2. For the
1-chain
system, the unit cell index n increases from left to right. For the
2-chain
system, the 2N basis states (emitter in |g⟩ and unit cell index n) of the first
chain
are shown first, followed by the
2N-1 basis states (emitter in |g⟩ and unit cell index n') of the
second
chain.
We first consider the dynamics of the
1-chain
system.
The excited emitter state |vac;e⟩
has a population of 1 at t=0 and then oscillates
with a frequency that can be obtained analytically using the 3-state model introduced in
Sec. <ref>.
Interestingly, while the dynamics of the
population of state |vac;e⟩ is qualitatively
similar for the two g/u values considered—albeit with different oscillation frequency—,
the population dynamics of the states |n,j;g⟩
shows a marked difference for the two different g values.
For g/u=10^-3, both end cavities of the chain have an enhanced population when the population of the
state |vac;e⟩ is smallest [see red arrows in Fig. <ref>(a)].
For g/u=10^-1, in contrast, the left end of the chain displays an enhanced population while the right end of the chain does not [white arrow in Fig. <ref>(c)]. These population dynamics are consistent with our
discussion in the previous section and, in particular, with our conclusion
that the excited emitter state hybridizes with the photonic components that live on the left arm of the chain
for g/u ≳ 10^-2.
The initial state has essentially zero overlap with the
gap state that has approximately zero energy [see blue solid lines in Figs. <ref>(d) and <ref>(h)] and can be, in the large G limit, approximated by
|ψ_edge,R;g ⟩ (see
Sec. <ref> and
Appendix <ref>).
As a consequence, the absence of population in the right arm, marked by the white arrow in Fig. <ref>(c), can be interpreted as a key fingerprint of the
change of the hybridization of the gap states
for sufficiently large G.
If we repeat the dynamical study for ħω_e=0 but otherwise identical parameters (not shown), we find that the photonic populations on the right arm undergo oscillations for g/u=10^-3 (the left arm has vanishing photonic populations) and those on the left arm undergo oscillations for g/u=10^-1 (the right arm has vanishing photonic populations); see the insets of Fig. <ref> for depictions of the corresponding ħω_e=0 gap states.
The dynamics of the cavity-shared
2-chain
system is analogous to that of the
1-chain
system, with the dynamics of the
2-chain
system being slightly slower than that of the
1-chain
system, as would be expected based on the small but visible changes of the energy spectra with the
number of chains (see the top row of
Fig. <ref>).
For g/u=10^-3, the population of the four end cavities is maximal when the population of the state |vac;e⟩ is minimal [red arrows in Fig. <ref>(b)].
For g/u=10^-1, in contrast, only the left ends of both chains
get populated appreciably [red and white arrows in Fig. <ref>(d)].
As in the
1-chain
case, the
excited emitter state hybridizes with the g=0 edge states such that, for sufficiently strong coupling, only cavities in the left arms of the chains are occupied.
In analogy to the
1-chain
case, the absence of population in two arms for sufficiently large G [white arrows in Fig. <ref>(d)] signals the
change in hybridization.
The behavior for the cavity-shared
3-chain
system
with open BCs
(not shown) is similar to that for the
1-chain and cavity-shared 2-chain
systems
with open BCs.
§ CONCLUSIONS
This work investigated static and dynamic properties of one, two, and three SSH chains,
which possess chiral symmetry, coupled weakly to a
2LE that breaks the chiral symmetry.
In the case of a single chain, the emitter was coupled to one of the lattice sites.
In the case of the
2- and 3-chain
systems, the emitter was either coupled to one lattice site of each
chain
(emitter-coupled
k-chain
system; k=2 or 3) or to a single lattice site that
was shared between the
chains
(cavity-coupled
k-chain
system). Since the rotating wave approximation was employed, coupling strengths were limited to g/u ≤ 10^-1.
Using open BCs and working in the single-excitation manifold, this work focused on the states that reside in the energy gap between the two nearly continuous energy bands. Throughout, the excitation energy of the emitter was chosen such that the emitter was in resonance with the band gap.
The number of states in the band was found to depend on the
chain
geometry. For all
chain
geometries considered, it was found that the eigen states in the gap could be grouped into three states that depend on the coupling strength g/u and zero or more states that, to a very good approximation, had zero population in the emitter. A fully analytical 3-state model was found to provide an excellent description of the g-dependent gap states with topological characteristics for all k-chain systems investigated.
This work exploited that a SSH chain
with open BCs
is characterized by a finite edge state energy E_edge,
leading to a g-dependent hybridization that is absent in the system with periodic BCs.
This finite energy scale modifies the role of the emitter from being perturbative for
|G/E_edge| ≲ 1
to
being non-perturbative for
|G/E_edge| ≳ 1. The hybridization of the excited emitter state and the g=0 edge states was analyzed in detail and the behavior was contrasted with that for the system with periodic BCs.
Generalizations and variants of the paradigmatic SSH model include studies of two nonreciprocal coupled SSH chains <cit.>, a bipartite lattice of domain wall states <cit.>, and topological synchronization <cit.>.
Our work adds to the growing body of emitters coupled to topological wave guides <cit.>. Extensions of the present work to 3-level emitters, which themselves support dark states, will open the door for coupling the dark state of the emitter and the dark states of cavity-coupled
k-chain
systems.
In hyperbolic lattices, which contain
chain- or
ring-like building blocks,
the relaxation dynamics of a
2LE
was recently proposed as a probe of the hyperbolic bath <cit.>.
Another intriguing
prospect is to work in the weak-coupling regime where the emitter plays a perturbative role and to devise sensing protocols by which the emitter dynamics, or that of two entangled emitters, can be used to probe topological matter.
Acknowledgement:
Support by the National Science Foundation through grant numbers PHY-2110158 and PHY-1950235 (REU/RET) is gratefully acknowledged.
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§ DARK STATES
This appendix determines the number of dark states, i.e., the number of eigen states of
Ĥ_Ck-2LE
that have vanishing energy.
The cavity-shared
k-chain
system
in the single-excitation sub-space is spanned by a total of
(kN-k+1)+(kN)+1=2kN-k+2 basis states.
It proves useful to reorder the site basis states as follows:
the
basis states
1,⋯, kN-k+1 are of type
|n,1;g⟩;
the basis states kN-k+2,⋯,2kN-k+1
are of type
|n,2;g⟩
(there are kN basis states of this type);
and
the
basis state
2kN-k+2 is equal to
|vac;e⟩.
With this ordering,
the Hamiltonian matrix
H_Ck-2LE
has the simple block structure
H_Ck-2LE
=
(
[ O V; V^† P ]),
where O, P, and V are matrices
of size (kN-k+1) × (kN-k+1), (kN+1) × (kN+1),
and
(kN+1)× (kN-k+1), respectively.
The matrix elements of O are all equal to zero since
Ĥ_Ck-2LE
does not couple basis states |n,1;g⟩ and |n',1;g⟩
for either n=n' or n n'.
The square matrix P arises from combining the states
|n,2;g⟩ and |vac;e⟩.
The matrix elements of P are all equal to zero, except for the element
P_2kN-k+2,2kN-k+2, which is equal to ħω_e.
A key point is that the basis states are ordered such that the matrix V accounts for all “couplings", i.e.,
the Hamiltonian terms proportional to u, v, v_N, and g are included in V.
Note in particular that, since the emitter is not coupled to sub-lattice 2 but to sub-lattice 1, the coupling constant g does not enter into P (i.e., P contains at most one non-zero entry).
To proceed, we first consider the special case where the detuning ħω_e is equal to zero. In this case,
all matrix elements of P are equal to zero.
Applying the results from the Appendix of Ref. <cit.>, it follows that
the number of dark states is given by the difference, in magnitude, between the number of rows and the number of columns of V.
The cavity Hamiltonian
Ĥ_Ck-2LE
hence supports k dark states for ħω_e=0.
For ħω_e 0, one of the dark states turns “bright", i.e., the energy
of this state is pushed away from zero; this behavior is clearly visible in the top row of Fig. <ref>.
Mathematically, the disappearance of the dark state follows since the right lower matrix element of P takes on a finite value.
To summarize, the cavity-coupled
k-chain
systems
with ħω_e 0 support k-1 dark states (see Fig. <ref>).
The arguments for determining the number of dark states for the emitter-shared
k-chain
systems proceed analogously.
Keeping the same grouping of the basis states,
the matrices
O, P, and V are of size (kN)×(kN), (kN+1) × (kN+1),
and
(kN+1)× (kN), respectively.
The key difference compared to the cavity-shared systems is that the number of basis states of type |n,1;g⟩ is kN in the emitter-shared system as opposed to kN-k+1.
Applying the results
from the Appendix of Ref. <cit.>, it follows that the emitter-shared
k-chain
systems support exactly one dark state if and only if ħω_e=0.
Figure <ref> indicates the absence of dark states since it summarizes the more
general
ħω_e 0 case (the special ħω_e=0 case is referred to in the caption).
§ 3-STATE MODEL FOR GAP STATES OF
1-CHAIN
SYSTEM
Section <ref>
discusses
selected properties of the 3-state Hamiltonian
H_3-st.(G).
This appendix presents analytical expressions for three special cases.
The eigen values of H_3-st.(G) can be obtained by solving the cubic
equation
λ^3 - ħω_e λ^2 - [ 2 G^2 +
(E_edge)^2 ] λ +
ħω_e (E_edge)^2=0.
Special case 1: When the detuning vanishes,
the eigen energies are equal to
λ|_ħω_e=0 =0
and
λ|_ħω_e=0=
±[ 2 G^2 +
(E_edge)^2 ]^1/2.
This shows that the splitting between the energetically lowest- and highest-lying gap states is approximately equal to
2|E_edge| and 2 √(2) G when
2 G^2 ≪
(E_edge)^2
and
2 G^2 ≫
(E_edge)^2, respectively.
These inequalities suggest that a “transition" occurs when
√(2) G is comparable to |E_edge|.
The unnormalized zero-energy eigen state reads (G/E_edge,-G/E_edge,1).
Special case 2:
Figure <ref> shows that the energy levels undergo two separate avoided crossings
when g/u is small [Figs. <ref>(a) and <ref>(b)]. When g/u is comparatively large [Fig. <ref>(d)],
in contrast, the two avoided crossings can no longer
be treated separately.
To identify the energy scale at which the crossings start to overlap, we
consider
a 2-state model, which removes the first row and first column from
H_3-st.(G).
The eigen energies λ_2-st. of the 2-state
model are given by
λ_2-st. =
ħω_e + E_edge/2±√((
ħω_e - E_edge/2)^2
+ G^2 )
and the energy splitting at the avoided crossing is equal to
2 G. Since the energies of
these two states have the same magnitude but opposite sign,
it is readily argued from this splitting that the two avoided crossings can no longer be treated separately if
2 G approaches |E_edge|. As expected, the
“competition scale"
obtained via the 2-state model is similar to that obtained from the 3-state model.
For the data shown in Fig. <ref>, the
competition
scale is reached roughly when g/u is equal to 10^-2, consistent with
what is concluded by visual inspection.
Special case 3:
To gain additional insight into the larger g/u regime [Fig. <ref>(d)],
we return to the 3-state model and consider the limit where |E_edge| is much smaller than
G. Setting E_edge=0,
the eigen values of H_3-st.(G) are
λ |_E_edge=0=0
and
λ |_E_edge=0=
ħω_e/2±√((
ħω_e/2)^2
+ 2 G^2
).
The eigen state corresponding to λ |_E_edge=0=0
is
equal to
(|ψ_+^C1⟩ - |ψ_-^C1⟩ )|g⟩/√(2)=|ψ_edge,R⟩|g⟩, i.e., this eigen state has non-vanishing amplitude only in one side of the
chain
and only in sub-lattice 2
(see Sec. <ref> for further discussion).
Altogether, the analysis outlined in this appendix
shows that the emitter acts as a perturbation
when g/u is much smaller than about |E_edge/(u c_n^*,1)|.
The state |vac;e⟩ hydridizes with the g=0 edge states (emitter in |g⟩) when the effective coupling G is comparable to |E_edge|. Where the transition occurs
can be tuned
by increasing g/u for fixed N or by increasing N for fixed g/u.
We note that the unit cell n^* at which the emitter is placed can also be used as a tuning knob.
We now present approximate analytical expressions for the IPR. We start with the photonic Hamiltonian (excluding the emitter Hilbert space).
The IPR for the
states |ψ_±^C1⟩
reads
IPR_|ψ_±^C1⟩=
N^2 [ 1+(u/v)^2N]/2 +2(
u/v)^2,
which can be simplified to
IPR_|ψ_±^C1⟩≈1-( u/v)^2/2+2( u/v)^2.
For u/v=2 and N=15, e.g., Eq. (<ref>)
evaluates to
3/10, which deviates from Eq. (<ref>) by less than 6 × 10^-10.
To obtain approximate analytical expressions for the IPR for the states
|ψ_l^gap⟩
(1-chain
system with finite g), we write
|ψ_l^gap⟩= d_+^(l) | ψ_+^C1⟩ |g⟩+
d_-^(l) | ψ_-^C1⟩ |g⟩
+
d_e^(l) | vac,e⟩,
where the expansion coefficients
d_+^(l), d_-^(l),
and d_e^(l)
are extracted from the eigen vectors of the 3-state model.
Evaluating the IPR for the states given in Eq. (<ref>), we find
IPR_|ψ_l^gap⟩
=
IPR_|ψ_±^C1⟩(
|d_+^(l)|^4 +
|d_-^(l)|^4
+
6 |d_+^(l)|^2 |d_-^(l)|^2
)+
|d_e^(l)|^4.
§ EMITTER-SHARED 2- AND
3-CHAIN
SYSTEMS
To construct a few-state model that describes the g-dependence of the energy levels that lie in the middle of the gap for k>1, we first consider the emitter-shared
2-chain
system. We introduce approximate expressions for the g=0 eigen states with energy close to zero.
Since the two
chains
are decoupled for g=0, the system supports two eigen states with energy E_edge
and two eigen states with energy -E_edge,
|ψ_±^C1⟩ | vac,C2 ⟩ |g ⟩
and
| vac,C1 ⟩ |ψ_±^C2⟩ |g ⟩,
where |vac,Ck ⟩
refers to the vacuum state of
chain
k and where the state |ψ_±^C2⟩ is defined analogously to |ψ_±^C1⟩ (see
Sec. <ref>).
To construct a few-state model, we form linear combinations of the two states that have energy E_edge as well as linear combinations of the two states that have energy -E_edge:
1/√(2)(
|ψ_+^C1⟩ | vac,C2 ⟩±
| vac,C1 ⟩ |ψ_+^C2⟩) |g ⟩
and
1/√(2)(
|ψ_-^C1⟩ | vac,C2 ⟩±
| vac,C1 ⟩ |ψ_-^C2⟩) |g ⟩.
The “+"-linear combinations couple to the state |vac,C1⟩ | vac,C2⟩|e⟩
while the “-"-linear combinations do not.
Correspondingly, we consider a 3-state model that is spanned by the two “+"-linear combinations and |vac,C1⟩ | vac,C2⟩|e⟩.
Calculating the coupling matrix elements, we find that the coupling strength is √(2)-times larger than that of the
1-chain
system, i.e., the 3-state Hamiltonian is given by
H_3-st.(√(2)G) [Eq. (<ref>) with G replaced by √(2)G].
The emitter-shared
3-chain
system supports seven states with energy close to zero. Forming appropriate linear combinations, we find that only three of these are shifted when g is turned on. Thus, we construct a 3-state model spanned by the states
1/√(3)(
|ψ_+^C1⟩
| vac,C2⟩
| vac,C3⟩
+
| vac,C1⟩|ψ_+^C2⟩
| vac,C3⟩
+
| vac,C1⟩| vac,C2⟩
|ψ_+^C3⟩)
|g⟩,
1/√(3)
(
|ψ_-^C1⟩
| vac,C2⟩
| vac,C3⟩
+
| vac,C1 ⟩|ψ_-^C2⟩
| vac,C3⟩
+
| vac,C1⟩| vac,C2⟩
|ψ_-^C3⟩
)
|g⟩,
and
| vac,C1⟩| vac,C2⟩| vac,C3⟩|e⟩.
The
3-state Hamiltonian for the emitter-shared
3-chain
systems is given by
H_3-st.(√(3)G) [Eq. (<ref>) with G replaced by √(3)G]. For the emitter-shared
k-chain
system, the effective coupling energy is √(k)G.
As in the
1-chain system, we can—analogously to Eq. (<ref>)—write the gap states |ψ_l^gap⟩ for the emitter-shared
k-chain
systems as a superposition of the three states that span the 3-state Hamiltonian.
Evaluating the IPR within the 3-state model, we find
IPR_|ψ_l^gap⟩
=
IPR_|ψ_±^C1⟩/j(
|d_+^(l)|^4 +
|d_-^(l)|^4
+
6 |d_+^(l)|^2 |d_-^(l)|^2
)+
|d_e^(l)|^4,
where d_+^(l), d_-^(l), and d_e^(l) are obtained from the eigen vectors of the 3-state Hamiltonian.
§ CAVITY-SHARED 2- AND
3-CHAIN
SYSTEMS
For the
cavity-shared
k-chain
systems, the g=0 eigen states with eigen energy close to zero
fall into three groups.
The first group contains, for the
2-chain system (3-chain
system), two (four) states with
energies that are finite but different from ± E_edge
and
that are, to a very good approximation, not affected when the coupling g is turned on.
The second group contains one (two) non-topological dark states (see Appendix <ref>).
The third group contains three states
with energies -E_edge,
E_edge, and ħω_e that couple to the emitter when g is non-zero.
The states with energies ± E_edge are essentially identical to those introduced in Appendix <ref>, with the exception that there only exists one basis state
|n^*,1⟩
as opposed to k basis states
|n^*,1; Ck⟩.
Using these two states together with |vac;e⟩, we find that
the
3-state
Hamiltonian
matrix
for the cavity-shared systems is identical to
that for the 1-chain systems
but with reduced coupling constant [Eq. (<ref>) with G replaced by G/√(k)].
The reduction of the coupling energy compared to the
1-chain
and emitter-shared systems is due to the fact that the
cavity that the emitter is coupled to is shared among all
chains.
|
http://arxiv.org/abs/2307.04812v1 | 20230710180255 | Probing single electrons across 300 mm spin qubit wafers | [
"Samuel Neyens",
"Otto Zietz",
"Thomas Watson",
"Florian Luthi",
"Aditi Nethwewala",
"Hubert George",
"Eric Henry",
"Andrew Wagner",
"Mohammad Islam",
"Ravi Pillarisetty",
"Roza Kotlyar",
"Kent Millard",
"Stefano Pellerano",
"Nathan Bishop",
"Stephanie Bojarski",
"Jeanette Roberts",
"James S. Clarke"
] | quant-ph | [
"quant-ph",
"cond-mat.mes-hall"
] |
Intel Corp., 2501 NE Century Blvd, Hillsboro, OR 97124, USA
*These authors contributed equally to this work
†Corresponding authors: [email protected]; [email protected]
Building a fault-tolerant quantum computer will require vast numbers of physical qubits. For qubit technologies based on solid state electronic devices <cit.>, integrating millions of qubits in a single processor will require device fabrication to reach a scale comparable to that of the modern CMOS industry. Equally importantly, the scale of cryogenic device testing must keep pace to enable efficient device screening and to improve statistical metrics like qubit yield and process variation. Spin qubits <cit.> have shown impressive control fidelities <cit.> but have historically been challenged by yield and process variation. In this work, we present a testing process using a cryogenic 300 mm wafer prober <cit.> to collect high-volume data on the performance of industry-manufactured spin qubit devices at 1.6 K. This testing method provides fast feedback to enable optimization of the CMOS-compatible fabrication process, leading to high yield and low process variation. Using this system, we automate measurements of the operating point of spin qubits and probe the transitions of single electrons across full wafers. We analyze the random variation in single-electron operating voltages and find that this fabrication process leads to low levels of disorder at the 300 mm scale. Together these results demonstrate the advances that can be achieved through the application of CMOS industry techniques to the fabrication and measurement of spin qubits.
Probing single electrons across 300 mm spin qubit wafers
James S. Clarke†
August 12, 2023
========================================================
Silicon quantum dot spin qubits <cit.> have recently demonstrated single- and two-qubit fidelities well above 99% <cit.>, satisfying thresholds for error correction <cit.>. Today, integrated spin qubit arrays have reached sizes of six quantum dots <cit.> with larger quantum dot platforms in 1D <cit.> and 2D <cit.> configurations also being demonstrated. To realize practical applications with spin qubit technology, physical qubit count will need to be increased dramatically <cit.>. This will require fabricating spin qubit devices with a density, volume, and uniformity comparable to those of classical computing chips, which today contain billions of transistors. The spin qubit technology has inherent advantages for scaling due to the qubit size (∼100 nm), as well as, in the case of Si-based devices, a native compatibility with complementary metal-oxide-semiconductor (CMOS) manufacturing infrastructure. It has therefore been posited that manufacturing spin qubit devices with the same infrastructure as classical computing chips can unlock spin qubits' potential for scaling and provide a path to building fault-tolerant quantum computers with the technology.
The scaling of classical chips according to Moore's Law has depended on significant advancements in process variation <cit.> as well as density and speed. For spin qubits today, process variation and yield are significant challenges. It has not yet been clearly shown that CMOS manufacturing infrastructure can bring the same improvements to variation and yield of quantum devices as have been made for classical devices. Spin qubits have been made with hybrid fabrication flows, where industry-standard techniques are interleaved with research techniques such as e-beam lithography and/or liftoff <cit.>. More fully industry-compatible devices in Si-MOS have also been demonstrated <cit.> but are currently limited by high levels of disorder due to the qubits being formed directly at the Si/SiO_2 interface. Spin qubits hosted in epitaxial group-IV heterostructures offer reduced disorder <cit.> but are less straightforward to integrate in an industry process, due to the 300 mm SiGe epitaxy and reduced thermal budget compared to CMOS.
In addition to fabrication challenges, the bottleneck of cryogenic electrical testing presents a barrier to scaling any solid state quantum technology, from spin qubits to superconducting <cit.> and topological <cit.> qubits. To improve process variation and yield in quantum devices, process changes must be combined with statistical measurements. This requires wafer-scale datasets of device performance measured at low temperature. Traditional test systems that cool down one device at a time introduce significant overhead through dicing, die attaching, bonding, and thermal cycling devices. This overhead limits the number of devices per wafer that can be tested to sample wafer-scale trends. One solution is device multiplexing, using either on-chip <cit.> or off-chip <cit.> circuitry to increase the sample capacity of a cryostat. Both approaches come with limitations. With off-chip multiplexing, the packaging time is still linear in the number of devices; with on-chip multiplexing, the area of the wafer being sampled is limited to a single die. By contrast, the standard technique in the semiconductor test industry is full wafer probing. This approach provides maximal flexibility, as all devices on the wafer are simultaneously accessible for electrical measurement. For quantum devices, wafer-scale probing requires additional cooling hardware to reach the required temperatures. For spin qubits based on Si/SiGe quantum dots, accessing the single electron operating regime typically requires temperatures ≲4 K. Only recently has wafer probing at such low temperatures become possible.
In this work we present two advancements. First, we develop a 300 mm cryogenic probing process to collect high volume data on spin qubit devices across full wafers. Second, we optimize an industry-compatible process to fabricate spin qubit devices on Si/SiGe heterostructures, combining low process variation with a low disorder host material. These two advancements are mutually reinforcing: the development of full-wafer cryogenic test capabilities enables the optimization of the complex 300 mm fabrication process, and the optimization of the fabrication process improves device reliability to enable significantly deeper automated measurements across wafers. As we will show, together these culminate in the automated probing of single electrons in spin qubit arrays across 300 mm wafers.
The spin qubit devices studied here are fabricated in Intel's D1 factory where the company's CMOS logic processes are developed. The host material is a Si/Si_0.7Ge_0.3 heterostructure <cit.> grown on 300 mm Si wafers. Fig. <ref>a shows an optical image of a completed spin qubit wafer. The quantum dots are defined by a planar architecture with two gate layers, one passive layer for screening/depletion and one active layer for controlled accumulation <cit.>. All patterning is done with optical lithography. The quantum dot gate patterning is done in a single pass with extreme ultraviolet (EUV) lithography, allowing us to explore gate pitches from 50-100 nm. The fabrication of all device sub-components is based on fundamental industry techniques of deposition, etch, and chemical-mechanical polish <cit.>. As we will demonstrate, this approach leads to high yield and low process variation across the 300 mm wafer.
The cryogenic wafer prober (cryo-prober) we use <cit.> was manufactured by Bluefors and AEM Afore and was developed in collaboration with Intel. The cryo-prober can cool 300 mm wafers to a base temperature of 1.0 K at the chuck and an electron temperature of 1.6 ± 0.2 K (see Extended Data Fig. <ref>) in ∼2 hrs. Fig. <ref> shows an overview of the wafer measurement process. After cooldown, thousands of spin qubit arrays and test structures on the wafer are available for measurement. An individual device is aligned to the probe pins using the wafer stage control and a machine vision algorithm. The wafer is brought into contact with the probe pins to electrically connect device pads to voltage sources and current and voltage detectors at room temperature. Measurements are taken with these instruments to extract a variety of metrics. These measurements are repeated on many devices across a wafer to generate wafer-scale statistics. The entire process, from alignment to device measurement, is fully automated and programmable, speeding up device data collection by several orders of magnitude compared to the measurement of singular devices in a cryostat.
The mask set used here produces many different device types on each wafer, including fully integrated spin qubit arrays and test structures. These test structures are designed to emulate sub-components of the complete devices and aid in both troubleshooting and targeting specific processes within the fabrication flow. All structures have the same pad design to match the probe pin array, allowing many different structures to be measured in situ. Switching among device types simply requires changes in software or minor changes at the electronics rack. Fig. <ref>a-c shows examples from the range of devices we test with the cryo-prober. These include gate line resistance test structures, Hall bar structures, and spin qubit arrays containing 3 to 12 quantum dots. For each case, the active device pads are highlighted and schematics of the measurement configuration are shown in Fig. <ref>a-c. The performance of all these structures is improved through process optimization, guided by feedback from the cryo-prober.
Improvements in gate line resistance across multiple wafers are shown in Fig. <ref>d. The DC gate line resistance, including both gate and interconnect layer, is an important factor in RF signal delivery during qubit control. Here gate line resistance is reduced through optimization of the gate fabrication process with normal-conducting materials and through the introduction of superconducting materials to the stack. Validating the superconducting process in particular is made possible by the 1.6 K base temperature of the cryo-prober.
Carrier mobility is another important metric for spin qubits. In the case of Si/SiGe devices, mobility is a direct measure of the quality of the Si quantum well where qubits are defined and provides a target for optimizing the heterostructure growth recipe. While a magnetic field is needed to measure mobility most accurately, we can generate a reasonable estimate to compare the quantum well quality of different wafers (see Methods for details). Estimated carrier mobility across multiple wafers is shown in Fig. <ref>e. These measurements show a significant increase in the median mobility with a change in the epitaxial growth process designed to reduce defect density. We also observe a similar mobility distribution before and after isotopic purification of the quantum well to ^28Si, confirming epitaxial quality is maintained with the purified growth precursor.
For quantum dot spin qubit arrays, process optimization involves many factors, including gross yield, quantum dot confinement, device stability, and voltage variation. To optimize these factors, we iterate through a wide variety of changes to the fabrication flow, including but not limited to fixed charge in the gate stack, thermal budget, etch impacts, and the integration of a screening gate layer. Through all these changes, a simple but useful metric for wafer quality is the cross-wafer spread in threshold voltage (), the voltage required to turn on and off current with a particular gate. Fig. <ref>f shows distributions for 15 wafers, highlighting three versions of the device stack: two intermediate stacks and the optimized stack. For each stack, ∼4,000 data points are shown. Before the process is optimized, distributions show large spread both within and between wafers. By comparison, the optimized stack shows tight distributions that are consistent from wafer to wafer. Additionally, quantum dot confinement can be characterized qualitatively through collection of “barrier-barrier scans,” a 2D sweep of the barrier gate voltages that define each quantum dot. These scans reveal the point of low tunnel coupling to source and drain where Coulomb blockade can occur <cit.>. Fig. <ref>g shows examples of these measurements from each of the three stacks featured in Fig. <ref>f. The intermediate stacks show significant disorder and/or instability in these measurements. By comparison, the optimized stack shows clean confinement with the barrier gates and stable current throughout the length of the scan.
After process optimization, we characterize the optimized process flow with measurements on 12-quantum-dot (12QD) devices. Measurements are again fully automated to maximize the speed and consistency of data collection (see Methods). The 12QD design is comprised of a linear array of twelve quantum dots with four opposing sensor dots isolated by a center screening gate. An in-line SEM image of this device with a schematic of the measurement configuration is shown in Fig. <ref>b. Quantum dots on both the qubit side and the sensor side are defined by three gates each: one plunger gate to control the electron number on the dot, and one barrier gate on each side to tune the tunnel coupling to the neighboring dot or charge reservoir. The array of twelve quantum dots can be operated as qubits in a variety of spin encodings, including single spin qubits <cit.> (in a 12-qubit array) or exchange-only qubits <cit.> (in a 4-qubit array). Depending on the spin qubit encoding, an optional micromagnet layer can be added to the device and the center screening gate can supply microwave electric fields to control the qubits with electron dipole spin resonance.
As in a CMOS logic process, improving qubit yield is a necessary part of scaling up quantum processors, as larger systems will depend on an increasing number of qubit components to function. To analyze the yield of this fabrication flow, we test 232 12QD devices on a wafer. These tests cover a map of 58 die across the wafer and include four nominally identical devices per die. We exclude the outer-most ring of die at the edge of the wafer as these are not targeted in all steps of fabrication. We calculate component yield for ohmic contacts, gates, quantum dots, and full 12QD devices. These yield metrics are summarized in Table <ref>. Both ohmic contact and gate yield are 100%. The large number of gates tested and working on this wafer (>10,000) highlights the consistency of the gate fabrication process. Quantum dot yield is 99.8%, which further emphasizes the reliability of electrostatic gate control. Lastly, the full device yield, including the linear array of 12 quantum dots and the 4 charge sensors, is 96%. (See Methods for more details.)
Fig. <ref>c shows a summary of gate values collected on 12QD devices across a wafer. The distributions are highly consistent across the 25-gate array. We also observe a systematic shift in median for the two outer-most gates in the array. The symmetry of this effect suggests it is electrostatic in nature, due to the proximity of the reservoir gates. While trends like this might be difficult to confirm through one-off device testing, they are readily observable with full-wafer statistics. The gate distributions also contain information on process variation. The standard deviation of for the 25 plunger and barrier gates ranges from 63 to 89 mV across the wafer. Standard deviation incorporates all causes of cross-wafer variation, including both random effects and systematic cross-wafer phenomena arising from processes like deposition and etch. To estimate the random variation in across gates and devices, we follow a standard CMOS industry method of analyzing matched pair differences <cit.>, calculated between mirror-symmetric pairs of gates. We subtract the mean from each gate-pair distribution to center them at zero and merge them into one distribution. The resulting distribution represents the random variation due to local contributions, factoring out systematic effects such as local geometry or cross-wafer processing phenomena. The resulting matched pair Δ distribution is plotted in Fig. <ref>d. The standard deviation of this distribution, reduced by a factor of √(2), is 58 mV, and represents the random component of variation between gates due to local contributions.
The measurements presented so far are all taken in the transport regime, where devices are operated as 1D transistors or many-electron quantum dots. Operating a device as a spin qubit processor requires tuning the electron occupancy to one electron (typically) per quantum dot. Accessing this regime can be challenging even for devices that perform well in the transport regime, since atomistic disorder that may be screened at many-electron occupation is laid bare at single-electron occupation. Confirming that devices can reliably reach this spin qubit operating point is therefore a crucial test of a spin qubit fabrication process. To characterize the single electron regime of these devices, we perform automated charge sensing measurements with each of the twelve quantum dots in the linear array. In each measurement, one quantum dot is tuned up on the qubit side and one on the sensor side. Changes in electron number are detected by modulating the voltage on an exterior screening gate and using lock-in detection of the charge sensor current at that frequency. A typical measurement is shown in Fig. <ref>a. In this 2D sweep, the horizontal axis is plunger voltage, and the vertical axis is the voltage of both barrier gates <cit.>. The sweep range is chosen to take each quantum dot from zero-electron to several-electron occupation along the plunger axis and from low tunnel rate (≪1 kHz) to high tunnel rate (≫1 GHz) along the barrier axis. Transition lines disappear at the bottom of the scan window where tunnel rate falls below the lock-in frequency (∼1 kHz) and at the top of the scan window where the lines become broadened by tunnel coupling energy.
Charge sensing scans are taken for all 12 quantum dot sites in the linear array, across 58 die on the wafer, for a total of 696 quantum dot sites. The “success” of each charge sensing scan depends on multiple factors: the relevant sensor dot must yield, the sensing signal must be high relative to noise, and the charge sensor must remain stable throughout the length of the scan. Over the 696 scans taken on a wafer with 50 nm SiGe barrier, we find a 91% success rate in observing clear transitions (as gauged by eye). This success rate represents highly consistent device performance and is primarily limited by the measurement algorithm. We expect the charge sensing success rate can be improved by reducing electron temperature to reduce the low-frequency charge noise <cit.> that gives rise to charge sensor shifts. Improvements could also come from incorporating active feedback into the measurement loop to analyze data quality <cit.> and re-take measurements after charge sensor shifts occur.
For further analysis on the 91% of successful scans on this wafer, we apply a numerical algorithm to detect transition curves in the 2D data and extract the coordinates for the first electron (1e) transition (see Methods). We define the “1e voltage” as the plunger voltage position of the 1e transition at the midpoint of the barrier voltage axis, indicated by the red star in Fig. <ref>a. We use the distance between the transition voltage and the left edge of the scan window to gain high confidence that these transitions represent the first electron in the quantum dot (see Methods).
A summary of plunger and barrier voltages at the 1e transition is shown in Fig. <ref>b. These data represent the voltages needed to set the 1e charge state in individual sites of 12QD arrays, sampled across a 300 mm wafer. They therefore can reveal how process variation translates to variation in the spin qubit operating point. Improving variation in spin qubit operating voltage has multiple benefits. Lower 1e voltage variation makes for easier automation, as operating voltages are more predictable. Also, many proposals for large-scale spin qubit processors rely on sharing voltages among spin qubit lines to alleviate the interconnect bottleneck <cit.>. Such voltage-sharing schemes will require extremely low levels of variation in 1e voltages across large arrays. In the same way threshold voltage variation must be reduced in a transistor process, variation at the single electron regime must be improved to enable the grandest visions for spin qubit scaling.
To analyze the variation in 1e transition voltage data, we repeat the same matched pair voltage difference analysis as above, taking differences between 1e voltages for mirrored pairs of plunger gates. The resulting distributions of voltage differences are shown in Fig. <ref>c-d for two wafers. The random variation in 1e voltage extracted from wafers with a 30 nm and 50 nm SiGe barrier are 59 mV and 60 mV, respectively. Both of these values closely agree with the random variation in gate , meaning the random variation of a transistor-like metric (gate ) is matched by the random variation of a quantum metric (1e voltage). This implies that these devices are not subject to significantly increased disorder at the single electron regime compared with the many electron regime. Also, while the 1e voltage variation is nearly the same between the two wafers, the variation in chemical potential is better reflected by the ratios between 1e voltage variation and 1e-2e addition voltage (Fig. <ref>e-f). These ratios are 1.0 ± 0.1 and 0.76 ± 0.08 for the 30 nm and 50 nm barrier wafer, respectively. The observation that the wafer with a deeper quantum well has a reduced ratio of this kind suggests that the 1e voltage variation is dominated by sources in the gate stack above the heterostructure. These sources could include charge defects (e.g., interface traps or fixed charge in the oxide), gate line edge roughness, gate work function variation, oxide thickness variation, or some combination. These possible sources of variation all have analogies in the transistor field and could be improved by borrowing similar strategies; for example, the impact of oxide charge defects could be reduced by decreasing the oxide thickness between the heterostructure and the gate <cit.>.
The charge sensing data can also be used to benchmark the compatibility of these devices with voltage-sharing protocols <cit.>. One basic requirement for such schemes could be that all quantum dots in an array be tuned to the same electron number using the same voltage. From the 1e and 2e voltages obtained here, we estimate that a median of 63% of quantum dots per 12QD device could be set to n=1e with a common voltage. (See Methods for more detail and Extended Data Fig. <ref>.) While this result is still far from the level of uniformity needed to tune an ensemble of spin qubits to their operating point with shared voltages, the 1e voltage variation results in Fig. <ref> highlight the device metrics that must be further improved in order for voltage sharing protocols to be feasible in large spin qubit processors.
To further assess variation at the single electron regime, we calculate the standard deviation of the difference between plunger and barrier voltages at the cutoff point of the 1e transition line <cit.>. Fig. <ref>g-h shows the distribution of this voltage difference across all gates and all devices tested on the wafer. We again compare datasets from two wafers, with a 30 nm and 50 nm SiGe barrier, respectively. The distributions for the two wafers have different means due to their different geometry, but their standard deviations are in close agreement at 0.12 (0.13) V for the 30 (50) nm barrier wafer. This standard deviation agrees with the values reported in Ref. <cit.> for six-dot devices with high exchange qubit fidelity <cit.>, confirming that the devices studied here can achieve low levels of disorder at the single electron regime while being fabricated in high volume with a 300 mm process.
In conclusion, these results demonstrate a spin qubit fabrication process based on a low disorder host material (Si/SiGe) and all CMOS industry-compatible techniques to achieve low process variation. We present a novel measurement system, a 300 mm cryo-prober at 1.6 K, as a solution to the bottleneck of low-temperature quantum device testing. We use this system to characterize a variety of device types including test structures and fully integrated spin qubit arrays. We demonstrate charge sensing at the single electron regime across full wafers, directly probing the operating point of spin qubits and extracting statistics to characterize process variation at this regime. We observe that, for these devices, variation at the single-electron regime closely agrees with standard transistor variation metrics, pointing the way to strategies that could reduce variation in spin qubit control parameters. While these measurements do not yet directly characterize qubit performance, they test the basic electrostatics framework on which any spin qubit encoding relies. This baseline level of performance is a necessary but not sufficient condition for successful spin qubit implementation, yet achieving it in any device has been historically challenging across the spin qubit field. By leveraging the tools and techniques of the CMOS industry, we achieve a major leap forward in the yield and variation of spin qubit electrostatics, produced at industry scale and characterized with a high-volume measurement system to match. These results set a new standard for what can be achieved with spin qubit devices today and pave the way for significantly larger and more complex spin qubit arrays of the future.
§ METHODS
§.§ Electron temperature measurement
Electron temperature in the cryo-prober is measured from a charge stability diagram, using a transition line that is tuned to avoid tunnel rate broadening. This stability diagram is shown in Extended Data Fig. <ref>a. A 1D measurement of the transition line is then taken to extract the width of the transition line. The lock-in data is integrated with respect to swept voltage and subtracted by a linear background. The resulting data is then fit to the model for a temperature-broadened charge sensor transition <cit.> to extract an electron temperature of 1.6 ± 0.2 K. The processed data and theoretical fit are shown in Fig. <ref>b. The uncertainty is estimated from the uncertainty of the lever arm (0.08 ± 0.01), which is measured from bias triangles.
§.§ Carrier mobility estimation
Carrier mobility is estimated from measurements of channel resistance in 4-probe Hall bar devices at zero magnetic field. The mobility calculation depends on knowing the carrier density, so we approximate a fixed carrier density (4×10^11 cm^2/Vs) by measuring the device and setting the gate voltage to V_T+Δ V where Δ V = eΔ n/c_g, e is the electron charge, Δ n is the approximated carrier density, and c_g is the estimated gate capacitance per area based on the gate stack. While the gate capacitance estimate is inexact, all the mobility estimations shown in Fig. <ref> are from wafers with nominally the same gate stack. Additional uncertainty comes from the unknown percolation density (n_p) at which the device first shows current. This leads to a systematic over-estimate of mobility by a factor of (1+n_p/Δ n), which we estimate to be at most ∼30%. While this is significantly less accurate than measurements made with magnetic field control, it is nevertheless a useful method for observing wafer-scale trends and comparing wafers with different heterostructure details and gate stack parameters held fixed. We note that all wafers contain a fraction of devices (10-20%) with significantly reduced mobility, as can be seen in Fig. <ref>e. This statistical phenomenon is confirmed with conventional Hall measurements and is not an artifact of the measurement method. Since a similar phenomenon is not observed in the quantum dot devices (manifesting in, e.g., anomalously high channel resistance), we attribute this to a discrete defect mode of the larger-area Hall bar devices.
§.§ Yield analysis
The component yield analysis present in Table <ref> uses the following definitions. Ohmic contact yield is defined as the fraction of contacts through which current in the Si quantum well can be linearly controlled. Gate yield is defined as the fraction of gates that can be used to turn on and pinch off their respective current channel. Quantum dot yield is defined as the fraction of quantum dot sites where a viable quantum dot tune-up point can be identified from barrier-barrier scans. Lastly, full device yield is defined as the fraction of devices where all sub-components (all ohmic contacts, gates, and quantum dots) yield.
Out of 3,712 quantum dot sites tested and summarized in Table <ref>, the nine that fail to tune up are also observed to have anomalously low pinch-off voltage (<0.2 V) on at least one of the three gates defining that quantum dot. These nine sites are also confined to the charge sensor side, where gate geometry is most complex. This indicates that this small number of non-yielding quantum dots is due to the processing of the 0.3% most marginal gates as opposed to, e.g., quantum well defects. We interpret these edge cases on the charge sensor side to a known failure mode in the gate lithography process. We note that the paths to improving the robustness of this process to fix these extreme outlier cases are well understood.
§.§ Automated device measurements
After a device is contacted with the probes, each current channel in the device (including the qubit channel and the four charge sensor channels) is turned on with all gates over that channel at the same voltage. Once each channel's is recorded, the gates of each channel are set to a fixed voltage relative to the channel . The qubit channel is then isolated from the sensor channels by reducing the center screening gate voltage until the cross-conductance between channels drops to zero (within the noise floor). The voltage of individual gates is then fine-tuned to set a roughly uniform carrier density across the channel. This is done through an iterative process where the transconductance of each gate is sampled and the voltage on that gate is increased (decreased) if the transconductance is above (below) a threshold value. This effectively sets the voltages of all gates so they are at roughly the same point on their pinch-off curves relative to their . The data for all gates are extracted from pinch-off curves taken with a source-drain bias of 1 mV. is identified as the voltage where current crosses 1 nA. The voltages needed to tune up a quantum dot at each site are identified by setting each plunger gate to a fixed voltage relative to its and varying the barrier gate voltages about their individual values in a 2D sweep (a barrier-barrier scan). A phenomenological 2D function is fitted to this data to extract the corner point, which combined with the plunger voltage is used to define the “tune-up” parameters for the quantum dot site.
The charge sensing measurements shown in Fig. <ref> are taken with one quantum dot tuned up on the qubit side. The closest charge sensor to that quantum dot is also tuned up, and neighboring charge sensor dots are pinched off with their respective plunger gates. To generate the charge sensing measurement, the plunger voltage is swept at a fixed range relative to its , and the two barrier gate voltages are stepped simultaneously. The barrier gates are stepped over the same voltage interval but with separate voltage values. The step values of each barrier gate are defined relative to that gate's individual “tune-up” voltage extracted from the barrier-barrier scan. In the example shown in Fig. <ref>a, the barrier voltage range displayed on the vertical axis is the voltage of the left barrier gate.
Charge sensing measurements can also be taken on double quantum dots. The three barrier gates that define each double quantum dot are first set to a fixed voltage relative to their individual values. The plunger gate voltages for each dot are then swept to generate a 2D charge stability diagram. While these scans are not analyzed quantitatively in this work, a demonstration of this type of measurement can be seen in Extended Data Fig. <ref>.
We note that the overall device measurement rate is predominately set by the speed of measurement hardware. Significant gains can therefore be made by implementing faster hardware (e.g., arbitrary waveform generators) and higher-bandwidth amplification (e.g., cryogenic amplifiers <cit.>) without any further changes to the tune-up procedure.
§.§ Charge sensing transition curve analysis
Transition line coordinates are extracted from charge sensing measurements using the following procedure. The raw lock-in amplifier data is first filtered with a first-order Gaussian filter to remove slowly-varying features. A maximum filter is then used to identify features of high signal in the pre-filtered data. An algorithm is then used to convert the set of “maximum points” into a set of “curve segments.” Curve segments are found by searching for groupings of maximum points that satisfy the following criteria: each point in the curve segment must be the closest maximum point to its nearest neighbor; the slope between each pair of neighboring points must be within a target window; and the set of points must span a minimum specified “length” in the vertical direction. Overlapping curve segments are then merged into transition curves. Transition curves are then further filtered to remove outlier curves and ordered by their coordinate means. The first and second transition curve generated from this algorithm are identified with the 1-electron and 2-electron transition, respectively. An example of the entire sequence is shown in Extended Data Fig. <ref>. The “1e (2e) voltage” is defined as the plunger voltage at which the 1e (2e) transition line crosses the midpoint of the barrier voltage axis. The 1e-2e addition voltage is calculated as the difference between these voltages. We note that in some cases (15%), the 1e (2e) transition in the scan window does not cross the midpoint of the barrier voltage axis, in which case no 1e (2e) transition voltage is extracted from that scan.
§.§ 1e transition validation
To validate that the 1e voltages we report are actually the first electron in the quantum dot, we extract the margin between the 1e transition voltage and the left edge of the scan window and compare it to the distribution of addition voltages between the 1e and 2e transitions. To have high confidence that the first transition represents the first electron, we require this “scan margin” be >2 times the typical addition voltage. For the 50 nm SiGe barrier wafer characterized in Fig. <ref>b, 98% of 1e voltage data points have a scan margin value above this threshold, giving us high confidence that the 1e transition data summarized in Fig. <ref>b is actually single-electron data. See Extended Data Fig. <ref> for histograms of the 1e-2e addition voltage and 1e scan margin data from this wafer.
§.§ Voltage sharing analysis
To estimate the proportion of quantum dots in each 12QD device that could be set to single-electron occupation with shared voltages, we analyze the 1e voltage and 2e voltage data from the 50 nm SiGe barrier wafer and search for a common voltage that best divides the 1e and 2e voltage distributions for each 12QD device. In this scheme, any 1e voltage value above the common voltage corresponds to n=0e, and any 2e voltage value below the common voltage corresponds to n≥2e. The remaining instances correspond to quantum dots tuned to n=1e. For each device, the optimal common voltage is found by minimizing the number of instances where n=0e or n≥2e. Extended Data Fig. <ref> shows a histogram of 1e and 2e voltage data points shifted relative to their assigned device-level common voltage. A scatter plot also shows the proportion of quantum dots in each category of electron number for all 12QD devices. We note that the data used in this analysis comes from measurements of quantum dots tuned one at a time and that this method does not take into account the individualized setpoints of other gates in the array during measurements. Nevertheless, we believe it gives a reasonable estimate of the success rate of using shared voltages across a device to set a common charge state.
§ DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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§ AUTHOR CONTRIBUTIONS
S. N., O. Z., and T. W. designed the automated measurements. S. N., O. Z., and A. N. performed the measurements. F. L. contributed to the measurement software. H. G., E. H., A. W., and M. I. fabricated the devices. S. N. and O. Z. analyzed the data. R. P., R. K., and S. P. contributed to the data analysis. O. Z., R. P., and K. M. enabled the cryo-prober installation. N. B., S. B., J. R., and J. S. C. supervised the project. S. N. and O. Z. wrote the manuscript with input from all authors.
§ EXTENDED DATA
|
http://arxiv.org/abs/2307.04085v1 | 20230709023446 | Vector Commitments with Efficient Updates | [
"Ertem Nusret Tas",
"Dan Boneh"
] | cs.CR | [
"cs.CR"
] |
Age of FGK Dwarfs Observed with LAMOST and GALAH: Considering the Oxygen Enhancement
Jinghua Zhang
Received August 12, 2023; accepted August 12, 2023
====================================================================================
Dynamic vector commitments that enable local updates of opening proofs
have applications ranging from verifiable databases with membership changes to stateless clients on blockchains.
In these applications, each user maintains a relevant subset of the committed messages and the corresponding opening proofs with the goal of ensuring a succinct global state.
When the messages are updated, users are given some global update information and update their opening proofs to match the new vector commitment.
We investigate the relation between the size of the update information and the runtime complexity needed to update an individual opening proof.
Existing vector commitment schemes require that either the information size or the runtime scale linearly in the number k of updated state elements.
We construct a vector commitment scheme that asymptotically achieves both length and runtime that is sublinear in k, namely k^ν and k^1-ν for any ν∈ (0,1).
We prove an information-theoretic lower bound on the relation between the update information size and runtime complexity that shows the asymptotic optimality of our scheme.
While in practice, the construction is not yet competitive with Verkle commitments,
our approach may point the way towards more performant vector commitments.
§ INTRODUCTION
A Vector Commitment (VC) scheme <cit.> enables a committer to succinctly commit to a vector of elements.
Later, the committer can generate an opening proof to prove that
a particular position in the committed vector is equal to a certain value.
VCs have found many applications in databases and blockchains <cit.> as they enable
a storage system to only store a commitment to the vector instead of the entire vector.
The data itself can be stored elsewhere along with opening proofs.
In a multiuser system, every user might store only one position of the vector along with the opening proof for that position.
Dynamic VCs <cit.> are vector commitments that support updates to the vector.
Suppose the committed vector is of length N and some k < N positions in the vector are updated,
so that a new vector commitment is published.
Then, every user in the system will need to update their local opening proof to match the updated commitment,
and this is done with the help of some global update information U that is broadcast to all users.
This information is typically generated and published by a manager who maintains the entire vector.
Applications of dynamic VCs include verifiable databases, zero-knowledge sets with frequent updates <cit.> and stateless clients for blockchains <cit.>.
The challenge is to design a VC scheme that minimizes the size of the update information U
as well as the computation work by each user to update their local opening proof.
For example, consider stateless clients on a blockchain as an important application for dynamic VCs.
The state of the chain can be represented as a vector of length N, where position i corresponds to the state of account number i.
Every user will locally maintain its own state (corresponding to some position in the vector)
along with an opening proof that enables the user to convince a third party
as to its current state.
Whenever a new block is published, the state of the chain changes.
In particular, suppose k out of the N positions in the vector need to be updated.
The block proposer will publish the update information U along with the new block,
and every user will update their opening proof to match the new committed state of the chain.
Thus, users can ensure that their opening proofs are up to date with respect to the latest committed state of the chain.
We stress that in this application, the data being updated, namely the updated positions and diffs, is published as part of the block.
The update information U only contains additional information that is needed to update the opening proofs.
When we refer to the size of U, we refer to its size, excluding the updated data (i.e., excluding the updated positions and diffs).
In this paper, we investigate the trade-off between the length |U| of the update information and the time complexity of proof updates.
Dynamic VCs can be grouped into two categories in terms of these parameters (Table <ref>).
Tree-based VCs <cit.> enable users to update their proofs in time O(N).
Each opening proof typically consists of (N) inner nodes,
and the update information U contains the changes in the inner nodes affected by the message updates.
Each user calculates its new opening proof by downloading the relevant inner nodes published as part of U.
When k positions are updated, a total of O(k log(N)) inner nodes in the tree are affected in the worst case.
Thus, when each inner node has length Θ(λ), proportional to the security parameter λ,
the update information consists of O(k log(N)λ) bits.
In contrast, algebraic VCs <cit.> enable users to update their opening proofs with only
knowledge of the updated data.
They do not require any additional update information U to be published beyond the indices and the `diffs' of the updated data.
Thus, the length of the update information needed to update the opening proofs is O(1).
However, algebraic VCs typically require each user to read all of the changed messages and incorporate the effect of these changes on their proofs, resulting in Θ(k) work per proof update.
To summarize, while tree-based VCs support efficient calculation of the new opening proofs by publishing a large amount of update information, linear in k,
algebraic VCs do not require any additional update information beyond the updated data, but suffer from a large runtime for proof updates, linear in k.
We formalize the dichotomy of VCs in Section <ref>.
§.§ Our Results
We propose a family of VCs that can support sublinear update, where both the length |U| of the update information and the complexity of proof updates are sublinear in k.
More specifically, our VCs can attain |U| = Θ(k^νλ), ν∈ (0,1), with a proof update complexity of Θ(k^1-ν) operations.
Our candidate construction with sublinear update is a homomorphic Merkle tree, first developed by <cit.>, where each inner node can be expressed as a sum of the partial digests of the messages underneath (Section <ref>).
The algebraic structure of these trees enable each user to calculate the effect of a message update on any inner node without reading other inner nodes or messages.
We identify homomorphic Merkle tree constructions based on lattices, from the literature <cit.>.
In Section <ref>, we provide the update algorithms (Alg. <ref>) for homomorphic Merkle trees, parameterized by ν∈ (0,1).
Our algorithm identifies a special subset of size Θ(k^ν) of the inner nodes affected by the message updates, and publish their new values as U; so that the users need not calculate these values.
These inner nodes are selected carefully to ensure that any inner node outside of U is affected by at most Θ(k^1-ν) updated messages.
Thus, to modify its opening proof, each user has to calculate the partial digests of at most Θ(k^1-ν) updated messages per inner node within its proof (that consists of Θ(log(N)) inner nodes).
Moreover, to calculate these partial digests, the user only needs the `diffs' of the updated messages.
This brings the asymptotic complexity of proof updates to Θ(k^1-ν) operations, while achieving an update information size of Θ(k^νλ) as opposed to Θ(kλ) on Merkle trees using SHA256.
In Section <ref>, we prove an information theoretic lower bound on the size of the update information given an upper bound on the runtime complexity of proof updates.
The bound implies the asymptotic optimality of our scheme with sublinear update.
Its proof is based on the observation that if the runtime complexity is bounded by O(k^1-ν), a user that wants to update its proof cannot read beyond O(k^1-ν) updated messages.
Then, to calculate the effect of the remaining k-O(k^1-ν) messages on its opening proof, the user has to download parts of the structured update information U.
Finally, to obtain the lower bound on |U|, we use Shannon entropy and lower bound the number of bits, namely O(k^νλ), required to capture the total information that will be downloaded by the users; while maintaining the security of the VC with parameter λ.
§.§ Applications
We identify three main applications for VCs with sublinear update.
§.§.§ Stateless clients for PoS Ethereum
Ethereum is the largest decentralized general purpose computation platform by market cap.
Ethereum state (, user accounts) is currently stored in the form of a Merkle tree <cit.> and grows approximately by half every year <cit.>.
Stateless clients <cit.> were proposed to mitigate the problem of state bloat and prevent the state storage and maintenance from becoming a bottleneck for decentralization.
Stateless clients maintain an opening proof to their account balances within the Ethereum state, thus can effortlessly prove the inclusion of their accounts within the latest state.
This enables the other Ethereum clients to verify the transactions that come with opening proofs without having to download the full state and check the validity of the claimed account balances.
Since block verification now requires downloading the proofs for the relevant state elements, Verkle trees <cit.> were proposed as a replacement for Merkle trees due to their short proof size.
Each new Ethereum block contains transactions that update the state elements and their opening proofs.
Archival nodes and block producers still maintain the full state so that they can inform the stateless clients about their new opening proofs.
For this purpose, block producers must broadcast enough information to the clients over the peer-to-peer gossip network of Ethereum.
As minimizing the proof size was paramount to decentralizing verification for blocks, minimizing the update information size becomes necessary for decentralizing the role of the block producer who has to disseminate this information.
However, reducing the length of the update information must not compromise the low overhead of stateless clients by requiring larger number of operations per proof update.
Therefore, the ideal VC scheme for stateless clients must strike a delicate balance between the size of the update information and the runtime complexity of proof updates.
In Section <ref>, we provide the update algorithms (Algs. <ref> and <ref>) for Verkle trees.
We observe that Verkle trees do not support sublinear update, and fall under the same category as tree-based VCs with update information length Θ(k λ).
Despite this fact, Verkle trees are highly practical in terms of updates.
In Section <ref>, we estimate that the update information size after a typical Ethereum block does not exceed |U| ≈ 100 kBytes (compared to the typical block size of <125 kBytes).
Moreover, each Verkle proof can be updated within approximately less than a second on commodity hardware.
In contrast, even the most efficient homomorphic Merkle tree construction <cit.> requires an update information size of 110.88 MBytes and an update time of 32.6 seconds when the trade-off parameter ν is 1/2, despite its asymptotic optimality (Section <ref>).
The large update information size is due to the lattice-based construction of these VCs.
Designing dynamic VCs that are both asymptotically optimal and practically efficient remains an open problem.
§.§.§ Databases with frequent membership changes
VCs with sublinear update can support databases with frequent membership changes.
When a user first registers, a message is updated to record the membership of the user.
The user receives this record and its opening proof, using which it can later anonymously prove its membership.
When the user leaves the system, the message is once again updated to delete the record.
In all these steps, membership changes result in updates to the opening proofs of other members.
When these changes are frequent, it becomes infeasible to distribute new proofs after each change.
VCs with sublinear update offer an alternative and efficient way to update the opening proofs of the users in the event of such changes.
§.§ Related Work
There are many VC constructions, each with different guarantees regarding the proof, commitment and public parameter sizes, verification time, updatability and support for subvector openings <cit.> (cf <cit.> for an SoK of VCs).
First formalized by <cit.>, almost all VCs allow some degree of updatability.
Whereas <cit.> enable updating the commitment and the opening proofs with only the knowledge of the old and the new messages, most VCs require some structured update information beyond the messages when the users do not have access to the internal data structures.
Among the lattice-based accumulators, vector commitments and functional commitments <cit.>, constructions amenable to sublinear update are presented in <cit.>.
Homomorphic Merkle trees were formalized and instantiated by <cit.> in the context of streaming authenticated data structures and parallel online memory checking.
The construction presented in <cit.> offers an alternative VC with sublinear update as it is not a Merkle tree, yet has the property that each inner node can be expressed as a sum of the partial digests of individual messages.
For dynamic accumulators that support additions, deletions and membership proofs, Camacho and Hevia proved that after k messages are deleted, Ω(k) bits of data must be published to update the proofs of the messages in the initial accumulated set <cit.>.
Their lower bound is information-theoretic and follows from a compression argument (Appendix <ref>).
Christ and Bonneau subsequently used a similar method to prove a lower bound on the global state size of a revocable proof system abstraction <cit.>.
As revocable proof systems can be implemented by dynamic accumulators and vector commitments, their lower bound generalizes to these primitives, , after k messages are updated in a dynamic VC, at least Ω(k) bits of data must be published to update the opening proofs (Appendix <ref> for the proof).
They conclude that a stateless commitment scheme must either have a global state with linear size in the number of accounts, or require a near-linear rate of local proof updates.
In our work, we already assume a linear rate of local proof updates, , after every Ethereum block or k messages in our parameterization, and that the message updates are publicized by the blockchain.
We instead focus on the trade-off between the global structured update information size (beyond the published messages) and the runtime complexity of proof updates.
§ PRELIMINARIES
§.§ Notation
We denote the security parameter by λ.
An event is said to happen with negligible probability, if its probability, as a function of λ, is o(1/λ^d) for all d>0.
An event happens with overwhelming probability if it happens except with negligible probability.
We denote the set {0,1,2,…,N-1} by [N].
When y = O(h(x) (x)), we use the shorthand y=O(h(x)) (similarly for Θ(.) and Θ(.)).
The function H(.) ℳ→{0,1}^λ represents a collision-resistant hash function.
We denote the binary decomposition of an integer x by (x), and for c>2, its base c decomposition by _c(x).
A vector of N elements (n_0, …, n_N-1) is shown as (n_i)_i.
The notation 𝐱[i:j] denotes the substring starting at the i^th index and ending at the j^th index within the sequence 𝐱.
In the subsequent sections, k will be used to denote the number of updated messages.
For a prime p, let 𝔽_p denote a finite field of size p.
We use 𝔾 to denote a cyclic group of prime order p with generator g.
The Lagrange basis polynomial for a given x ∈𝔽_p is denoted as L_x(X):
C
L_x(X) = ∏_i ∈𝔽_p
i ≠x X-i/x-i
We will use |G| and |H| to denote the maximum size of the bit representation of a single group element and a single hash value respectively.
We will use T_G and T_f to denote the time complexity of a single group operation and a single function evaluation for the hash functions in Section <ref>.
§.§ Vector Commitments
A vector commitment (VC) represents a sequence of messages such that each message can be proven to be the one at its index via an opening proof.
A dynamic vector commitment allows updating the commitment and the opening proofs with the help of an update information when the committed messages are changed.
Dynamic (updateable) vector commitments can be described by the following algorithms:
KeyGen(1^λ, N) → pp Given the security parameter λ and the size N=(λ) of the committed vector, the key generation algorithm outputs public parameters pp, which implicitly define the message space ℳ.
Commit_pp(m_0, …, m_N-1) → (C, ) Given a sequence of N messages in ℳ and the public parameters pp, the commitment algorithm outputs a commitment string C and the data required to produce the opening proofs for the messages.
Here, contains enough information about the current state of the VC's data structure (, the current list of committed messages) to help generate the opening proofs.
Open_pp(m, i, ) →π_i The opening algorithm is run by the committer to produce a proof π_i that m is the i^th committed message.
Verify_pp(C, m, i, π_i) →{0,1} The verification algorithm accepts (, outputs 1) or rejects a proof.
The security definition will require that π_i is accepted only if C is a commitment to some (m_0, …, m_N-1) such that m = m_i.
Update_pp(C, (i, m_i)_i ∈ [N], (i, m'_i)_i ∈ [N], ) → (C', U, ') The algorithm is run by the committer to update the commitment C when the messages (m_i_j)_j ∈ [k] at indices (i_j)_j ∈ [k] are changed to (m'_i_j)_j ∈ [k]. The other messages in the vector are unchanged. It takes as input the old and the new messages, their indices and the data variable . It outputs a new commitment C', update information U and the new data variable '.
ProofUpdate_pp(C, p((i, m_i)_i ∈ [N], (i, m'_i)_i ∈ [N]), π_j , m', i, U) →π_j'
The proof update algorithm can be run by any user who holds a proof π_j for some message at index j and a (possibly) new message m' at that index.
It allows the user to compute an updated proof π'_j (and the updated commitment C') such that π'_j is valid with respect to C', which contains m'_i, i ∈ N, as the new messages at the indices i ∈ N (and m' as the new message at index i).
Here, p(.) specifies what portion of the old and the new messages is sufficient to update the opening proof.
For instance, the proof update algorithm often does not need the old and the new messages in the open; but can carry out the proof update using only their differences.
In this case, p((i, m_i)_i ∈ [N], (i, m'_i)_i ∈ [N]) = (i, m'_i-m_i)_i ∈ N.
Correctness of a VC requires that ∀ N = (λ), for all honestly generated parameters pp KeyGen(1^λ, N), given a commitment C to a vector of messages (m_0, …, m_N-1) ∈ℳ^N, generated by Commit_pp (and possibly followed by a sequence of updates), and an opening proof π_i for a message at index i, generated by Open_pp or ProofUpdate_pp, it holds that Verify_pp(C, m_i, i, π_i)=1 with overwhelming probability.
Security of a VC is expressed by the position-binding property:
A VC satisfies position-binding if ∀ i ∈ [N] and for every PPT adversary 𝒜, the following probability is negligible in λ:
C
[Verify_pp(C, m, i, π_i) = 1
Verify_pp(C, m', i, π'_i) = 1 m ≠m' pp KeyGen(1^λ, N)
(C, m, m', π_i, π'_i) 𝒜(pp)]
We relax the succinctness assumption of <cit.> and denote a value to be succinct in x if it is (x).
§.§ KZG Polynomial Commitments
The KZG commitment scheme <cit.> commits to polynomials of degree bounded by ℓ using the following algorithms:
KeyGen(1^λ, ℓ) → pp outputs pp = (g, g^τ, g^(τ^2), …, g^(τ^ℓ)) as the public parameters, where g is the generator of the cyclic group 𝔾 and τ is a trapdoor (pp[i] = g^τ^i).
Commit(pp, ϕ(X)) → (C, )
The commitment to a polynomial ϕ(X) = ∑_i=0^ℓ-1 a_i X^i is denoted by [ϕ(X)],
and is computed as [ϕ(X)] = ∏_i=0^ℓ (pp[i])^a_i.
The commitment algorithm outputs C = [ϕ(X)] and = ϕ(X).
Open_pp(m, i, ) →π:
outputs the opening proof π_i that ϕ(i) = m, calculated as the commitment to the quotient polynomial (ϕ(X)-ϕ(i)) / (X-i).
Verify(C, m, i, π) accepts if the pairing check
e(C/g^m, g) = e(π, pp[1]/g^i )
holds.
We refer to <cit.> for the security analysis of this scheme.
§.§ Merkle Trees
Merkle Tree is a vector commitment using a collision-resistant hash function.
In a Merkle tree, hashes of the committed messages constitute the leaves of a c-ary tree of height h = log_c(N), where each inner node is found by hashing its children.
The depth of the root is set to be 0 and the depth of the leaves is ⌈log_c(N) ⌉.
The commitment function outputs the Merkle root as the commitment C and the Merkle tree as .
The opening proof for a message m_x at some index x is the sequence of h(c-1) hashes consisting of the siblings of the inner nodes on the path from the root to the hash of the message m_x.
We hereafter consider binary Merkle trees (c=2) and assume N=c^h = 2^h unless stated otherwise.
Let u_b_0, b_1,…,b_i-1, b_j ∈{0,1}, j ∈ [i], denote an inner node at depth i-1 that is reached from the root by choosing the left child at depth j if b_j=0 and the right child at depth j if b_j=1 (b_0= and u_ is the root).
By definition, for a message m_x at index x, H(m_x) = u_,(x).
§.§ Verkle Trees
A Verkle tree <cit.> is similar to a Merkle tree except that each inner node is calculated as the hash of the KZG polynomial commitment to its children.
Let b_j ∈ [c], j=1, …, h, denote the indices of the inner nodes on the path from the root to a leaf at index x, _c(x) = (b_1, …, b_h), relative to their siblings.
Define f_b_0,…,b_j, j ∈ [h], as the polynomials determined by the children of the inner nodes on the path from the root to the leaf, where f_b_0=f_ is the polynomial determined by the children of the root.
Let C_b_0,…,b_j = [f_b_0,…,b_j], j ∈ [h], denote the KZG commitments to these polynomials.
By definition, u_b_0,…,b_j = H(C_b_0,…,b_j), and the value of the polynomial f_b_0,…,b_j at index b_j+1 is u_b_0,…,b_j+1 for each j ∈ [h].
Here, u_b_0 = H(C_b_0) is the root of the tree, and u_b_0,…,b_h equals the hash H(m_x) of the message at index x.
For consistency, we define C_b_0,…,b_h as m_x.
For example, given h = 3 and c = 4, the inner nodes from the root to the message m_14 have the indices b_0 = 0, b_1 = 3 and b_2 = 2, and they are committed by the polynomials f_, f_,0 and f_,0,3 respectively.
The commitment function Commit_pp(m_0, …, m_N-1) outputs the root u_b_0 as the commitment C and the Verkle tree itself as .
The Verkle opening proof for the message m_x, (x) = (b_1, …, b_h), consists of two parts: (i) the KZG commitments (C_b_0,b_1, …, C_b_0,…, b_h-1) on the path from the root to the message, and (ii) a Verkle multiproof.
The goal of the Verkle multiproof is to show that the following evaluations hold for the inner nodes from the root to the message: f_b_0,…,b_j(b_j+1)=u_b_0,…,b_j+1 = H(C_b_0,…,b_j+1), j ∈ [h].
It has two components: (i) the commitment [g(X)] and (ii) the opening proof π' for the polynomial h(X)-g(X) at the point t=H(r,[g(X)]), where
C
g(X)=∑_j=0^h-1 r^j f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1, h(X)=∑_j=0^h-1 r^j f_b_0,…,b_j(X)/t-b_j+1,
and r=H(C_b_0,..,C_b_0,…,b_h-1,u_b_0,b_1,..,u_b_0,…,b_h,b_1,..,b_h).
Thus, Open_pp(m, i, ) outputs ((C_b_0,b_1, …, C_b_0,…,b_h-1), ([g(X)], π')).
To verify a Verkle proof π = ((C_b_0,b_1, …, C_b_0,…,b_h), (D,π')), the algorithm Verify_pp(C, m, x, π) first computes r and t using u_b_0,…,b_j = H(C_b_0,…,b_j), j ∈ [h], and u_b_0,…,b_h = H(m).
Then, given the indices (x) = (b_1, …, b_h) and the commitments (C_b_0,b_1, …, C_b_0,…,b_h), it calculates
C
y = ∑_j=0^h-1 r^j C_b_0,…,b_j/t-b_j+1 E = ∑_j=0^h-1 r^j/t-b_j+1 C_b_0,…,b_j.
Finally, it returns true if the pairing check e(E-D-[g(X)],[1]) = e(π', [X-t]) is satisfied.
As the degree c of a Verkle tree increases, size of the opening proofs and the runtime of the verification function decreases in proportion to the height h = log_cN of the tree.
This enables Verkle trees to achieve a short opeining proof size for large number of messages (as in the case of the Ethereum state trie) by adopting a large degree (, c=256).
In comparison, each Merkle proof consists of (c-1) log_cN inner nodes, which grows linearly as c increases.
§ FORMALIZING THE DICHOTOMY OF VCS
We first analyze the trade-off between the number of operations required by proof updates and the size of the update information U by inspecting different types of dynamic VCs.
Recall that the number of updated messages is k ≤ N.
§.§ Updating KZG Commitments and Opening Proofs
In the subsequent sections, we assume that each user has access to a dictionary of KZG commitments to the Lagrange basis polynomials L_i(X), i ∈𝔽_p, and for each polynomial, its opening proofs at each point j ∈𝔽_p, j < N.
With the help of this table, one can instantiate a KZG based VC to the messages (m_i)_i ∈ [N], by treating them as the values of the degree N polynomial ϕ(X) at inputs i ∈𝔽_p, i<N.
We next analyze the complexity of the update information and the proof updates in this VC.
The update and proof update algorithms are described by Alg. <ref> in Appendix <ref>.
§.§.§ Update Information
Suppose the vector (i, m_i)_i ∈ [N] is updated at some index i such that m'_i m_i + δ for some δ∈𝔽_p.
Then, the polynomial ϕ(X) representing the vector is replaced by ϕ'(X) such that ϕ'(X) = ϕ(X) if X ≠ i, and ϕ'(i) = ϕ(i) + δ at X = i.
Thus, the new KZG commitment C' to ϕ'(X) is constructed from the commitment C to ϕ(X) as follows:
rCl
C' = [ϕ'(X)] = [ϕ(X)+δL_i(X)]
= [ϕ(X)][L_i(X)]^δ = C ·[L_i(X)]^δ = C ·[L_i(X)]^m'_i-m_i.
If the vector is modified at k different indices i_1,...,i_k from message m_i_j to m'_i_j, j ∈ [k], then
the new commitment C' = [ϕ'(X)] becomes
rCl
[ϕ(X)+∑_j=1^k (m'_i_j-m_i_j) L_x_i_j(X)]
= [ϕ(X)] ∏_j=1^k[L_i_j(X)]^(m'_i_j-m_i_j) = C ∏_j=1^k[L_i_j(X)]^(m'_i_j-m_i_j).
Thus, the commitment can updated given only the old and the new messages at the updated indices, besides the table.
§.§.§ Proof Update
Let π_x denote the opening proof of a polynomial ϕ(X) at a point (x,m_x).
When k messages are updated, the new opening proof π'_x can be found as a function of the old proof π_x and the opening proofs π_i_j,x of the Lagrange basis polynomials L_i_j(X), j ∈ [k], at the index x (m'_x = m_x+∑_j=1^k (m'_i_j-m_i_j) · 1_x=i_j is the new value of m_x after the k updates):
rCl
π'_x = [ϕ'(X)-m_x-∑_j=1^k δ_j ·1_x=i_j/X-x]
= π_x ∏_j=1^k [L_i_j(X)-L_i_j(x)/X-x]^m'_i_j-m_i_j = π_x ∏_j=1^k π^m'_i_j-m_i_j_i_j,x
Thus, the proof can updated given only the old and the new messages at the updated indices, besides the table.
The update information is set to be the empty set, , U = ∅.
§.§.§ Complexity
The size of the update information is constant, , Θ(1).
Each user can update its proof after k accesses to the dictionary, and in the worst case, Θ(k log|ℳ|) = Θ(k) group operations as log(m'_i-m_i)≤log|ℳ| for all i ∈ [N].
§.§ Updating Merkle Trees and Opening Proofs
We next consider a Merkle tree and analyze the complexity of the update information size and the runtime for proof updates.
A simple update scheme would be recalculating the new Merkle tree given all of the old messages or the old inner nodes of the Merkle tree, and the message updates.
However, this implies a large complexity for the runtime of the proof update algorithm that scales as Ω(k) when users keep track of the inner nodes, and as Ω(N) when the users recalculate the tree from scratch at each batch of updates.
Moreover, in many applications, the users do not have access to any messages or inner nodes besides those that are part of the Merkle proof held by the user.
Hence, in the following sections, we describe update and proof update algorithms
that reduce the runtime complexity of the proof updates at the expanse of larger update information (Alg. <ref> in Appendix <ref>).
§.§.§ Update Information
Suppose the vector (i, m_i)_i ∈ [N] is updated at some index x, (b_1,…,b_h) = (x), to m'_x.
Then, the root C=u_b_0 and the inner nodes (u_b_0,b_1, …, u_b_0,b_1,…,b_h), (b_1,…,b_h) = (i), must be updated to reflect the change at that index.
Given the old inner nodes, the new values for the root and these inner nodes, denoted by C'=u'_b_0 and (u'_b_0,b_1, …, u'_b_0,b_1,…,b_h), are calculated recursively as follows:
rCl
u'_b_0,b_1,…,b_h H(m'_x),
u'_b_0,b_1,…,b_j ^
H(u'_b_0,b_1,…,b_j,0, u_b_0,b_1,…,b_j,1) if b_j+1 = 0, j<h
H(u_b_0,b_1,…,b_j,0, u'_b_0,b_1,…,b_j,1) if b_j+1 = 1, j<h
When the messages are modified at k different points i_j, j ∈ [k], the calculation above is repeated k times for each update.
As the updated inner nodes are parts of the Merkle proofs, the update information consists of the new values at the inner nodes listed from the smallest to the largest depth in the canonical left to right order.
For instance,
U = ((, u'_), (0, u'_0), (1, u'_1), (00, u'_00), (10, u'_10), …)
implies that the root u_ and the inner nodes u_0, u_1, u_00 and u_10 were updated after k messages were modified at the leaves of the Merkle tree.
We reference the updated inner nodes using their indices (, U[b_0, b_1 … b_j] = v, when (b_1 … b_j, v) ∈ U).
§.§.§ Proof Update
The Merkle proof π_x for a message at index
x, (b_1, …, b_h) = (x), is the sequence (u_b_1, u_b_1,b_2, …, u_b_1,b_2,…,b_h).
When k messages are updated, some of the inner nodes within the proof might have changed.
A user holding the Merkle proof for index x can find the new values of these inner nodes by querying the update information with their indices.
§.§.§ Complexity
Upon receiving the update information U, each user can update its proof in Θ(log^2(N)+|H| log(N)) = Θ(1) time by running a binary search algorithm to find the updated inner nodes within U that are part of its Merkle proof, and reading the new values at these nodes.
Since modifying each new message results in h = log(N) updates at the inner nodes and some of the updates overlap, |U| = Θ(k log(N/k) (log(N)+|H|)) = Θ(k)|H|, as each updated inner node is represented by its index of size Θ(log(N)) and its new value of size |H| in U.
§.§ Dichotomy of VCs
In the case of KZG commitments, |U| = Θ(1), and there is no information overhead on top of the message updates.
For Merkle trees with an efficient proof update algorithm, |U| = Θ(k)|H|,
thus there is an extra term scaling in Θ(k)|H| = Θ(k)λ, since |H| = Ω(λ) for collision-resistant hash functions.
In contrast, for KZG commitments, each user has to do Θ(k) group operations to update its opening proof; whereas in Merkle trees, each user can update its proof in Θ(1) time, which does not depend on k.
Hence, KZG commitments outperform Merkle trees in terms of the update information size, whereas Merkle trees outperform KZG commitments in terms of the time complexity of proof updates.
Table <ref> generalizes this observation to a dichotomy between algebraic VC schemes and tree-based ones favoring shorter runtimes for proof updates.
The algebraic and tree-based ones outperform each other in terms of the update information size and runtime complexity respectively.
§ VECTOR COMMITMENTS WITH SUBLINEAR UPDATE
We would like to resolve the separation in Table <ref> and obtain a vector commitment, where both the size of the update information and the complexity of proof updates have a sublinear dependence on k.
In particular, |U| = Θ(g_1(k)λ) in the worst case,
and the proof update algorithm requires at most Θ(g_2(k)) operations,
where both g_1(k) and g_2(k) are o(k).
We say that such a VC supports sublinear update.
In this section, we describe a family of VCs with sublinear update, parameterized by the values ν∈ (0,1) and characterized by the functions (g_1,g_2) = (k^ν, k^1-ν).
§.§ Homomorphic Merkle Trees
We first introduce homomorphic Merkle trees where messages placed in the leaves take values in a set ℳ.
We will use two collision-resistant hash functions f̃𝒟×𝒟→ℛ and f ℳ→ℛ,
where both ℳ and 𝒟 are vector spaces over some field 𝔽,
and ℛ is an arbitrary finite set.
We will also need an injective mapping g: ℛ→𝒟, which need not be efficiently computable.
We use g^-1: 𝒟→ℛ to denote the inverse of g,
meaning that g^-1(g(x)) = x for all x ∈ℛ.
We require that g^-1 be efficiently computable.
Now, for j ∈ [h], where h is the height of the tree, every node u_b_0,…,b_j∈𝒟 of the homomorphic Merkle tree is characterized by the following expressions:
llCl
a leaf node: g^-1(u_b_0,(i)) = f(m_i)
an internal node: g^-1(u_b_0,…,b_j) = f̃(u_b_0,…,b_j,0, u_b_0,…,b_j,1) for j < h
The homomorphic property of the Merkle tree refers to the fact that
there are efficiently computable functions
h_i,j: 𝒟→𝒟 for i ∈ [N] and j ∈ [h],
such that every inner node u_b_0,…,b_j∈𝒟 can be expressed as
rCl
u_b_0 = ∑_i ∈[N] h_i,0(m_i)
u_b_0,…,b_j = ∑_i(i)[0:j-1]=(b_1,…,b_j) h_i,j(m_i).
We refer to the function h_i,j as a partial digest function
and refer to h_i,j(m_i) as the partial digest of m_i.
In a homomorphic Merkle tree, every internal node is the sum of the partial digests of the leaves under that node.
We will show in Section <ref> that each function h_i,j can be expressed
as an iterated composition of the functions f and f̃.
Evaluating h_i,j requires evaluating the functions f and f̃ exactly h-j times.
Opening proof for a message consists of both children of the internal nodes on the path from the message to the root (as opposed to Merkle opening proofs that contain only the siblings of the internal nodes on the path).
For instance, the opening proof for the message m_i at leaf index i, with (i) = (b_1,…,b_h), is (i, (u_b_0,…,b_j,0,u_b_0,…,b_j,1)_j=0,…,h-1).
Opening proofs are verified using the functions f and f̃ (not by using the functions h_i,j).
To verify an opening proof (i, (u_b_0,…,b_j,0,u_b_0,…,b_j,1)_j=0,…,h-1) for a message m_i with respect to the root u_b_0, the verifier checks if the following equalities hold:
llCl
for the leaf: g^-1(u_b_0,(i)) = f(m_i)
for the internal nodes: g^-1(u_b_0,…,b_j) = f̃(u_b_0,…,b_j,0, u_b_0,…,b_j,1) for j = h-1, …, 0.
If so, it accepts the proof, and otherwise it outputs reject.
As an example, consider a homomorphic Merkle tree that commits to four messsages m_0,m_1,m_2,m_3.
Then, its root u_ and inner nodes u_,0, u_,1, u_,0,0, u_,0,1, u_,1,0, u_,1,1 can be calculated as follows:
rClrCl
u_ = h_0,0(m_0) + h_1,0(m_1) + h_2,0(m_2) + h_3,0(m_3) ; u_,0,0 = h_0,2(m_0)
u_,0 = h_0,1(m_0) + h_1,1(m_1) ; u_,0,1 = h_1,2(m_1)
u_,1 = h_2,1(m_2) + h_3,1(m_3) ; u_,1,0 = h_2,2(m_2)
u_,1,1 = h_3,2(m_3)
The opening proof for m_3 is given by (3, ((u_,0, u_,1), (u_,1,0, u_,1,1))), and verified by checking the following equations:
llCl
for u_,1,1: g^-1(u_,1,1) = f(m_i)
for u_,1: g^-1(u_,1) = f̃(u_,1,0, u_,1,1)
for u_: g^-1(u_) = f̃(u_,0, u_,1)
It now follows that
when a message m_i is updated to m'_i, each inner node on the path from the leaf to the root can be updated from u_b_0,…,b_j to u'_b_0,…,b_j using the functions h_i,j as follows:
u'_b_0,…,b_j =
h_i,j(m'_i) + ∑_x ≠ i
(x)[0:j-1]= (b_1,…,b_j) h_x,j(m_x) =
u_b_0,…,b_j + h_i,j(m'_i) - h_i,j(m_i)
When the partial digest functions are linear in their input, the expression h_i,j(m'_i) - h_i,j(m_i) can be written as
h_i,j(m'_i) - h_i,j(m_i) = sign(m'_i-m_i)h_i,j(|m'_i-m_i|).
This lets us calculate the updated internal node using only the knowledge of the message diff m_i'-m_i.
We provide examples of homomorphic Merkle tree constructions in Section <ref> with linear partial digest functions h_i,j.
Homomorphic Merkle proofs in these constructions consist of the two siblings of the inner nodes on the path from the proven message to the root (Section <ref>).
Unlike in Section <ref>, homomorphic Merkle trees enable calculating the new inner nodes after message updates using only the new and the old updated messages, in particular using only their difference.
Hence, we can construct a tree that achieves the same complexity for the update information size as algebraic VCs, albeit at the expanse of the proof update complexity, without requiring the users to keep track of the old messages or to calculate the tree from scratch given all messages (Appendix <ref> for further discussion).
This is in contrast to Merkle trees based on SHA256.
The update and proof update algorithms of such a homomorphic Merkle tree with no structured update information and the same asymptotic complexity as algebraic VCs is described in Appendix <ref>.
Since the homomorphic Merkle trees can achieve both extremes in terms of update information size and update runtime (Table <ref>), with a smart structuring of the update information, they can support sublinear update.
We show how in the next subsection.
§.§ Structuring the Update Information
We now describe the new update and proof update algorithms that enable homomorphic Merkle trees to achieve sublinear complexity as a function of the parameter ν (Alg. <ref>).
§.§.§ Update Information
When the messages (i_j, m_i_j)_j ∈ [k] change to (i_j, m'_i_j)_j ∈ [k], the update information U is generated recursively using the following algorithm:
* Start at the root u_b_0. Terminate the recursion at an inner node if there are k^1-ν or less updated messages under that node.
* If there are more than k^1-ν updated messages with indices ≥ N/2, , under the right child, then publish the new right child of the root as part of U, and apply the same algorithm to the subtree rooted at the right child, with u_b_0 and N replaced by u_b_0,1 and N/2 respectively.
* If there are more than k^1-ν updated messages with indices less than N/2, , under the left child, then publish the new left child of the root as part of U, and apply the same algorithm to the subtree rooted at the left child, with u_b_0 and N replaced by u_b_0,0 and N/2 respectively.
The new values of the inner nodes included in U are again listed from the smallest to the largest depth in the canonical left to right order.
§.§.§ Proof Update
When the messages (i_j, m_i_j)_j ∈ [k] are updated to (i_j, m'_i_j)_j ∈ [k], a user first retrieves the inner nodes within its Merkle proof that are published as part of the update information.
It then calculates the non-published inner nodes within the proof using the partial digests.
For instance, consider a user with the proof (u_b_1, u_b_1,b_2, …, u_b_1,b_2,…,b_h) for some message m_x, (b_1, …, b_h) = (x).
To update the proof, the user first checks the update information U and replaces the inner nodes whose new values are provided by U:
u'_b_1,…,b_d U[b_1 …b_d], d ∈ [h], if U[b_1 …b_d] ≠.
Otherwise, the user finds the new values at the nodes u_b_1,…,b_d, d ∈ [h], using the functions h_x,d:
rCl
u'_b_1, …, b_d-1,b_d = u_b_1, …, b_d-1,b_d
+ ∑_j ∈[k] 1_(i_j)[:d] = (b_1, …, b_d-1,b_d) (sign(m'_i_j-m_i_j)h_i_j,d(|m'_i_j -m_i_j|)))
§.§.§ Complexity
Finally, we prove bounds on the complexity given by these algorithms:
Complexity of the update information size and the runtime of proof updates are as follows: g_1(k) = k^ν and g_2(k) = k^1-ν.
We finally show that this VC publishes O(k^ν) new inner nodes in the worst case.
Let 𝒰 denote the subset of the inner nodes published by the algorithm as part of U such that no child of a node u ∈𝒰 is published.
Then, there must be over k^1-ν updated messages within the subtree rooted at each node u ∈𝒰.
Since there are k updated messages, and by definition of 𝒰, the subtrees rooted at the nodes in 𝒰 do not intersect at any node, there must be less than k/k^1-ν = k^ν inner nodes in 𝒰.
Since the total number of published inner nodes is given by 𝒰 and the nodes on the path from the root to each node u ∈𝒰, this number is bounded by k^νlog(N) = Θ(k^ν).
Hence,
|U| = Θ(k^νlog(N)(log(N)+|H|)) = Θ(k^ν)|H| = Θ(k^ν) λ,
which implies g_1(k) = k^ν.
For each inner node in its Merkle proof, the user can check if a new value for the node was provided as part of U, and replace the node if that is the case, in at most Θ(log(N)+|H|) time by running a binary search algorithm over U.
On the other hand, if the new value of a node in the proof is not given by U, the user can calculate the new value after at most k^1-νlog(N) function evaluations.
This is because there can be at most k^1-ν updated messages within the subtree rooted at an inner node, whose new value was not published as part of U.
This makes the total time complexity of a proof update at most
C
Θ(log(N)(log(N)+|H|+k^1-νlog(N)T_f)) = Θ(k^1-ν) T_f,
which implies g_2(k) = k^1-ν.
§.§ Constructions for Homomorphic Merkle Trees
Homomorphic Merkle trees were proposed by <cit.>.
These hash functions are lattice-based, and their collision-resistance is proven by reduction to the hardness of the gap version of the shortest vector problem (𝖦𝖠𝖯𝖲𝖵𝖯_γ), which itself follows from the hardness of the small integer solution problem.
We next describe the construction introduced by <cit.>, which is similar to those proposed by later works <cit.> (an alternative construction is provided in Appendix <ref>).
Its correctness and security follow from <cit.>.
Let L(𝐌) denote the lattice defined by the basis vectors 𝐌⊂ℤ^k × m_q for appropriately selected parameters k,m,q, where m = 2 k log q.
Consider vectors u ∈{0, …, t}^k log q, where t is a small integer.
The homomorphic hash functions f ℤ^k log q→ L(𝐌) and f̃ℤ^k log q×ℤ^k log q→ L(𝐌) used by <cit.> are defined as f(x) = 𝐌x and f̃(x,y) = 𝐌𝐔 x + 𝐌𝐃 y respectively.
Here, 𝐔 and 𝐃 are special matrices that double the dimension of the multiplied vector and shift it up or down respectively.
The remaining entries are set to zero.
For convenience, we define 𝐋 = 𝐌𝐔 and 𝐑 = 𝐌𝐃.
Since the domain and range of the hash functions are different, to ensure the Merkle tree's homomorphism, authors define a special mapping g ℤ^k_q →ℤ^k logq_q from the range of the hash functions to their domain.
Here, g(.) takes a vector 𝐯∈ℤ_q as input and outputs a radix-2 representation for 𝐯.
However, as there can be many radix-2 representations of a vector, to help choose a representation that yields itself to homomorphism, authors prove the following result: for any x_1, x_2, …, x_t ∈ℤ_q, there exists a short radix-2 representation g(.) such that g(x_1 + x_2 + … + x_t q) = b(x_1) + b(x_2) + … + b(x_t) q ∈{0, …, t}^k log q, where the function b ℤ^k_q →{0,1}^klogq returns the binary representation of the input vector.
This equality enables the mapping g(.) to preserve the hash functions' original homomorphic property.
Then, given an inner node u_b_0,…,b_j as input, the homomorphic Merkle tree uses the short radix-2 representation g(.) that enforces the following equality: g(u_b_0,…,b_j) = g(𝐋 u_b_0,…,b_j,0 + 𝐑 u_b_0,…,b_j,1 q) = b(𝐋 u_b_0,…,b_j,0) + b(𝐑 u_b_0,…,b_j,1) q.
Finally, this enables calculating the value of each inner node as a sum of the partial digests h_i,j(.) of the messages m_i under the node u_b_0,…,b_j (, (m_i)_(i)[0:j] = (b_0,…,b_j)) as outlined in Section <ref>, i.e., u_b_0,…,b_j equals
rCl
𝐋g(u_b_0,b_1,…,b_j,0) + 𝐑g(u_b_0,b_1,…,b_j,1)
= 𝐋g(𝐋g(u_b_0,…,b_j,0,0) + 𝐑g(u_b_0,…,b_j,0,1)) + 𝐑g(𝐋g(u_b_0,…,b_j,1,0) + 𝐑g(u_b_0,…,b_j,1,1))
= 𝐋b(𝐋g(u_b_0,…,b_j,0,0)) + 𝐋b(𝐑g(u_b_0,…,b_j,0,1))
+ 𝐑b(𝐋g(u_b_0,…,b_j,1,0)) + 𝐑b(𝐑g(u_b_0,…,b_j,1,1))
= ∑_i(i)[0:j-1]=(b_1,…,b_j) h_i,j(m_i),
where h_i,j(.) is expressed in terms of the bits (i)[j:h-1] = (b'_1, …, b'_h-j):
C
h_i,j(m_i) = f_b'_1(f_b'_2(…f_b'_h-j(f(m_i))))
Here, f_0(.) and f_1(.) are defined as 𝐋b(.) and 𝐑b(.) respectively.
Since b(.), binary expansion, is a linear operation and matrix multiplication is linear, h_i,j(.) is linear in its input.
Opening proof of a message m consists of its index and g(α_i) and g(β_i), i ∈ [h], h = log(N), where α_i and β_i are the children of the inner nodes on the path from m to the root.
The proof can be verified in log(N) time by iteratively checking if f(m) = g^-1(α_h) (or = g^-1(β_h)) and f̃(g(α_i),g(β_i)) = g^-1(α_i-1) (or =g^-1(β_i-1) depending on the message index), where g^-1 returns a number given its radix-2 representation <cit.>.
Note that both f and f̃ are homomorphic hash functions <cit.>.
Other examples of homomorphic hash functions include Pedersen hashes and KZG commitments.
However, the homomorphic property of the hash function is not sufficient for constructing a homomorphic Merkle tree when the function is combined with the output of other functions in a serial manner as in Merkle trees.
For the lattice-based function, this was possible because of repeated linearity <cit.>, which refers to the existence of a linear mapping g(.) from the range to the domain of the hash function.
This mapping enabled the iterative hashing within the Merkle tree to preserve the linearity of the hash function.
Such repeated linearity does not exist for Pedersen hashes and KZG commitments as a linear mapping from the range to the domain would imply the violation of the discrete log assumption.
That is why Verkle trees based on KZG commitments are not homomorphic and cannot support sublinear update.
§.§ A Concrete Evaluation
Suppose the Ethereum state is persisted using the homomorphic Merkle tree construction of <cit.> with the trade-off parameter ν = 1/2.
We next estimate the size of the update information and the proof update time after observing an Ethereum block with ERC20 token transfers.
Suppose the block has the target size of 15 million gas <cit.>, and each token transfer updates the balance of two distinct accounts stored at separate leaves of the homomorphic Merkle tree.
Since each ERC20 token transfer consumes approximately 65,000 gas, there are ∼ 230 such transactions in the block, and the block updates k = 460 accounts.
Suppose the homomorphic Merkle tree has degree 2 and commits to N = 256^3 = 2^24 accounts.
For comparison, 256^3 ≈ 16.7 million, matching in magnitude the total number of cumulative unique Ethereum addresses, which is 200 million as of 2023 <cit.>.
Each opening proof consists of 2log(N) = 48 inner nodes.
When 460 accounts are updated, in the worst case, the update information consists of ⌈√(k)⌉log(N) = 528 inner nodes.
To evaluate its size, we use the parameters calculated by <cit.> for secure instantiations of the homomorphic Merkle trees from both their paper and <cit.>.
Since the parameters for <cit.> result in a large inner node size on the order of hundreds of MBs, our evaluation takes the size of an inner node as that of <cit.>, namely |H| = 0.21 MB (which is equal to the key size in <cit.>).
This implies an update information size of |U| = 110.88 MBytes and an opening proof size of |π| = 10.08 MBytes.
As for update time, in the worst case, each user has to calculate the partial digests of 44 updated messages at each height of the homomorphic Merkle tree, , the effect of these updated messages on each inner node of its opening proof.
Calculating the partial digest of a message at height h measured from the leaves requires h evaluations of the hash function.
This implies a proof update complexity of 2 ∑_i=0^logN-1 i min(⌈√(k)⌉, 2^i) = 11,900 hash evaluations.
To find numerical upper bounds for the update time, we use the hash function evaluation times, namely T_f = 26.84 and T_f = 2.74 ms, published by <cit.> for both the hash function in <cit.> and their new and more performant function (these times are for commodity hardware; <cit.> for the details).
This gives an upper bound of 319.4 and 32.6 seconds for the update time using the hash functions in <cit.> and <cit.> respectively.
Based on the benchmarks for the practical hash function introduced in <cit.>, Table <ref> compares the number of published inner nodes ⌈ k^ν⌉log(N), the total update information size ⌈ k^ν⌉log(N) |H| (assuming that the size of each inner node is |H| upper bounded by 0.21 MBytes), the number of hash function evaluations per proof update 2 ∑_i=0^logN-1 i min(⌈ k^1-ν⌉, 2^i) and the proof update time 2 ∑_i=0^logN-1 i min(⌈ k^1-ν⌉, 2^i) T_f (assuming that each hash evaluation takes less than T_f = 2.74 ms) at ν = 0, 1/4, 1/2, 3/4, 1.
The degree of the homomorphic Merkle tree and the opening proof size are fixed at 2 and 48 inner nodes (|π| = 10.08) respectively.
§ UPDATING VERKLE TREES AND OPENING PROOFS
We now describe the update and proof update functions for Verkle trees (Algs. <ref> and <ref> respectively).
Since Verkle trees were proposed to support stateless clients,
we describe an update scheme that minimizes the runtime complexity of proof updates and does not require the users to download the updated messages or have access to old inner nodes.
As Verkle trees do not support sublinear update, we numerically estimate the size of the update information and the complexity of proof updates in Section <ref>.
§.§ Update Information
Suppose the vector (i, m_i)_i ∈ [N] is modified at some index x, (b_1, …, b_h) = (x) to be m'_x.
Since each inner node is the hash of a KZG commitment, the new inner nodes u'_b_0,…,b_j = H(C'_b_0,…,b_j), j ∈ [h], can be found as a function of the old commitments at the nodes and the powers of the Lagrange basis polynomials as described in Section <ref>:
C
C'_b_0,…,b_h m'_x, C'_b_0,…,b_j C_b_0,…,b_j [L_b_j+1]^(u'_b_0,…,b_j+1-u_b_0,…,b_j+1)
When k messages are updated, the above calculation is repeated k times for each update.
Update information U consists of the new values of the KZG commitments on the path from the updated messages to the Verkle root akin to the Merkle trees, ordered in the canonical top-to-bottom and left-to-right order.
§.§ Verkle Proofs
Let π_x denote the Verkle proof of some message m_x at index x, (b_1,…,b_h) = (x): π_x = ((C_b_0,b_1, …, C_b_0,…,b_h-1), ([g(X)], π)).
We define π^f_x as the opening proof for index x within polynomial f.
We observe that the commitment [g(X)] and the proof π can be expressed as functions of the opening proofs of the inner nodes u_b_0,b_1, …, u_b_0,…,b_h at the indices b_1,…,b_h within the polynomials f_b_0, …, f_b_0,…,b_h-1, respectively:
rCl
[g(X)] = [∑_j=0^h-1 r^j f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]
= ∏_j=0^h-1 [f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]^r^j = ∏_j=0^h-1 (π^f_b_0,…,b_j_b_j+1)^r^j.
Similarly, the opening proof π=π^(h-g)_t for index t within the polynomial h(X)-g(X) can be expressed as follows (Appendix <ref>):
rCl
[h(X)-g(X)-(h(t)-g(t))/X-t] = ∏_j=0^h-1 [f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]^r^j/t-b_j+1
= ∏_j=0^h-1 (π^f_b_0,…,b_j_b_j+1)^r^j/t-b_j+1
We assume that each user holding the Verkle proof π_x for some index x, (b_1,…,b_h) = (x), also holds the opening proofs π^f_b_0,…,b_j_b_j+1, j ∈ [h], in memory.
As we will see in the next section, the user also holds the KZG commitments at the children of the inner nodes on the path from the root to the message m_x, C_b_0,…,b_j,i for all j ∈ [h] and i ∈ [c] in memory.
These opening proofs and KZG commitments are not broadcast as part of any proof; however, they are needed for the user to locally update its Verkle proof after message updates.
§.§ Proof Update
When the messages (i_j, m_i_j)_j ∈ [k] are updated to (i_j, m'_i_j)_j ∈ [k], to calculate the new Verkle proof π'_x, the user must obtain the new commitments C'_b_0, …, C'_b_0,…,b_h-1 on the path from the root to message m_x, the new commitment [g'(X)] and the new opening proof π' for the polynomial h'(X)-g'(X) at index t'= H(r',[g'(X)]).
Message updates change the commitments at the inner nodes, which in turn results in new polynomials f_b_0,…,b_j, j ∈ [h].
Suppose each polynomial f_b_0,…,b_j, j ∈ [h], is updated so that
C
f'_b_0,…,b_j(X) = f_b_0,…,b_j(X) + ∑_i=0^c-1(f'_b_0,…,b_j(i)-f_b_0,…,b_j(i)) L_i(X),
where, by definition, f'_b_0,…,b_j(i)-f_b_0,…,b_j(i) = u'_b_0,…,b_j,i-u_b_0,…,b_j,i = H(C'_b_0,…,b_j,i)-H(C_b_0,…,b_j,i).
Then, given the new and the old commitments (C_b_0,…,b_j,i,C'_b_0,…,b_j,i) for i ∈ [c] and j ∈ [h], the table of Lagrange basis polynomials, and using the technique in Section <ref>, the new opening proofs π̃^f_b_0,…,b_j_b_j+1 after the message updates can be computed as follows for j ∈ [h]:
C
π̃^f_b_0,…,b_j_b_j+1 = π^f_b_0,…,b_j_b_j+1 ∏_i=0^c-1[L_i(X)-L_i(b_j+1)/X-b_j+1]^(H(C'_b_0,…,b_j,i)-H(C_b_0,…,b_j,i)),
where [L_i(X)-L_i(b_j+1)/X-b_j+1] is the opening proof of the Lagrange basis polynomial L_i(X) at index b_j+1.
Once the new opening proofs are found, the new commitment [g'(X)] and the new proof π' become
C
[g'(X)] = ∏_j=0^h-1 (π̃^f_b_0,…,b_j_b_j+1)^r'^j, π' = ∏_j=0^h-1 (π̃^f_b_0,…,b_j_b_j+1)^r'^j/t'-b_j+1
where r'=H(C'_b_0,b_1,..,C'_b_0,…,b_h-1,u'_b_0,b_1,..,u'_b_0,…,b_h,b_1,..,b_h) and
t'=H(r',[g'(X)]).
Note that both r' and t' can be calculated by the user given the new KZG commitments C'_b_0,…,b_j,i for all i ∈ [c] and j ∈ [h].
Finally, to retrieve the new KZG commitments C'_b_0,…,b_j,i for all i ∈ [c] and j ∈ [h], the user inspects the commitments published as part of the update information U:
C'_b_0,b_1,…,b_j-1,i U[b_0,b_1,…,b_j-1,i] if U[b_0,b_1,…,b_j-1,i] ≠ and C'_b_0,b_1,…,b_j-1,i C_b_0,b_1,…,b_j-1,i otherwise, for all i ∈ [c] and j ∈ [h].
In Verkle trees, the user cannot calculate the effect of an updated message on an arbitrary inner node without the knowledge of the inner nodes on the path from the message to the target node.
For instance, suppose U[b_0,b_1,…,b_j-1,i] = for some i ∈ [c] and j ∈ [h], and the user wants to calculate the effect of an update from m_x to m'_x on C'_b_0,…,b_j-1,i,b̃_j+1,…,b̃_h, (x) = (b_1,…,b_j-1,i,b̃_j+1,…,b̃_h) and b̃_j = i.
Then, for each ℓ∈{j,…,h-1}, the user have to find
rCl
C'_b_0,…,b̃_j,…,b̃_h m'_x
C'_b_0,…,b̃_j,…,b̃_ℓ C_b_0,…,b̃_j,…,b̃_ℓ [L_b̃_ℓ+1]^(u'_b_0,…,b̃_j,…,b̃_ℓ+1-u_b_0,…,b̃_j,…,b̃_ℓ+1),
where C'_b_0,…,b̃_j,…,b̃_ℓ are the commitments on the path from the target commitment C_b_0,b_1,…,b_j-1,i to the message m_x.
Hence, the user has to know the original commitments on the path from the message to the target commitment, , keep track of inner nodes, which contradicts with the idea of stateless clients.
This shows the necessity of publishing all of the updated inner nodes as part of the update information.
§.§ Complexity
Suppose each KZG commitment is of size |G| and each hash H(C) of a KZG commitment, each inner node, has size |H|.
Then, updating a single message results in one update at each level of the Verkle tree and requires Θ(h|H|) group operations.
Thus, when k messages are updated, the new Verkle root can be found after Θ(kh|H|) group operations.
As U consists of the published KZG commitments at the inner nodes and their indices, |U| = Θ(k log_c(N)(log(N)+|G|)) = Θ(k)|G|, which implies g_1(k) = k.
The user can replace each KZG commitment at the children of the inner nodes from the root to its message in Θ(log(N)+|G|) time by running a binary search algorithm over U.
Since there are ch such commitments to be updated, , C_b_0,…,b_j,i, i ∈ [c] and j ∈ [h], updating these commitments takes Θ(c h (log(N)+|G|)) = Θ(1) time.
Upon obtaining the new commitments C'_b_0,…,b_j-1,i, i ∈ [c], j ∈ [h], with access to the table of Lagrange basis polynomials, the user can update each opening proof π_b_j+1 (for the function f_b_0,…,b_j),
j ∈ [h], with Θ(c|H|) group operations.
Since there are h such proofs, updating them all requires Θ(c h |H|) group operations.
Given the new proofs, computing the new commitment [g'(X)] and proof π' requires Θ(h |H|) group operations.
This makes the total complexity of updating a Verkle proof Θ(c h + 2 h) |H| T_G + Θ(c h (log_c(N)+|G|)).
For a constant c and h = log_c(N), this implies a worst-case time complexity of Θ(1) |H| T_G for Verkle proof updates, , g_2(k) = 1.
§.§ A Concrete Evaluation
We now estimate the size of the update information and the number of group operations to update an opening proof after observing an Ethereum block consisting of ERC20 token transfers.
As in Section <ref>, suppose the block has the target size of 15 million gas <cit.>, and each token transfer updates the balance of two distinct accounts stored at separate leaves of the Verkle tree.
Then, there are ∼ 230 such transactions in the block, and the block updates k = 460 accounts.
We assume that the Verkle tree has degree 256 ( <cit.>) and commits to 256^3 accounts as in Section <ref>.
Then, each proof consists of 2 KZG commitments, C_,b_1 and C_,b_1,b_2 and a multiproof consisting of the commitment [g(X)] and proof π'.
These components are elements of the pairing-friendly elliptic curve BLS12_381 and consist of |G| = 48 bytes <cit.>.
This implies a proof size of (log_c(N)+1)|G| = 192 bytes (excluding the message at the leaf and its hash value; adding those makes it 272 bytes).
When 460 accounts are updated, in the worst-case, the update information has to contain k log_c(N) (log(N)+|G|) = 460 × 3 × (24+48) Bytes, , 99.4 kBytes.
This is comparable to the size of the Ethereum blocks, which are typically below 125 kBytes <cit.>.
Hence, even though the update information of Verkle trees is linear in k, it does not introduce a large overhead beyond the block data.
Note that the runtime of the proof updates are constant and do not scale in the number of updated messages k, or the Ethereum block size.
On the other hand, in the worst case, an opening proof can be updated after c log(c) |H| + 2 log_c(N) |H| group operations.
Then, with |H|=256, the number of bits output by SHA256, as many as c log_c(N) |H| + 2 log_c(N) |H| = (c + 2) log_c(N) |H| = 774 × 2256 ≈ 200,000 elliptic curve multiplications might have to be made.
Following the benchmarks published in <cit.> for the specified curve, these operations can take up to (c + 2) log_c(N) 0.000665471 ns = 0.52 seconds on commodity hardware, given a runtime of 665471 nanoseconds per exponentiation of a group element with a message hash value.
This is again comparable to the 12 second inter-arrival time of Ethereum blocks.
Table <ref> compares the Verkle proof size |π| = (log_c(N)+1) |G|, update information size |U| = k log_c(N) (log_cN+|G|), the upper bound (c + 2) log_cN |H| on the number of group operations needed for a single proof update and the estimated time it takes to do these operations on a commodity hardware for different values of c, the Verkle tree degree, while keeping the number of accounts and the updated accounts fixed at 2^24 and 460 respectively.
The table shows the trade-off between the Verkle proof and update information size on one size and update complexity on the other.
Comparing Table <ref> with Table <ref> shows that the Verkle tree with any given degree c, 1 < c ≤ 256, significantly outperforms the existing homomorphic Merkle trees in Section <ref> in terms of almost all of proof size, update information size and proof update time.
§ LOWER BOUND
Finally, we prove the optimality of our VC scheme with sublinear update by proving a lower bound on the size of the update information given an upper bound on the complexity of proof updates.
The lower bound is shown for VCs that satisfy the following property.
It formalizes the observation that for many dynamic VCs (, Merkle trees <cit.>, Verkle trees <cit.>, KZG commitments <cit.>, RSA based VCs <cit.>), the opening proof for a message at some index can often act as a commitment to the vector of the remaining messages.
A VC scheme is said to be if the following probability is negligible in λ for all PPT adversaries 𝒜:
C
[Verify_pp(C, m_i^*, i^*, π) = 1
Verify_pp(C', m_i^*, i^*, π) = 1 pp KeyGen(1^λ, N);
π, m_i^*, (m_0, …, m_i^*-1, m_i^*+1, …, m_N-1),
(m'_0, …, m'_i^*-1, m'_i^*+1, …, m'_N-1) 𝒜(pp);
(m_0, …, m_i^*-1, m_i^*+1, …, m_N-1)
≠(m'_0, …, m'_i^*-1, m'_i^*+1, …, m'_N-1);
Commit_pp(m_0, …, m_i^*-1, m_i^*, m_i^*+1, …, m_N-1) = C;
Commit_pp(m'_0, …, m'_i^*-1, m_i^*, m'_i^*+1, …, m'_N-1) = C' ]
To prove the lower bound, we first show that implies that (i^*, m_i^*, π) is a binding commitment to the rest of the vector.
Consider a dynamic and VC, where π is the correctly generated opening proof for the message m_i at some index i.
Then, for any i ∈ [N], it holds that the tuple (i, m_i, π) is a binding commitment to the vector of messages m_j, j ∈ [N], j ≠ i.
Since the VC is , with overwhelming probability, no PPT adversary 𝒜 can find an opening proof π^*, an index i^*, a message m^* and two sequences of messages
such that
C
(m_1, …, m_i^*-1, m_i^*+1, …, m_N-1) ≠(m'_1, …, m'_i^*-1, m'_i^*+1, …, m'_N-1)
and Verify_pp(C, m_i^*, i^*, π) = Verify_pp(C', m_i^*, i^*, π) = 1,
where C and C' are commitments to the message sequences (m_1, …, m_i^*-1, m_i^*, m_i^*+1, …, m_N-1) and (m'_1, …, m'_i^*-1, m_i^*, m'_i^*+1, …, m'_N-1).
Thus, it holds that the tuple (i, m_i, π) is a binding commitment to the vector of messages m_j, j ∈ [N], j ≠ i, with the following new commitment function:
C
NewCommit_pp((m_j)_j ∈[N], j ≠i) = (i, m_i, Open_pp(m_i, i, )),
where = Commit_pp(m_0, …, m_N-1)..
The following lemma shows that all randomized VCs can be derandomized to obtain a deterministic and secure VC as we do not use hiding commitments in this work.
Consider a VC , where the commitment is a random function of the public parameters pp and the committed messages.
Let ' denote the VC that is the same as , except that the randomness is fixed.
Then, ' is a correct and secure VC with at most the same upper bound on the error probability.
Let R denote the sequence of bits sampled uniformly at random from the set ℛ to instantiate the VC .
Since is binding, no PPT adversary 𝒜 can find two different sequences of messages 𝐦 and 𝐦' such that (𝐦, R) = (𝐦', R') for some R,R' ∈ℛ, except with negligible probability.
This implies that for any fixed R^* ∈ℛ, no PPT adversary 𝒜 can find two different sequences of messages 𝐦 and 𝐦' such that (𝐦, R^*) = (𝐦', R^*), except with negligible probability.
Hence, the commitment scheme '(.) = (., R^*) is a position-binding, , secure VC.
Its correctness follows from the correctness of .
Finally, equipped with Lemmas <ref> and <ref>, we can prove the following lower bound for dynamic and VCs.
Consider a dynamic and VC such that for every PPT adversary 𝒜, it holds that
C
[Verify_pp(C, m, i, π_i) = 1
Verify_pp(C, m', i, π'_i) = 1 m ≠m' pp KeyGen(1^λ, N)
(C, m, m', π_i, π'_i) 𝒜(pp)] ≤e^-Ω(λ).
Then, for this VC, if g_2(k) = O(k^1-ν), then g_1 = Ω(k^ν) for all ν∈ (0,1).
Suppose the messages m_i_j, j ∈ [k], are updated to m'_i_j.
Define 𝒮 as the sequence (m'_i_j)_j ∈ [k], and let m'_i = m_i for i ∉{i_j j ∈ [k]}.
Let 𝒫_i, i ∈ [N], denote the user that holds the opening proof π_i for the message m_i at index i, and aims to calculate the new proof π'_i for the message m'_i using π_i, the update information U and the old and the new sequences of messages m_i, m'_i, i ∈ [N].
Suppose g_2 = O(k^1-ν).
Then, there exists a constant α such that each user can read at most α k^1-ν of the updated messages while updating its opening proof.
Let 𝒮_i ⊆ (m'_i_j)_j ∈ [k] denote the sequence of updated messages and their indices, which were not observed by 𝒫_i, and 𝒮_i = 𝒮∖𝒮_i denote the sequence read by 𝒫_i.
Here, |𝒮| denotes the number of messages within the sequence 𝒮.
Since 𝒫_i is assumed to know m'_i, it must be that m'_i ∈𝒮_i.
We next show that each user 𝒫_i that successfully updates its opening proof must download enough bits of U to generate a binding, deterministic commitment to the set 𝒮_i.
By Lemma <ref>, the tuple (i, m'_i, π'_i) is a binding commitment to the sequence of messages (m'_j)_j ∈ [N], j ≠ i.
This implies that the tuple (i, 𝒮_i, π'_i) is a binding commitment to the sequence 𝒮_i.
By Lemma <ref>, the commitment (i, 𝒮_i, π'_i) can be de-randomized to obtain a deterministic commitment C_i to the sequence 𝒮_i (with at most the same upper bound on the error probability).
Let denote the deterministic VC scheme such that C_i = (𝒮_i).
Since is a deterministic function given the public parameters, and the updated messages are sampled independently and uniformly at random, then I(𝒮_i;{m_i}_i ∈ N,𝒮_i|pp) = 0, where I(.;.) is the mutual information.
Moreover, as π_i is a function of the old messages {m_i}_i ∈ N and the randomness of the original VC, I(C_i; π_i|pp) = 0.
Hence, C_i = f(U, i, {m_i}_i ∈ N, π) is a deterministic function of the update information U.
For all i ∈ [k], it holds that |𝒮_i| ≥ k - α k^1-ν and m'_i ∉𝒮_i.
Given these constraints, the minimum number of distinct sequences 𝒮_i is k/α k^1-ν = k^ν/α.
For an appropriately selected β that will be defined later, without loss of generality, let 𝒮_0, …, 𝒮_M-1 denote the first
C
M = min(⌊k^ν/β - α/β - λ/βk^1-ν ⌋, k^ν/α)
distinct sequences.
Since C_i is a deterministic function of U for all i ∈ N, it holds that the Shannon entropy H(.) of U satisfies the following expression:
C
H(U) ≥H(C_0, …, C_M-1)
≥H(C_0) + ∑_i=1^M-1 H(C_i | C_0, …, C_i-1)
As g_2(k) = O(k^1-ν), there exists a constant β such that each user can download at most β k^1-ν bits of data from U.
Then, for all i ∈ [k], it must be that H(C_i) ≤ H(U) ≤β k^1-ν since C_i is a deterministic function of U for each i ∈ [N].
Finally, we show that H(C_0), H(C_i | C_0, …, C_i-1) = Ω(λ) for all i=1, …, M-1.
Towards contradiction, suppose ∃ i^* H(C_i^* | C_0, …, C_i^*-1) = o(λ).
Note that
rCl
H(C_0, …, C_i^*-1) ≤ ∑_i=0^M-1 H(C_i)
≤ min(k^ν/β - α/β - λ/βk^1-ν, k^ν/α) βk^1-ν ≤k-αk^1-ν-λ.
Now, consider an adversary 𝒜 that tries to break the binding property of the VC scheme .
Due to the upper bound on the entropy of (C_0, …, C_i^*-1), it holds that
H(𝒮_i^* | C_0, …, C_i^*-1) ≥λ;
since H(𝒮_i^*) ≥ k-α k^1-ν, and
rCl
H(𝒮_i^*) - H(𝒮_i^* | C_0, …, C_i^*-1) = I(𝒮_i^*; (C_0, …, C_i^*-1))
≤ H(C_0, …, C_i^*-1) ≤k-αk^1-ν-λ.
However, when H(C_i^* | C_0, …, C_i^*-1) = o(λ), for sufficiently large λ, given (C_0, …, C_i^*-1), the adversary can find a collision such that (𝒮_i^*)=(𝒮'_i^*) for two 𝒮_i^*≠𝒮'_i^*, with probability 2^-o(λ).
As this is a contradiction, it must be that H(C_0) and H(C_i | C_0, …, C_i-1) = Ω(λ) for all i < M, and thus, H(U) = Ω(k^νλ) and g_1(k) = Ω(k^ν).
Theorem <ref> shows that the update information length scales as Θ(k^νλ) when the runtime complexity for proof updates is Θ(k^1-ν) and the error probability for the security of the VC is e^-Ω(λ) for a PPT adversary.
When the error probability is just stated to be negligible in λ, then the same proof can be used to show that the update information length must scale as Ω(k^ν(λ)) for any polynomial function of log(λ).
§ CONCLUSION
Dynamic VCs with sublinear update are the key to reducing the size of the global update information while minimizing the runtime of clients synchronizing with the latest commitment.
In this work, we propose a construction that can achieve an update information size of Θ(k^ν) and a proof update time of Θ(k^1-ν) in the number of changed messages k.
Our construction combines a novel update algorithm (Alg. <ref>) with homomorphic Merkle trees <cit.> that allow each inner node to be expressed as a linear function of the underlying messages.
It achieves the smallest asymptotic complexity for the update information size and proof update time.
We also provide update algorithms for the Verkle trees proposed for stateless clients on Ethereum.
The existing instantiations of homomorphic Merkle trees are based on lattices and require relatively large parameters for security.
Consequently, despite the appealing asymptotic complexity of our construction,
its performance for concrete parameters is dominated by Verkle trees.
As such, designing asymptotically optimal and practically efficient dynamic VCs remains an open problem.
An interesting direction is to design a more preferment homomorphic Merkle tree system.
Acknowledgments.
This work was partially funded by NSF, DARPA, the Simons Foundation, and NTT Research.
Additional support was provided by the Stanford Center for Blockchain Research.
Opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do not
necessarily reflect the views of DARPA.
plain
§ LOWER BOUND ON THE SIZE OF THE UPDATE INFORMATION
Consider a dynamic accumulator, where k out of N messages m_i,j are updated to m'_i,j≠ m_i_j, j ∈ [k].
Suppose |M| = (λ).
Then, Ω(k (log(N|ℳ|))) bits of information must be published to enable updating the opening proofs after these k updates.
The proof idea is very similar to those presented in <cit.>.
Namely, the update information must contain a minimum amount of bits for the VC to remain correct and secure after the update.
Consider a game between a platform 𝒫 maintaining the data structures of the VC and an adversary 𝒜.
The platform 𝒫 updates k out of N messages m_i,j to m'_i,j≠ m_i_j, j ∈ [k], in a way not known to user 𝒜, and publishes the update information U along with the new commitment value C' (let m'_i = m_i for i ∉{i_j j ∈ [k]}).
Before receiving the update information, 𝒜 knows the old sequence of messages m_i, i ∈ [N], and their opening proofs π_i.
Upon receiving the update information, 𝒜 updates the opening proofs for each message to π'_i.
Then, it must be that for all j ∈ [k], Verify_pp(C', m'_i_j, i_j, π'_i_j) = 1, and for all i ∉{i_j j ∈ [k]}, Verify_pp(C', m_i, i, π'_i) = 1.
Otherwise there would be messages among m'_i, i ∈ [N], for which an updated witness cannot be computed, violating correctness.
Similarly, for all j ∈ [k], Verify_pp(C', m̃, i_j, π'_i_j) = 0 for any m̃≠ m'_i_j; as otherwise the position binding property, thus security, would be violated.
Hence, by calling the function Verify_pp(C', m_i, i, π'_i_j) for each index, 𝒜 can figure out the indices i_j, j ∈ [k], where the messages were updated.
Similarly, by evaluating the function Verify_pp(C', m̃, i_j, π'_i_j) for the |ℳ| possible messages m̃ for each j ∈ [k], 𝒜 can identify the new value m'_i_j of the message at each such index i_j.
Hence, the adversary can recover the sequence (i_j, m'_i_j)_j ∈ [k].
As there are N!/(N-k)!log^k|ℳ| possible sequences (i_j, m'_i_j)_j ∈ [k], it holds that
|U| ≥log(N!/(N-k)!log^k|ℳ|) = Ω(k (logN+log|ℳ|)).
When |M| = Ω((λ)), the minimum number of bits to be published depends on the error probability for the security of the PPT adversary.
As in Remark <ref>, if |ℳ| = Θ(2^λ) and the error probability is e^-Ω(λ), then the same proof can be used to show that Ω(kλ) bits of information must be published.
When the error probability is just stated to be negligible in λ, then the number of bits must scale as Ω(kλ) for any polynomial function of log(λ).
§ HOMOMORPHIC MERKLE TREES WITH NO UPDATE INFORMATION
§.§ Update Information
When k messages are updated, the new commitment, the new Merkle root can be calculated just like any other inner node, by incorporating the effect of the old and the new messages, (i_j, m'_i_j-m_i_j)}_j ∈ [k]:
rCl
C' = u'_b_0 = u_b_0 + ∑_j ∈[k] (h_i_j,0(m'_i_j) - h_i_j,0(m_i_j))
= C + ∑_j ∈[k] sign(m'_i_j-m_i_j)h_i_j,0(|m'_i_j - m_i_j|)
As in KZG commitments, the update information is U=∅.
§.§ Proof Update
When the messages are modified at k points, each user holding a Merkle proof π_x for index x can calculate the new values of the inner nodes within the proof using the old and the new messages and modify the proof respectively.
§.§ Complexity
Calculating each partial digest h_x,j takes at most logN function evaluations.
Then, each user can update each inner node within its Merkle proof after at most klog(N) operations, making the total number of operations Θ(klog^2N) = Θ(k) in the worst case.
The size of the update information U is Θ(1).
Hence, this scheme matches the algebraic VCs in terms of complexity.
§ WHY ARE HOMOMORPHIC MERKLE TREES NEEDED?
Merkle trees based on SHA256 can also achieve complexity sublinear in k, for both the update information and the runtime of proof updates, if the users have access to the old messages and inner nodes of the Merkle tree.
In this case, homomorphism is not needed since the nodes can find the effect of the updated messages on the inner nodes within their Merkle proofs by hashing these messages together with the old inner nodes.
However, this is possible for only a single batch of updates.
Indeed, if this scheme is to be repeated, the assumption of having access to the old inner nodes requires the users to keep track of changes throughout the Merkle tree, by calculating the effect of all updated messages on all inner nodes.
This implies a runtime linear in k per proof updates.
In contrast, homomorphic Merkle trees can maintain a sublinear complexity for future proof updates since they do not require access to the old messages and inner nodes for finding the partial digests of the updated messages.
§ AN ALTERNATIVE CONSTRUCTION
An alternative tree-based VC is proposed by <cit.>, where each inner node is itself a lattice-based VC to its children (akin to Verkle trees <cit.>).
Opening proof for a message consists of the inner nodes (commitments) on the path from the message to the root, along with the opening proofs for these inner nodes with respect to their parent nodes.
The construction again enables expressing each inner node as a sum of partial digests of the messages underneath.
Using the public parameters and the updated inner nodes, users can then derive their updated opening proofs at different heights of the tree.
This construction supports trees of large degrees c without a linear increase in the proof size as would be the case for Merkle trees; this however comes at the cost of a larger runtime complexity for proof updates, proportional to the degree.
Section <ref> describes similar steps in the context of Verkle trees, and exposes the dependence of the runtime complexity of proof updates on the tree degree c.
§ DERIVATION OF THE OPENING PROOF Π^(𝐡-𝐠)_𝐭
Since
rCl
h(X)-g(X) = ∑_j=0^h-1 r^j (f_b_0,…,b_j(X)/t-b_j+1 - f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1)
= ∑_j=0^h-1 r^j (X-t)f_b_0,…,b_j(X)+u_b_0,…,b_j+1(t-b_j+1)/(t-b_j+1)(X-b_j+1),
the opening proof π=π^(h-g)_t for index t within the polynomial h(X)-g(X) is
rCl
[h(X)-g(X)-(h(t)-g(t))/X-t]
= [∑_j=0^h-1 r^j/X-t ((X-t)f_b_0,…,b_j(X)+u_b_0,…,b_j+1(t-b_j+1)/(t-b_j+1)(X-b_j+1)-u_b_0,…,b_j+1/t-b_j+1) ]
= [∑_j=0^h-1 r^j/t-b_j+1 f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]
= ∏_j=0^h-1 [f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]^r^j/t-b_j+1 = ∏_j=0^h-1 (π^f_b_0,…,b_j_b_j+1)^r^j/t-b_j+1
§ UPDATE AND PROOF UPDATE ALGORITHMS FOR KZG COMMITMENTS AND MERKLE TREES
|
http://arxiv.org/abs/2307.07252v1 | 20230714095539 | Analysis of the higher twist GTMD $F_{31}$ for proton in the light-front quark-diquark model | [
"Shubham Sharma",
"Harleen Dahiya"
] | hep-ph | [
"hep-ph"
] |
=6.0in =8.25in
=-0.3in =-0.20in
#1
#1
#1
#1
#1
#1
and
#1
Submitted to #1
Abstract
Presented
PRESENTED AT
|
http://arxiv.org/abs/2307.04253v1 | 20230709193405 | The equality case in the substatic Heintze-Karcher inequality | [
"Stefano Borghini",
"Mattia Fogagnolo",
"Andrea Pinamonti"
] | math.DG | [
"math.DG",
"math.AP",
"49Q10 (Primary) 53C24, 58J32, 53E10, 53C21 (Secondary)"
] |
We provide a rigidity statement for the equality case for the Heintze-Karcher inequality in substatic manifolds. We apply such result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle's characterization of constant mean curvature hypersurfaces in warped products.
Relativistic time dilation as a quantum mechanism
Esteban Martínez Vargas
August 12, 2023
=================================================
MSC (2020):
49Q10, 53C24 , 58J32, 53E10, 53C21.
Keywords:
Heintze-Karcher inequality, substatic manifolds, constant mean curvature, Alexandrov theorem.
Relativistic time dilation as a quantum mechanism
Esteban Martínez Vargas
August 12, 2023
=================================================
§ INTRODUCTION AND MAIN STATEMENTS
The Heintze-Karcher inequality usually denotes the geometric inequality that, in its more simple form for domains Ω sitting in ^n, with smooth and strictly mean-convex boundary Σ, reads as
n-1/n∫_Σf/ d σ≥Ω.
This inequality, that was essentially contained in the seminal earlier paper by Heintze-Karcher <cit.> was first pointed out in this very form by Ros <cit.>, where it was also observed that it holds in a general manifold with nonnegative Ricci curvature. Moreover, he showed that equality in <ref> is in force only if Ω is a flat Euclidean balls.
Li-Xia <cit.> very vastly generalized Ros' Heintze-Karcher inequality to the setting of substatic Riemannian manifolds with horizon boundary. We recall that with this locution we mean a Riemannian manifold (M, g) endowed with a nonnegative smooth function f (the substatic potential) satisfying
f - ∇∇ f + Δ f g ≥ 0,
and where ∂ M = {f = 0} is a compact, minimal, regular level set of f (i.e. with ∇ f≠ 0 on ∂ M). In particular, the boundary of such a manifold is empty if and only if f is strictly positive. We are occasionally referring to the tensor on the left-hand side of <ref> as substatic Ricci tensor. Very interestingly, it can be in fact checked to arise as Ricci tensor of a suitable affine connection on (M, g), see <cit.>. The condition <ref> stems naturally from the Einstein Fields Equations of General Relativity, and it is easily observed to hold for initial data sets of static spacetimes. We leave the interested reader to <cit.> and <cit.> for the explicit computations.
Letting Σ be a smooth strictly mean-convex hypersurface homologous to ∂ M, and Ω the bounded set enclosed by Σ and ∂ M, the substatic Heintze-Karcher inequality <cit.> has been sharpened
in <cit.> as
n-1/n∫_Σf/ d σ≥∫_Ω f d μ + c_∂ M∫_∂ M∇ f dσ,
where
c_∂ M =n-1/n∫_∂ M∇ f dσ/∫_∂ M∇ f[Δ f/f - ∇∇ f/f(∇ f/∇ f, ∇ f/∇ f)] dσ .
The above constant has been shown to be well defined and strictly positive in <cit.>, given the existence of a strictly mean-convex Σ homologous to M as in our case.
Our first result shows that a strong rigidity is triggered when <ref> holds with equality sign.
Let (M, g) be a substatic Riemannian manifold with connected horizon boundary ∂ M, such that the substatic potential f satisfies
∇∇ f/f∈ C^0, α(M ∪∂ M)
for α∈ (0, 1).
Let Σ be a connected, smooth strictly mean-convex hypersurface homologous to ∂ M Then, the Heintze-Karcher inequality <ref> holds with equality if and only if the domain Ω such that ∂Ω = Σ⊔∂ M is isometric to
([s_0, s] ×∂ M, ds ⊗ ds/f(s)^2 + s^2 g_∂ M).
A version of inequality <ref> was originally obtained by Brendle <cit.> as the crucial step for obtaining an Alexandrov-type Theorem in warped product manifolds. For proving <ref>, such warped products were assumed to satisfy a set of assumptions (H1)-(H3), recalled and discussed in <ref> below. As we are going to recall, if
they are satisfied, then in particular the warped product is substatic with horizon boundary.
Thus, <ref> directly yields a rigidity statement for Brendle's Heintze-Karcher inequality if the additional, technical <ref> is satisfied.
Such assumption was fundamental in the elliptic proof of <ref> conceived by Li-Xia <cit.> and reworked by the second and third named authors <cit.>.
Exploiting a new synergy between <ref> and the geodesic flow proof worked out in <cit.> to provide <ref> in the warped product setting, we are able to remove assumption <ref> in the special warped product geometry, endowing <cit.> of an optimal rigidity statement.
Let (M, g) be a substatic warped product with cross section ∂ M = N, of the form
([s_0, s) × N, ds ⊗ ds/f(s)^2 + s^2 g_N).
Let Σ be a connected, smooth, strictly mean-convex hypersurface homologous to N satisfying <ref> with the equality sign. Then, Σ = {s = c} for some c ∈ (s_0, s).
We address the reader to the beginning of <ref> for a more detailed presentation of the very peculiar proof of the above result.
It was already observed in <cit.> that a constant mean curvature hypersurface must fulfil the identity in <ref>, as a consequence of a straightforward Minkowski identity <cit.>.
Thus, <ref> directly provides the following characterization of hypersurfaces of constant mean curvature in substatic warped product, improving on <cit.>.
Let (M, g) be a substatic warped product with a connected horizon boundary, of the form
([s_0, s) × N, ds ⊗ ds/f(s)^2 + s^2 g_N).
Let Σ be a connected smooth hypersurface homologous to ∂ M = N of constant mean curvature. Then, Σ = {s = c} for some c ∈ (s_0, s).
As clarified with additional details in <ref>, the substatic warped products of the form <ref> correspond precisely to the family of warped products considered in the Alexandrov-type Theorem <cit.>.
On the other hand, such result was proved under an additional assumption, (H4), substantially prescribing the Ricci curvature being smallest in the radial direction.
While (H4) is verified on a number of known model solutions, such as the Schwarzschild–de Sitter and Reissner–Nordström manifolds mentioned as applications in <cit.>, there are important examples where (H4) does not hold.
Indeed, the Schwarzschild–Anti de Sitter warped product
M=[s_0,+∞)× N , g=ds⊗ ds/f^2+s^2 g_N , f=√(-1+s^2-2ms^2-n) ,
with cross-section N satisfying _g_N≥ -(n-2)g_N and such that -√((n-2)^n-2/n^n)<m≤ 0 is a substatic manifolds with horizon boundary that does not satisfy (H4). In the special case where _g_N= -(n-2)g_N, the warped product (<ref>) is a well known vacuum static solution of the Einstein Fields Equations that has been investigated to some extent in the literature (see e.g. <cit.> and references therein) and constituted the model for the Lee-Neves Riemannian Penrose Inequality <cit.>.
<ref>, stemming from our novel proof, allows to fully drop the extra hypothesis (H4), hence in particular also applies, for example, to the metric <ref>.
§.§ Further directions and remarks
We conclude mentioning, without any attempt to be complete, a couple of papers where the extra assumption <ref> or (H4) is added in connection with <cit.> and <cit.>. Namely, in <cit.>, the authors provided a quantitative version of Brendle's Alexandrov Theorem by exploiting the alternative proof through elliptic techniques devised in <cit.>; consequently they assume <ref>. In <cit.>, a far-reaching nonsmooth version of Brendle's geodesic flow technique is worked out, leading to a characterization of sets with finite perimeter with a distributional notion of constant mean curvature; as in <cit.>, (H4) is assumed, or some suitable weaker variant (see <cit.> and <cit.>). It may then be fruitful to elaborate on our arguments leading to <ref>, based on the combination of the elliptic and geodesic flow techniques, in order to go beyond <ref> and (H4) also in these kinds of more technical results.
§.§ Structure of the paper
In <ref> we provide the results of an elementary but, to our knowledge, not yet available analysis of warped product manifolds, and furnish a comparison with Brendle's set of assumptions (H1)-(H4). In <ref> we prove a generalized version of <ref>, where hypersurfaces Σ with several connected components as well as disconnected horizons are taken into account. In <ref> we prove <ref>, and deduce <ref>. We conclude the work with an Appendix containing the proofs of the computational results gathered in <ref>.
§.§ Acknowledgements
This work was initiated during a visit of A. P. at the Centro De Giorgi in Pisa. He warmly thanks the institution for the excellent working conditions offered.
Part of this work has been carried out while S. B. and M. F. were attending the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry that took place at the Fields Institute in Toronto.
They warmly thank the staff, the organizers and the colleagues for the wonderful atmosphere and the excellent working conditions set up there.
During the preparation of the work, M. F. was supported by the European Union – NextGenerationEU and by the University of Padova under the 2021 STARS Grants@Unipd programme “QuASAR".
The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), which is part of the Istituto
Nazionale di Alta Matematica (INdAM), and are partially funded by the GNAMPA project “Problemi al bordo e applicazioni geometriche".
A.P. and S.B. are also supported by MIUR and the University of Trento, Italy.
The authors are grateful to Luciano Mari, Lorenzo Mazzieri, Mario Santilli, Alessandro Savo and Mingxuan Yang
for the many useful conversations had during the preparation of the work, that substantially helped to improve the quality.
§ SUBSTATIC WARPED PRODUCTS
We consider warped product manifolds
(I × N, dr ⊗ dr + h^2(r) g_N),
for (N,g_N) a closed (n-1)-dimensional Riemannian manifold,
and I = [0, r), with r > 0. We will always assume that {r=0} is either a horizon boundary or a single point representing the origin of the polar coordinates. Of course, in the latter case, in order for the metric to be smooth at the point with r=0, the cross section (N,g_N) must be homothetic to a round sphere.
When ḣ≠ 0, we will find convenient to write warped products also in the equivalent form (ds⊗ ds)/ḣ(r)^2+s^2 g_N, for s=h(r). Since we will see in <ref> and <ref> below that the models we are interested in satisfy f=ḣ,
an advantage of this form is that the metric now depends directly on the substatic potential f, without the need of the auxiliary warping function h. Furthermore, more importantly, the new coordinate s allows to write the function f and the metric g in the explicit form (<ref>).
Let (M, g) be a substatic warped product of the form <ref> with positive nondecreasing h, with either a empty boundary or a horizon boundary.
If the substatic potential f is a function of the coordinate r only and
[f - ∇∇ f + Δ f g](∇ r, ∇ r) = 0
then, up to multiplying f and/or g_N by a positive constant,
the manifold (M,g) and the substatic potential f satisfy one of the following:
(i) there exists c∈ such that f=f(r) satisfies f̈+(n-2)c f≥ 0 and
g=dr⊗ dr+ g_N , _g_N≥ (n-2)c g_N ,
(ii) there exist c∈ and a function η:[h(r̅)^-n,h(0)^-n]→ with η”≥ 0 such that
g=ds⊗ ds/f^2(s)+s^2 g_N , _g_N≥ (n-2)c g_N ,
f = √(c+s^2η(s^-n)) .
In particular, s=h(r) and f(s)=ḣ(r).
We point out that the family of warped products considered by Brendle <cit.> correspond to the family in (ii) above, see <ref> for more details on this.
In the proof of <ref> we are going to exploit the following strengthening of Proposition <ref>, in force when the substatic Ricci tensor vanishes in an additional direction.
Under the assumptions of <ref>, if we also assume that h is not constant and that for every t∈[a,b], a,b∈ there exist x∈ N and a nontrivial X∈ T_xN⊂ T_(t,x)M such that
[f - ∇∇ f + Δ f g](X, X) = 0 ,
then, up to multiplying f by a positive constant, in the domain [a,b]× N the metric g and the function f have the form
g=ds⊗ ds/f^2+s^2 g_N , _g_N≥ (n-2)c g_N ,
f = √(c-λ s^2-2m s^2-n),
where f = ḣ and λ, m ∈.
The proofs of <ref> and <ref> involve elementary but lengthy calculations and have been included in the Appendix.
The potential f given in <ref> coincides with that of the de Sitter/Anti de Sitter–Schwarzschild manifold. When the cross-section is Einstein, these are known to be, together with cylinders, the only static warped product manifolds with compact horizon boundary, that is, with vanishing substatic Ricci tensor (see <cit.> or <cit.>). <ref> constitutes thus a more general warped product classification result.
It is worth discussing in some detail the regularity of η at the horizon boundary. Let s_0 be the value of s corresponding to the horizon. We can write η in terms of f as η(s^-n)=s^-2(f^2-c), hence in particular the value η(s_0^-n)=-cs_0^-2 at the boundary is finite. We can also show that η'(s_0^-n) is well defined. In fact, we can easily compute
| f|=f(s)f'(s)=sη(s^-n)-n/2s^1-nη'(s^-n) ,
hence the regularity of η' up to the boundary follows from the smoothness of f up to the boundary. On the other hand, there does not seem to be an easy way to show the regularity of the second and higher derivatives of η up to the boundary.
This is the very issue that in the end does not allow us to infer that <ref> holds in the warped product case. We are able to show that it holds under the assumption that η is C^2, α up to the boundary.
Let (M, g) be a substatic warped product of the form (<ref>). If the function η appearing in (<ref>) is C^2,α up to the boundary then ∇∇ f/f ∈ C^0,α(M ∪∂ M).
We compute
∇∇ f/f = (f'(s)^2/f(s)^2 + f”(s)/f(s)) ds ⊗ ds + s f(s) f'(s) g_N = [f'(s)^2 + f”(s) f(s)] dr ⊗ dr + s f(s) f'(s) g_N.
Moreover, for any function f having the form (<ref>), it holds
sf(s)f'(s) = s^2η(s^-n)-n/2s^2-nη'(s^-n) ,
f'(s)^2 + f”(s) f(s) = η(s^-n)+n(n-3)/2s^-nη'(s^-n)+n^2/2s^-2nη”(s^-n) .
Since s=s_0>0 at the horizon, from these computations we immediately see how the assumed regularity of η implies that of ∇∇ f / f up to the boundary ∂ M.
The warped products in <ref> satisfy η(t) = -λ -2mt, hence η” = 0. In particular, if such splitting takes place up to the horizon boundary, then <ref> implies that <ref> holds.
§.§ Comparison with Brendle's assumptions.
In the case of nonempty boundary, in <cit.>, warped products of the form <ref> with _g_N≥ (n-2) c g_N for some c∈ are assumed to satisfy the following set of assumptions.
(H1) ḣ(0) = 0, ḧ(0) > 0,
(H2) ḣ(r) > 0 for any r ∈ (0, r)
(H3) The function
F(r) = 2 ḧ(r)/h(r) -(n-2) c -ḣ(r)^2/h^2(r)
is nondecreasing,
(H4) It holds
ḧ(r)/h(r)+c-ḣ(r)^2/h(r)^2 > 0 .
Assumptions (H1)-(H3) correspond precisely to case (ii) in <ref>. Indeed, since f = ḣ(r), assumption (H1) implies that {f= 0} coincides with the boundary ∂ M, that its mean curvature =(n-1)ḣ(0)/h(0) vanishes and that | f|=ḧ(0)≠ 0 at the boundary. In other words, (H1) is equivalent to the request that the boundary is of horizon type.
(H2) entails the request that f is positive in the interior of M, while (H3) is instead equivalent to the substaticity of g, with substatic potential f = ḣ, as shown in <cit.>. It is finally worth pointing out that, remarkably, for a warped product <ref>, equation <ref> is always satisfied for f = ḣ. This is due to the following formula, contained in the proof of <cit.>
f - ∇∇ f + Δ f g = f(_g_N - (n-2) c g_N) + 1/2ḣ(r)^2 Ḟ(r) g_N.
As pointed out in the introduction, we will never need (H4) in our analysis. For warped products having the form (<ref>), (H4) is equivalent to sf(s)f'(s)>-c+f(s)^2. Substituting the formula for f into (<ref>) this is in turn equivalent to η'<0. In particular, the model solutions described in (<ref>) satisfy (H4) if and only if m>0.
§ WARPED PRODUCT SPLITTING OF SUBSTATIC MANIFOLDS
In this section, we are going to prove the following result, more general than <ref>, since it deals with possibly disconnected hypersurfaces and explicitly treats the case of components of Σ that are homologous to a point. In particular, it fully encompasses the case of an ambient M with empty boundary.
Let (M, g) be a substatic Riemannian manifold with possibly empty horizon boundary ∂ M such that the substatic potential f satisfies
∇∇ f/f∈ C^0, α(M ∪∂ M)
for α∈ (0, 1).
Let Σ be a smooth strictly mean-convex hypersurface homologous to a possibly empty union N = N_1 ⊔…⊔ N_l of connected components of ∂ M. Let Σ_1, …, Σ_k, k ≥ l, the connected components of Σ. Assume that, for 1 ≤ j ≤ l, each Σ_j is homologous to the component N_j of ∂ M, while for j > l Σ_j is null-homologous. Let Ω_j be the connected region enclosed by Σ_j and N_j if 1 ≤ j ≤ l, and the connected region enclosed by Σ_j if l < j ≤ k. Let also f_j the restriction of f on Ω_j. Then, the Heintze-Karcher inequality <ref> holds with equality if and only if the following hold.
(i) For 1≤ j≤ l, (Ω_j, g) is isometric to
([s_0^j, s_1^j] × N_j, ds ⊗ ds/f_j(s)^2 + s^2 g_N_j).
(ii) For l < j ≤ k , (Ω_j, g) is isometric to
([0, s_1^j] × N_j, ds ⊗ ds/f_j(s)^2 + (s/f_j(0))^2 g_^n-1).
(iii) We have f_1(s_0^1)f_1'(s_0^1)/s_0^1=… =f_l(s_0^l)f_l'(s_0^l)/s_0^l.
To prove <ref> we start from the full statement of the Heintze-Karcher inequality <ref>, given in <cit.>. It involves the solution u to the boundary value problem
Δ u = - 1 + Δ f/f u Ω
u=c_N N
u = 0 Σ,
where ∂Ω = Σ⊔ N, with N union of connected hypersurfaces N_1 ⊔…⊔ N_l, l ∈ and c_N is the constant given by <ref>.
It reads
n-1/n∫_Σf/ dσ -∫_Ω f dμ - ∑_j ∈ J c_j ∫_N_j∇ f dσ ≥ n/n-1∫_Ω |∇∇ u - Δ u/n g - u(∇∇ f/f - Δ f/nf g)|^2
+ Q(∇ u - u/f∇ f, ∇ u - u/f∇ f) dμ,
where
Q = f - ∇∇ f + Δ f g.
It yields in particular the following.
Let (M, g) be a substatic Riemannian manifold with horizon boundary satisfying <ref>, and let Σ be a smooth hypersurface homologous to N = N_1 ⊔…⊔… N_l for l ∈. Let c_N be given by <ref>, and let u be the solution to <ref>. Then, if equality holds in <ref>, then
∇∇ u - Δ u/ng - u (∇∇ f/f - Δ f/nf g) = 0
and
[f - ∇∇ f + Δ f g](∇ u - u/f∇ f, ∇ u - u/f∇ f) = 0
in Ω∖ N.
The following basic yet fundamental observation provides a conformal warped product splitting for the metric in Ω. It exploits (<ref>) only. In the remainder of this section, we agree to define N_j = ∅ when j > l.
In the assumptions and notations of <ref>,
let ϕ = u/f on Ω∖ N. Then, there exists a coordinate ρ on Ω_j∖ N_j such that ϕ depends on ρ alone and Ω_j∖ N_j splits as [0, ρ_j) ×Σ_j endowed with the metric
g_j = f^2_j (dρ⊗ dρ + ϕ_j^2(ρ)/f_j^2(0, θ) g_Σ_j),
where g_Σ_j is the metric induced on Σ_j by g, ϕ_j is the restriction of ϕ on Ω_j∖ N_j, and θ = (θ^1, …, θ^n-1) are coordinates on Σ_j. Moreover, ρ_j = +∞ for 1≤ j ≤ l, and finite for l< j ≤ k.
We focus on a single Ω_j∖ N_j, and drop for notational convenience the subscript j.
Consider the conformal metric g̃ = f^-2 g.
Then, it is readily checked that (<ref>) is equivalent to
∇∇_g̃ ϕ - Δ_g̃ϕ/n g̃ = 0
in Ω∖ N.
We recall moreover that a Hopf Lemma holds for u on Σ <cit.>, and consequently Σ is a regular level set for ϕ. Then,
by classical results (<cit.>, see otherwise <cit.> or <cit.>), there exists a coordinate ρ such that ϕ is a function of ρ alone and such that (Ω, g̃) splits as [0, ρ) ×Σ endowed with
g̃ = dρ⊗ dρ + ϕ^2(ρ)/ϕ^2(0)g̃_Σ,
with ρ being infinite if (and only if) N is nonempty. In <ref>, g̃_Σ is the metric induced on Σ by g̃. This proves <ref>.
Before providing the proof of <ref>, we point out, as another fundamental, although almost trivial, consequence of the assumption <ref>, that ∇ f is constant on each connected component of ∂ M.
Let (M, g) be a substatic Riemannian manifold with potential f and nonempty horizon boundary. Assume that (∇∇ f)/f is continuous up to the boundary ∂ M. Then, ∇ f is constant on each connected component of ∂ M.
Let N be a connected component of ∂ M, and X any vector field in TN. We have
⟨∇∇ f^2 , X⟩ = 2 ∇∇ f(∇ f, X).
on N. On the other hand, since ∇∇ f /f is continuous up to the boundary {f=0}, we necessarily have ∇∇ f = 0 on N. By (<ref>), we conclude that ∇ f is constant on N.
In order to complete the proof of <ref>, we are substantially left to prove that the substatic potential f depends on ρ alone. This information, plugged into <ref>, implies that (Ω, g) is in fact a warped product, and the conclusion follows from <ref>. To achieve this goal, we are going to exploit <ref>.
Again, we drop the dependency on j. We
consider again the conformal setting g̃ = f^-2 g.
Observe that the substatic Ricci tensor just translates into
f - ∇∇ f + Δ f g = f _g̃ - (n-1) ∇∇_g̃ f + 2(n-1)df ⊗ df/f.
Let T be the tensor in the right-hand side above, and observe that substaticity amounts to T ≥ 0. Moreover, letting again ϕ = u/f,
the condition <ref> reads
T (∇ϕ, ∇ϕ) = (ϕ'(ρ))^2 T (∇ρ, ∇ρ) = 0,
where ∇ϕ =ϕ'(ρ) ∇ρ is due to ϕ being a function of ρ alone, by <ref>.
Inspired by the proof of <cit.>, consider now, for θ^i, i ∈{1, …, n-1} a local coordinate on Σ and λ∈, the vector field Y_i = ∇ρ + λ∂_i, where we denoted ∂_i = ∂ /∂θ^i. The condition T(Y_i, Y_i) ≥ 0, coupled with (<ref>), yields, at any fixed point p ∈Ω,
T(Y_i, Y_i) = 2λ T_iρ + λ^2 T_ii≥ 0
for any λ∈. This can actually happen only if T_iρ = 0. Such condition reads
(∇∇)^g̃_iρ f = 2 ∂_ρ f ∂_j f.
Computing the g̃-Hessian of f from the expression <ref>, the above identity becomes
∂^2_iρ f - ϕ”(ρ)/ϕ'(ρ)∂_i f = 2 ∂_ρ f/f.
One can now directly check that, as a consequence of the above relation, we have
∂_ρ(∂_i f/f^2 ϕ'(ρ)) = 0,
so that, given 0 ≤ρ_1 < ρ_2 < ρ,
we get
∂_i (1/f) (ρ_2, θ) = ϕ'(ρ_2)/ϕ'(ρ_1)∂_i (1/f) (ρ_1, θ).
Integrating both sides along θ^i in (θ^i_0, θ^i_1), and omitting to explicate the dependencies on θ^m for m ≠ i, we finally get
ϕ'(ρ_2) f(ρ_2, θ^i_0) - ϕ'(ρ_2)f(ρ_2, θ^i_1)/(ϕ'(ρ_2))^2f(ρ_2, θ^i_0)f(ρ_2, θ^i_1) = ϕ'(ρ_1) f(ρ_1, θ^i_0) - ϕ'(ρ_1)f(ρ_2, θ^i_1)/(ϕ'(ρ_1))^2f(ρ_1, θ^i_0)f(ρ_1, θ^i_1).
We are now going to show that the left hand side converges to 0 as ρ_2 →ρ. Since we will need to tell between N_j and N = N_1 ⊔…⊔ N_l, we restore the dependencies on j when dealing with a specific connected component N_j of N. Recall also that we say that N_j is empty when j > l.
If N_j is empty, then the limit ρ_2 →ρ corresponds to approaching a particular point p in each connected component of Ω, and so the numerator on the left hand side of <ref> converges to zero, while the denominator stays bounded away from zero.
To understand the case of a nonempty N,
recall first that ϕ = u/f, and observe that ∇ρ = 1/f. Then, we have
(f ϕ')^2(ρ_2, θ) = f^4
(∇ u^2/f^2 + u^2/f^4∇ f^2 - 2u/f^3⟨∇ u, ∇ f⟩)(ρ_2, θ),
where all the quantities are understood in terms of g. Since u attains smoothly the datum on N_j (see <cit.>), and f → 0 on N_j, that we are approaching as ρ_2 →∞, we deduce that the limit of the above quantity is given by c_N^2 ∇ f^2_| N_j > 0, that crucially is constant by <ref>. Thus, again, this implies that the left hand side of <ref> vanishes in the limit as ρ_2 →∞, since so does the numerator, while the denominator tends to c_N^2 ∇ f^2_| N_j > 0.
We conclude, finally, that the numerator of the right hand side of <ref> vanishes for any ρ_1 ∈ [0, ρ), implying that f does not depend on θ^i. Being i ∈{1, …, n-1} arbitrary, we deduce that f depends on ρ only. This also implies that f depends on the g-distance from Σ only, and thus allows to write g on Ω as a warped product based at N_j if this is nonempty, and on a spherical cross-section otherwise.
We can thus invoke our characterization of substatic warped products <ref>, since <ref> just amounts to <ref>.
Moreover, since f is also constant on Σ, the cylindrical situation <ref> cannot arise either, since this would imply that the mean curvature of Σ is zero, against the assumption of strict mean-convexity. We are thus left with <ref>. Since the above argument works for any j, it follows that every connected component Ω_j must have the structure prescribed by points (i) and (ii) of the theorem.
To prove point (iii), we start by observing that all pieces having the form (i) or (ii) also saturate the Heintze-Karcher inequality individually. Imposing equality in (<ref>) in all connected components Ω_j for 1≤ j≤ l as well as equality in the whole domain Ω, we deduce
c_N ∫_N ∇ f dσ = ∑_j=1^l c_N_j∫_N_j∇ f dσ.
Using the explicit expression of the metric in the domain Ω_j, provided by point (ii) of the theorem proved above, we also compute
c_N = 1/n∑_j=1^l k_j|N_j|/∑_j=1^lk_j^2/s_0^j|N_j| , c_N_j = 1/ns_0^j/k_j ,
where k_j is the (constant) value of | f| on N_j={s=s_0^j}∩Ω_j. More explicitly k_j=f(s_0^j)f'(s_0^j). Equation (<ref>) can then be rewritten as
(∑_j=1^l k_j|N_j|)^2 = (∑_j=1^lk_j^2/s_0^j|N_j|)(∑_j=1^l s_0^j|N_j|) .
We now show that this equality forces k_1/s_0^1=… =k_l/s_0^l, concluding the proof of point (iii) of the Theorem. To this end, we actually prove the following more general statement: given positive numbers α_1,…,α_l,β_1,…,β_l, if it holds
(∑_j=1^lα_j)^2 = (∑_j=1^lα_jβ_j)(∑_j=1^lα_j/β_j) ,
then β_1=…=β_l. A way to show this is via a direct computation: expanding the above terms we have
∑_j=1^lα_j^2+∑_i≠ jα_iα_j = ∑_j=1^lα_j^2+∑_i≠ jβ_i/β_jα_iα_j .
Simplifying the equal terms on the left and right hand side and exploiting the symmetry in the indexes i and j, the above formula gives
2∑_i< jα_iα_j = ∑_i< j(β_i/β_j+β_j/β_i)α_iα_j = ∑_i< jβ_i^2+β_j^2/β_iβ_jα_iα_j .
Since β_i^2+β_j^2≥ 2β_iβ_j, with equality if and only if β_i=β_j, the wished result follows at once.
§ HEINTZE-KARCHER RIGIDITY IN SUBSTATIC WARPED PRODUCTS
We start by exploiting Brendle's monotonicity formula to deduce some useful geometric properties along an evolution of a hypersurface fulfilling the identity in <ref>. We focus on the case of a nonempty horizon boundary ∂ M = N and of Σ homologous to it; the case of Σ null-homologous and in particular the case of an ambient M with empty boundary is already fully encompassed by <ref>.
Consider the conformal metric g̃ = f^-2 g, and let Ω_t = {x ∈Ω | ρ(Σ, x) ≥ t}, where ρ is the g̃-distance, and Ω as above is the region enclosed between Σ and ∂ M. Let Σ_t = ∂Ω_t ∖∂ M. Crucially, the mean curvature of Σ_t is easily seen to remain strictly positive if the initial Σ is strictly mean-convex (see <cit.>).
Let
Q(t) = ∫_Σ_tf/ dσ,
where all the integrated quantities are expressed in terms of the original metric g. Then, utilizing, as in <cit.>,
classical evolution equations (see e.g. <cit.>) for our normal flow of speed f, and plugging in the identity
Δ_Σ_t f = Δ f - ∇∇ f (ν, ν) - (∇ f, ν)
(again in terms of g), one gets
Q'(t) = - n/n-1∫_Σ_t f^2 dσ - ∫_Σ_t(f/)^2 [^2 + ( - ∇∇ f/f +Δ f/f g)(ν, ν)] dσ.
Let now t ∈ (0, ∞) be such that Σ_τ is smooth for any τ∈ [0, t]. One has, applying the coarea formula,
Q(0) -Q(t) = -∫_0^t Q'(τ) dτ
= n/n-1∫_Ω∖Ω_t f dμ + ∫_0^τ∫_Σ_τ(f/)^2 [^2 + ( - ∇∇ f/f +Δ f/f g)(ν, ν)] dσ.
As long as Q(t) is smooth, Brendle's Heintze-Karcher inequality <cit.> states that
Q(t) ≥n/n-1∫_Ω_t f dμ + c_N∫_∂ M∇ f dμ.
Since equality holds in the Heintze-Karcher inequality for the initial Σ,
that is
Q(0) = n/n-1∫_Ω f dμ + c_N∫_∂ M∇ f dμ,
we get, applying <ref> and <ref> to the left hand side of <ref>,
that
∫_Σ_τ(f/)^2 [^2 + ( - ∇∇ f/f +Δ f/f g)(ν, ν)] dσ = 0
for any τ∈ [0, t]. We deduce the following information on the evolution of Σ, as long as it remains smooth.
Let (M, g) be a substatic warped product of the form <ref>, with a nonempty connected horizon boundary ∂ M. Let Σ = ∂Ω∖∂ M be a smooth, embedded, connected hypersurface homologous to ∂ M, such that <ref> holds with equality sign. Let Σ_t = {x ∈Ω | ρ(Σ, x) = t}, where ρ is the distance in the conformal metric g̃ = f^-2 g. Then, Σ_t is a totally umbilic hypersurface such that
[f - ∇∇ f + Δ f g] (ν, ν) = 0,
as long as Σ_t evolves smoothly.
We now illustrate how we are going to get <ref>. We first show that Σ_t remains smooth for all of its evolution, in <ref>. This is fundamentally due to the total umbilicity of the evolution coupled with the Heintze-Karcher inequality itself, preventing the second fundamental form to blow up, see <ref>.
Then, we adapt to the substatic setting an argument of Montiel <cit.>, yielding in our case a very peculiar dichotomy: if a totally umbilic hypersurface satisfying <ref> is not a cross-section of the warped product, then a vector field X tangent to ∂ M is found on the region spanned by Σ such that the condition <ref> in <ref> is also satisfied (see <ref>). But then, having showed that Ω is foliated by such hypersurfaces, this region must split as prescribed by <ref>. However, as observed in <ref>, this metric satisfies <ref>, and we conclude that the only possibility in the dichotomy is that in fact the initial Σ was isometric to a cross-section.
§.§ The g̃-flow remains smooth.
In order to show that the second fundamental form does not blow up along a smooth evolution Σ_t starting at a hypersurface Σ fulfilling the equality in Heintze-Karcher, we first observe that the diameters remain bounded. In this subsection, we are always denoting with C_t some positive constant possibly depending on t ∈ (0, +∞).
In the assumptions of <ref>, let t < +∞ be such that Σ_τ is smooth for any τ∈ [0, t).
Then, the metric g_τ induced by g on Σ_τ satisfies
|g_τ| ≤C_t ,
for any τ∈ [0, t)
where the norm of g_τ is induced by the norm of the diffeomorphic surface g_Σ_0. Moreover, the intrinsic diameter of Σ_τ satisfies
diam_g_τ(Σ_τ) ≤C_t
for any τ∈ [0, t).
Both <ref> and <ref> holds with g_τ replaced by g̃_τ, corresponding to the metric induced by the underlying conformal metric g̃ = f^-2 g.
As long as the flow is smooth, each level Σ_τ is diffeomorphic to Σ=Σ_0. In particular, for all τ∈[0,t), there exists a metric g_τ on Σ such that (Σ,g_τ) is isometric to Σ_τ endowed with the metric induced on it by g. Obviously the same holds for the conformal metrics g̃_τ induced by g̃.
This allows us to work on a fixed hypersurface Σ, letting the metrics g_τ, g̃_τ vary in time.
Notice that =f-(n-1)⟨ f | ν⟩>-(n-1)| f|≥ -K, where K>0 is the (finite) maximum value of (n-1)| f| in Ω, and where we have used that is strictly positive along the flow by <cit.>. By the evolution _τ (g̃_τ)_ij=- (g̃_τ)_ij we have _τlog|(g̃_τ)_ij|=-< K, hence
|(g̃_τ)_ij| < e^Kτ|(g̃_0)_ij|≤C_t .
Since f is bounded in the compact domain Ω enclosed by Σ, the above bound implies a fully equivalent one in terms of the metric g_τ induced by g on Σ_τ. This proves (<ref>).
For a fixed τ∈[0,t), let x_τ,y_τ∈Σ be two points realizing the diameter diam_g_τ(Σ) that we want to estimate, and let γ:[0,ℓ]→Σ be the g_0-unit length geodesic minimizing the distance between x_τ and y_τ, with respect to the starting conformal metric g_0.
By <ref>, the length of γ is directly estimated as follows:
|γ|_g_τ = ∫_0^ℓ |γ̇(s)|_g_τds
= ∫_0^ℓ√((g_τ)_ijγ̇^iγ̇^j)(γ(s))ds ≤ℓ C_t
By construction, the diameter diam_g_τ(Σ) coincides with the g_τ-distance between the endpoints of γ, so diam_g_τ(Σ) must be less than or equal to the g_τ-length of γ. Moreover, by construction, we have ℓ≤ diam_ g_0(Σ), and so we have shown
diam_g_τ(Σ) ≤γ_g_τ≤C_t.
This provides the desired uniform bound on the diameter diam_g_τ(Σ).
The following is the main observation triggering the smooth long time existence along the g̃-distance flow.
In the assumptions of <ref>, let t be such that Σ_τ is smooth for any τ∈ [0, t). Then, the second fundamental form _τ of Σ_τ satisfies
_τ≤C_t
for any τ∈ [0, t).
Assume by contradiction that there exists a sequence τ_j → t < +∞ as j → +∞ and points x_τ_j∈Σ_τ_j such that _τ_j(x_τ_j) blows up as j → +∞. Then, since the Σ_τ's are totally umbilical, the mean curvature _τ_j(x_τ_j) blows up too. We first show that, then,
inf_x ∈Σ_τ_j_τ_j (x) → +∞.
Indeed, by the Gauss-Codazzi equations and exploiting the total umbilicity we immediately get
∇_i (τ_j) = -n-2/n-1_iν
for any i ∈{1, …, n-1}. Since the right hand side is uniformly bounded in Ω, we deduce that ∇ is uniformly bounded along the evolution. Let then x ∈Σ_τ_j different from x_τ_j. We have
(x) ≥(x_τ_j) - diam(Σ_τ_j)sup_y∈Σ_τ_j∇(y) ≥(x_τ_j) - C_t,
where the bound on the diameter is <ref>; <ref> follows. On the other hand, recall that by the Heintze-Karcher inequality
we have
∫_Σ_τ_jf/ dσ≥ c_∂ M∫_∂ M∇ f dσ.
Now, the evolution equations for Σ_τ imply that Σ_τ≤Σ, while, since t<+∞ and ρ(Σ, ∂ M) = +∞, sup_Σ_τ_j f ≤C_t for some finite C_t > 0. Exploiting this information, we get at once from <ref> that
inf_x ∈Σ_τ_j(x) ≤C_t Σ (c_∂ M∫_∂ M∇ f dσ)^-1,
yielding a contradiction with <ref> that completes the proof.
Concluding from the above that Σ_t remains smooth for any t ∈ (0, +∞) turns out to be slightly technical, but very classical in nature. The arguments employed were pioneered by Hamilton <cit.> in an intrinsic flow setting, and adapted to extrinsic flows by Huisken <cit.>.
Let (M, g) be a substatic warped product of the form <ref>, with a nonempty connected horizon boundary ∂ M. Let Σ = ∂Ω∖∂ M be a smooth, embedded, connected hypersurface homologous to ∂ M, such that <ref> holds with equality sign. Let Σ_t = {x ∈Ω | ρ(Σ, x) = t}, where ρ is the distance in the conformal metric g̃ = f^-2 g. Then, Σ_t is a smooth, embedded, totally umbilic hypersurface such that
[f - ∇∇ f + Δ f g] (ν, ν) = 0,
for any t ∈ [0, ∞).
We are going to show that the nonempty set T⊆ [0, +∞) defined by
T = {t∈ [0, +∞) | Σ_τ is smooth and embedded for τ∈ [0, t]}
is both open and closed in [0, +∞), inferring the eternal smoothness of the flow. The identity <ref> is then a direct consequence of <ref>.
The openness of T is well-known in general; if a closed hypersurface Σ is smooth and embedded, then so are the equidistant hypersurfaces Σ_r = {x ∈ M | dist(Σ, x) = r} for any Riemannian metric-induced distance dist, see e.g. <cit.>. In our case, such result is applied to Σ_t with t ∈ T and with respect to the distance induced by g̃.
The closedness of T constitutes the bulk of the proposition, and will be substantially ruled by <ref> only. We are repeatedly employing the evolution equations for Σ_τ along the g̃-distance flow, in the conformal background metric g̃. Indeed, in this setting such equations are simpler to handle, and, since we are staying away from ∂ M = {f = 0} the estimates we are inferring will automatically hold also in terms of g, and viceversa.
In the remainder of this proof, all the quantities taken into account are thus understood as referred to g̃, even when not explicitly pointed out.
Let T ∋ t_j → t^- as j → +∞. Then, Σ_τ is smooth and embedded for any τ∈ [0, t) We want to show that Σ_t is smooth and embedded.
To accomplish this task, we are going to show that
∇^(k)_τ≤C_t
for an arbitrary k ∈,
where ∇^(k) is the k-th covariant derivative induced by g̃ on Σ_τ. Indeed, if this holds, then all the derivatives of the functions whose graphs describe Σ_τ would be uniformly bounded as τ→ t, implying that Σ_t would be actually smooth.
We are going to prove <ref> by induction. The case k = 0 corresponds to the <ref>, and we assume
∇^(l)_τ≤C_t
holds for any l ∈{0, …, k-1}. We employ the concise notation T * Q to indicate, at some fixed point, linear combinations of contractions of a tensor T with a tensor Q through the metric tensor. The uniform bound on the evolving metric tensors g_τ was observed in <ref>. We have
∂/∂τ∇^(k)_τ = ∇^(k)∂/∂τ_τ + ∇^(l_1)∂/∂τΓ*∇^(l_2)_τ,
where l_1, l_2 ∈ satisfy l_1 + l_2 = k-1, and the components of Γ are the Christoffel symbols of the evolving metric that g̃ induces on Σ_τ. We recall that the variation of the components of Γ are in fact components of a tensor, and that it holds
∂/∂τΓ^i_jm = 1/2g^ir(∇_j∂/∂τg_mr + ∇_m ∂/∂τ g_jr - ∇_r ∂/∂τ g_jm),
for i, j, l, r ∈{1, …, n-1} and where we meant with g the metric induced by g̃ on Σ_τ. Moreover, the second fundamental from _τ induced by g̃ on Σ_τ roughly evolves by (see e.g. <cit.>)
∂/∂τ_τ = _τ * _τ + Riem,
where Riem denotes some component of the Riemann tensor of the ambient g̃. Observe that, since t is finite, Riem, as well as any of its g̃-covariant derivative, remains bounded on Σ_τ as τ→ t^-. Plugging <ref> and <ref> into <ref>, and directly estimating by means of <ref>, we get that
∂/∂τ∇^(k)_τ = ∇^(k)_τ * T + Q,
where T and Q are tensors uniformly bounded on Σ_τ also as τ→ t^-. Then, taking into account once again that ∂_τ g_τ = -_τ g_τ, and that is an uniform bounded quantity as τ→ t^- thanks to <ref>, we deduce from <ref>
∂/∂τ∇^(k)_τ^2 ≤C_1∇^(k)_τ^2 + C_2,
where both C_1 and C_2 are constants uniformly bounded as τ→ t^-. Integrating <ref> for τ∈ [0, t) provides the claimed <ref>, inferring the smoothness of Σ_t.
We are left to discuss the embeddedness of Σ_t. Since Σ_t is compact, it is sufficient to show that Σ_t has no self intersections. Without loss of generality, suppose that t is the first time such that Σ_t has a self-intersection x. Then, in a neighborhood of x, by smoothness and compactness, Σ_t is a finite union of smooth embedded hypersurfaces S_1,…, S_k.
First of all, if two of these hypersurfaces, say S_1 and S_2, have different tangent spaces, then it is easily seen by continuity of the flow that Σ_t- also has self-intersections for times t- close to t (see Figure <ref>), against our assumption that t is the first value having them.
Thus, it only remains to analyze the case where the tangent space to S_1,…,S_k is the same. If the outward pointing normal is the same for S_1 and S_2, then it is easy to see that the level set Σ_t- must intersect Σ_t (this case is exemplified in Figure <ref>). This is impossible since Σ_t must be contained in the interior of the compact domain Ω_t- enclosed by Σ_t- for >0.
The only case that is left to rule out is the case where there are exactly two hypersurfaces S_1 and S_2, with outward normals pointing in opposite directions. With similar reasonings as in the previous cases, we can conclude that the situation is as in Figure <ref>. Namely, we can find coordinates (x^1,…,x^n) centered at x such that S_1={x^1= 0} and its outward normal points towards {x^1≤ 0}, whereas S_2 is contained in {x^1≥ 0} and its outward normal points towards {x^1≥ 0}. Recalling that Σ_t is mean convex, this configuration is clearly ruled out by the maximum principle.
§.§ Montiel-type argument and conclusion.
Thanks to <ref> and <ref>, we have two directions along which the tensor f- f+Δ f g vanishes. A crucial observation is that, if Σ is not a cross-section, these two directions are distinct at almost all points of Σ. This follows from an argument of Montiel <cit.>. In our substatic setting, thanks to <ref>, this forces the region Σ lives in to be of the special form <ref>.
Let (M, g) be a substatic warped product of the form <ref>, with a nonempty connected horizon boundary ∂ M. Let Σ = ∂Ω∖∂ M be a smooth, embedded, orientable, connected hypersurface homologous to ∂ M. Suppose that Σ is totally umbilical, that it holds
[f - ∇∇ f + Δ f g](ν,ν) = 0
and that Σ is not a cross-section.
where ν is a unit normal to Σ.
Then, the function f has the form (<ref>) in the region [s_ min,s_ max]× N, where
s_ min = min{s(x), x ∈Σ},
s_ max = max{s(x), x ∈Σ}.
The function f depends on the coordinate s only, so it is well defined (up to a constant) the function ϕ=∫ (s/f) ds. We can compute rather easily ϕ=f g. In other words, the vector Y=ϕ=sf/ s satisfies Y =ϕ=fg. Let Y^⊤=Y-g(Y,ν)ν be the projection of Y on Σ. If ∇_Σ is the covariant derivative induced by ∇ on Σ, then Y^⊤=∇_Σϕ.
We are assuming that Σ is not a cross section, that is, Y^⊤ does not vanish pointwise on Σ. Following the argument in <cit.>, for every vector field Z∈Γ(TΣ) we compute
_Σ_Σϕ(Z,·) =∇_Σ Y^⊤(Z,·)
=[∇_Z(Y-g(Y,ν)ν)]^⊤
=(∇_Z Y)^⊤-[∇_Z g(Y,ν)ν+g(Y,ν)∇_Zν]^⊤,
where we are using the notation ^⊤ to denote the projection on Σ (namely, for a vector field X∈Γ(TM), we denote by X^⊤ the vector X-g(X,ν)ν∈Γ(TΣ)).
Since ∇ Y=fg, ν^⊤=0, and (∇_Zν)^⊤=(Z,·), where is the second fundamental form of Σ, we deduce
∇_Σ∇_Σϕ=fg_Σ-g(Y,ν).
In particular, if Σ is umbilical, then =/(n-1)g_Σ and we obtain
∇_Σ_Σϕ=[f-g(Y,ν)/n-1]g_Σ.
Since Σ is compact and Y^⊤ does not vanish pointwise (meaning that ϕ is nontrivial), it is then well known <cit.> that Σ must be a warped product with spherical cross-sections. Namely
Σ=[0,R]×𝕊^n-2, g_Σ=dρ⊗ dρ+λ^2g_𝕊^n-2,
where ρ is the coordinate on [0,R] and λ=λ(ρ) is positive in (0,R), λ(0)=λ(R)=0, λ'(0)=-λ'(R)=1.
Furthermore the following relations hold:
λ'=f-g(Y,ν)/n-1, λ=/ρϕ_|_Σ, Y^⊤=∇_Σϕ=λ/ρ.
In particular, Y^⊤ is different from zero on Σ∖{x, y}, with x, y being the two points corresponding to ρ=0 and ρ=R.
Since we are assuming that the tensor f - ∇∇ f + Δ f g vanishes in the ν direction and we know this holds also in the Y direction (recall <ref>), we must also have
[f - ∇∇ f + Δ f g](Y^⊤,Y^⊤)=0
at all points of Σ.
We can then apply <ref> with X = Y^⊤
to conclude.
The above proof intriguingly shows also that a hypersurface Σ as in the statement of <ref> is itself a warped product.
We are now ready to conclude the proof of <ref>.
We can restrict our attention to the case of nonempty boundary ∂ M = {f =0} with Σ homologous to ∂ M, since the empty boundary (or null-homologous) case is fully covered by <ref>.
We consider the evolution of Σ given by the Σ_t ⊂Ω at g̃-distance t. By <ref>, Σ_t is smooth for any t ∈ [0, +∞). Suppose by contradiction that Σ is not a cross section. Then, Σ_t is not a cross-section for any t ∈ (0, +∞), for otherwise all of its g̃-equidistant hypersurfaces would be cross-sections, including Σ. Moreover, as recalled in <ref>, the Σ_t's are totally umbilical
and satisfy
[f - ∇∇ f + Δ f g](ν,ν)=0 .
We can then apply <ref> to any Σ_t, and deduce that in the region foliated by such evolution f can be written as (<ref>).
Since, as t → +∞, Σ_t by construction gets closer and closer to ∂ M, this holds in the whole of Ω. But then, as observed in <ref>, by <ref> the condition <ref> is satisfied.
We can then apply <ref> to conclude that Σ is a cross-section, a contradiction that concludes the proof.
§ APPENDIX: PROOF OF PROPOSITION <REF> AND LEMMA <REF>
We consider warped products
M=I× N , g=dr⊗ dr+h^2g_N ,
with h=h(r) positive, satisfying the substatic condition
-∇∇ f/f+Δ f/fg≥ 0
for some function f=f(r) that is assumed to be nonnegative and zero exactly on the (possibly empty) boundary of M. Our aim will be that of proving Proposition <ref>. To this end, we start by writing down the components of the relevant quantities in the substatic condition.
The Ricci tensor of a warped product is known to satisfy
_rr = -(n-1)ḧ/h ,
_ir = 0 ,
_ij = ^N_ij-[hḧ+(n-2)ḣ^2]g^N_ij .
Since both f and h are functions of the coordinate r only, the Hessian and Laplacian are given by the following formulas
^2_rrf = f̈ ,
^2_irf = 0 ,
^2_ijf = hḣḟ g^N_ij ,
Δ f = f̈+(n-1)ḣ/hḟ .
Substituting in (<ref>), we find out that the substatic condition is equivalent to the following two inequalities:
ḣḟ/f ≥ ḧ ,
_g_N ≥
h^2[ḧ/h-f̈/f+(n-2)ḣ^2/h^2-(n-2)ḣ/hḟ/f]g_N .
We are now ready to provide the proof of <ref>, telling between the case of a constant h, that is the cylindrical splitting of, and of a nonconstant h.
The product case.
We impose ḣ=0. Up to a rescaling of g_N we can then just set h≡ 1. The first identity in (<ref>) is trivial when ḣ=0.
The second inequality in (<ref>) instead reduces to
_g_N ≥
-f̈/fg_N .
In particular, if c is the minimum value such that there exists X∈ TN with _g_N(X,X)=(n-2)c |X|^2_g_N, we must have
f̈+(n-2)cf ≥ 0 .
For any f satisfying the above inequality and any (n-1)-dimensional Riemannian manifold (N,g_N) with _g_N≥ (n-2)cg_N the product manifold (I× N,dr⊗ dr+g_N) is substatic.
The warped product case.
We now consider the case where h is not constant. Since we are assuming
[f - ∇∇ f + Δ f g] ( r, r) = 0,
the first inequality in (<ref>) is saturated, forcing f=kḣ, for some constant k∈. Letting also c∈ be the minimum value such that there exists X∈ TN with _g_N(X,X)=(n-2)c |X|^2_g_N, we can rewrite the second inequality in (<ref>) as follows
⃛ h/ḣ+(n-3)ḧ/h-(n-2)ḣ^2-c/h^2 ≥ 0 .
This inequality also appears in <cit.>.
Since f is assumed to be positive in M, notice that in particular this forces ḣ to have a sign. Up to changing the sign of the coordinate r, we can assume ḣ>0. In particular, h is a monotonic function of r, which means that we can use h as coordinate in place of r. We use ' to denote derivative with respect to h.
Considering then the function
ψ = ḣ^2-c/h^2 ,
observing that ψ'=ψ̇/ḣ, we compute
(h^n+1ψ')' = 2(h^n-1ḧ-h^n-2(ḣ^2-c))'
= 2h^n-1(⃛ h/ḣ+(n-3)ḧ/h-(n-2)ḣ^2-c/h^2) .
Therefore,
inequality (<ref>) gives
(h^n+1ψ')' ≥ 0 ,
which in turn tells us that
ψ = ∫μ/h^n+1dh ,
where μ=μ(h) satisfies μ'≥ 0.
This is equivalent to asking ψ=η(h^-n), with η”≥ 0.
More explicitly, the substatic potential f and the warping function h must satisfy
f = kḣ = k√(c+h^2η(h^-n)) , η”≥ 0 .
Summing all up, we have found that all substatic warped products (I× N,g,f) with f radial and such that f - ∇∇ f + Δ f g vanishes in the radial direction are isometric to a solution having the following form
M=[a,b]× N , g=k^2ds⊗ ds/f^2+s^2 g_N , _g_N≥ (n-2)c g_N ,
f = k√(c+s^2η(s^-n)) ,
where 0<a<b, c>0, k>0 are constants, s is a coordinate on [a,b], and η:[b^-n,a^-n]→ satisfies η”≥ 0.
We conclude with the proof of <ref>.
If we further assume, as in the statement, that at the point p with coordinates (s_0,x) there exists a vector X∈ T_pN such that [f - ∇∇ f + Δ f g](X,X)=0, then it easily follows that the second inequality in (<ref>) is saturated. Retracing the computations above one then deduces that η”=0 at s=s_0. If it is possible to find such a vector X for every s in an interval [s_0,s_1]⊂ [a,b], then we can integrate the identity η”=0 in [s_1^-n,s_0^-n], obtaining η(t)=-λ-2mt for constants m,λ. Substituting in the formula for f we then obtain
f = k√(c-λ s^2-2m s^2-n) ,
as claimed.
|
http://arxiv.org/abs/2307.04555v1 | 20230710134037 | CIP-stabilized Virtual Elements for diffusion-convection-reaction problems | [
"L. Beirao da Veiga",
"C. Lovadina",
"M. Trezzi"
] | math.NA | [
"math.NA",
"cs.NA"
] |
| | |
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1,2]L. Beirão da Veiga [email protected]
3]C. Lovadina [email protected]
4]M. Trezzi [email protected]
[1]Dipartimento di Matematica e Applicazioni,
Università degli Studi di Milano Bicocca,
Via Roberto Cozzi 55 - 20125 Milano, Italy
[2]IMATI-CNR, Via Adolfo Ferrata 5 - 27100 Pavia, Italy
[3]Dipartimento di Matematica “F. Enriques”,
Università degli Studi di Milano,
Via Cesare Saldini 50 - 20133 Milano, Italy
[4]Dipartimento di Matematica “F. Casorati”,
Università di Pavia,
Via Adolfo Ferrata 5 - 27100 Pavia, Italy
CIP-stabilized Virtual Elements for diffusion-convection-reaction problems
[
August 12, 2023
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The Virtual Element Method for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the Virtual Element Method, the stability and the error analysis requires particular care, especially in treating the advective term. Some numerical tests are presented to support the theoretical results.
§ INTRODUCTION
The Virtual Element Method (VEM) is a fairly recent methodology for the discretization of problems in partial differential equations <cit.>, which can be interpreted as a generalization of classical Finite Elements (FEM) to meshes of much more general shape. Since its birth, the VEM has enjoyed a large success and been applied to a very wide range of problems; we here limit ourselves in mentioning the recent special issue <cit.> and the review paper <cit.>.
The focus of the present article is on the classical diffusion-reaction-advection scalar problem.
Under suitable assumptions on the data, this is a standard “textbook” elliptic problem without any particular difficulty. On the other hand, it is well known that, whenever the advective term dominates (in particular over the diffusive one) a classical FEM approach will lead to very large errors and oscillations in the discrete solution, unless an extremely fine mesh is adopted. There is a large FEM literature on the subject, offering a list of possible stabilized methods which are robust in this respect. From the theoretical standpoint, a method is typically called quasi-robust if, assuming sufficiently regular solution and data, it yields error estimates which are uniform with respect to the diffusion parameter in a norm including also some direct control on the convective term. Some well known approaches are upwind Discontinuous Galerkin schemes <cit.>, Streamline Upwind Petrov-Galerkin and variants <cit.>, Continuous Interior Penalty (CIP) <cit.>, Local Projection Stabilization <cit.>. Finally, one must note that the diffusion-reaction-advection problem serves also as a model for more complex problems in fluid mechanics, such as the Navier-Stokes equation.
The Virtual Element Method is particularly suitable in the context of advection dominated problems due to the flexibility of the mesh construction and its handling. For instance, VEM allows more local refinement procedures and easy an gluing of fine meshes with coarser ones (this latter feature is very useful in the presence of layers, for example). In addition, VEM offers a more efficient discretization of complex domains, which is greatly useful in applications such as reservoir <cit.> and fracture-network simulations <cit.>, where diffusion-reaction-advection equations play a crucial role. Unfortunately, due to the presence of projection operators which may alter the structure of the convective term, it is not easy to devise and analyze quasi-robust VEM schemes. Exceptions are the SUPG and LPS approaches detailed in <cit.> and <cit.>, respectively (regarding other polygonal technologies, see for instance <cit.>).
Since three of the most popular stabilization techniques, namely SUPG, LPS and CIP, have their own strongly defined set of assets/drawbacks, broadening the available approaches with CIP schemes is important for the VEM technology.
The purpose of the present contribution is exactly to fill this gap and develop CIP (Continuous Interior Penalty) stabilized VEM method, providing also a theoretical error analysis. Of course, our method combines VEM stabilization terms (to deal with polygonal meshes) and CIP-like terms (to deal with the avdection-dominated regime).
Furthermore, it is worth noticing that the backstage complex nature of CIP, which is a “minimal stabilization” as it adds the minimal positive term guaranteeing control on piecewise polynomial convection, makes the analysis in the VEM setting particularly interesting and challenging. Assuming, as it happens in most publications on the subject, a uniformly positive reaction term, we are able to develop quasi-robust error estimates for our method. In the absence of reaction, we are able to show some improved error estimates (over a non-stabilized scheme), but only under a piecewise polynomial convection data assumption. The paper ends with a set of numerical tests showing the actual robustness of the method and comparing it with the non-stabilized approach.
The paper is organized as follows. After presenting the continuous and discrete problems in Section <ref>, we develop the stability and convergence analysis in Section <ref>. Finally, numerical tests are shown in Section <ref>.
Throughout the paper, we use standard notations for Sobolev norms and semi-norms. Moreover, C and C_i will denote quantities, independent of the meshsize h, which may vary at each occurrence. We will make extensive use of the notation a ≲ b (a and b being non-negative quantities) to mean a≤ C b.
§ THE CONTINUOUS AND THE DISCRETE PROBLEMS
In this Section we deal with the continuous problem and its discretization by means of the Virtual Element Method.
§.§ Continuous Problem
We consider the following steady advection-diffusion-reaction problem:
{
- ϵΔ u + ·∇ u + σ u
= f in Ω ,
u_|Γ = 0 ,
.
where Ω⊂^2 is a polygonal domain of boundary Γ. Above, ϵ > 0 is the diffusion coefficient (assumed to be constant), while
∈ [W^1,∞(Ω)]^2 is the advection field such that = 0. Moreover,
σ>0 is the reaction constant (except for Section <ref>, where σ=0); we remark that we assume σ to be a positive constant since the extension to the case 0<σ∈ L^∞(Ω), with σ^-1∈ L^∞(Ω), is trivial.
Finally, f ∈ L^2(Ω) is the source term.
The domain boundary will be split into two non-overlapping regions:
Γ_in{∈Γ | (() ·) < 0 } and Γ_out{∈Γ | (() ·) ≥ 0 } ,
where is the outward unit normal vector to the boundary.
A variational formulation of problem (<ref>) reads as follows:
{ find u ∈ V H^1_0(Ω) such that:
ϵ a(u,v) + (u,v) + σ c(u,v) = ∫_Ω f v dΩ .
.
The bilinear forms
a(·, ·) V × V → ,
(·, ·) V × V →
and
c(·, ·) V × V →
are defined as
a(u, v) ∫_Ω∇ u ·∇ v dΩ for all u, v ∈ V,
(u,v) 12(b(u, v) - b(v, u)) with b(u, v) := ∫_Ω (·∇ u) v dΩ for all u, v ∈ V,
c(u, v) := ∫_Ω u v dΩ for all u, v ∈ V.
It is well known that when ϵ is small with respect to and/or to σ, standard discretizations of (<ref>) typically return unsatisfactory numerical solutions with spurious oscillations.
To overcome these difficulties, several strategies are available in the literature.
In this paper we take advantage of the so-called Continuous Interior Penalty (CIP) strategy, introduced in
<cit.> in a Finite Element framework.
From now on, we assume that the material parameters are scaled so that we have
=1 .
§.§ Preliminary notations and results
We start considering a sequence Ω_h_h of tessellations of Ω into non-overlapping polygons E.
We denote with e a general edge of E, while
|E| and h_E are the area and the diameter of E, respectively.
Furthermore, ^E is the unit outward normal vector to the boundary ∂ E.
As usual, h sup_E∈Ω_hh_E denotes the mesh parameter.
We suppose that Ω_h_h fulfils the following assumption:
(A1) Mesh assumption.
There exists a positive constant ρ such that for any E ∈Ω_h_h
* E is star-shaped with respect to a ball B_E of radius ≥ ρ h_E;
* any edge e of E has length ≥ ρ h_E;
* the mesh is quasi-uniform, any polygon has diameter h_E ≥ρ h.
We now introduce some basic tools and notations useful in the construction and the theoretical analysis of Virtual Element Methods.
Using standard VEM notations, for n ∈, m∈ and p∈ [1, +∞], and for any E ∈Ω_h,
let us introduce the spaces:
* _n(E): the set of polynomials on E of degree ≤ n (with _-1(E)={ 0 }),
* _n(Ω_h) := {q ∈ L^2(Ω) s.t q|_E ∈_n(E) for all E ∈Ω_h},
* W^m_p(Ω_h) := {v ∈ L^2(Ω) s.t v|_E ∈ W^m_p(E) for all E ∈Ω_h} equipped with the broken norm and seminorm
v^p_W^m_p(Ω_h) := ∑_E ∈Ω_hv^p_W^m_p(E) ,
|v|^p_W^m_p(Ω_h) := ∑_E ∈Ω_h |v|^p_W^m_p(E) ,
if 1 ≤ p < ∞,
v_W^m_p(Ω_h) := max_E ∈Ω_hv_W^m_p(E) ,
|v|_W^m_p(Ω_h) := max_E ∈Ω_h |v|_W^m_p(E) ,
if p = ∞,
and the following polynomial projections:
* the L^2-projection Π_n^0, E L^2(E) →_n(E), given by
∫_Eq_n (v - Π_n^0, E v) d E = 0 for all v ∈ L^2(E) and q_n ∈_n(E),
with obvious extension for vector functions Π^0, E_n [L^2(E)]^2 → [_n(E)]^2;
* the H^1-seminorm projection Π_n^∇,E H^1(E) →_n(E), defined by
{ ∫_E q_n · ( v - Π_n^∇,E v) d E = 0 for all v ∈ H^1(E) and q_n ∈_n(E),
∫_∂ E(v - Π_n^∇, E v) ds= 0 ,
.
with global counterparts
Π_n^0 L^2(Ω) →_n(Ω_h) and
Π_n^∇ H^1(Ω_h) →_n(Ω_h)
defined by
(Π_n^0 v)|_E = Π_n^0,E v ,
(Π_n^∇ v)|_E = Π_n^∇,E v ,
for all E ∈Ω_h.
We finally mention one classical result for polynomials on star-shaped domains (see for instance <cit.>).
Under the assumption (A1), for any E ∈Ω_h and for any smooth enough function ϕ defined on E, it holds
ϕ - Π^0,E_n ϕ_W^m_p(E)≲ h_E^s-m |ϕ|_W^s_p(E) s,m ∈, m ≤ s ≤ n+1, p=1, …, ∞,
ϕ - Π^∇,E_n ϕ_m,E≲ h_E^s-m |ϕ|_s,E s,m ∈, m ≤ s ≤ n+1, s ≥ 1,
∇ϕ - Π^0,E_n∇ϕ_m,E≲ h_E^s-1-m |ϕ|_s,E s,m ∈, m+1 ≤ s ≤ n+1, s ≥ 1.
§.§ Virtual Element spaces
Given a polygon E and a positive integer k, we define the local “enhanced” virtual element space as
V_h(E) =
{
v_h ∈ H^1(E) ∩ C^0(∂ E) s.t.
v_h|_e ∈_k(e) for all e ∈∂ E, .
.
Δ v_h ∈_k(E) ,
(v - v, p_k ) = 0 for all p_k ∈_k(E) / _k-2(E)} .
For the finite dimensional space V_h(E), one can check that the following linear operators are a set of DoFs:
* : the pointwise values of v_h at the vertexes of the polygon E,
* : the values of v_h at k-1 internal points of a Gauss-Lobatto quadrature for every edge e ∈∂ E,
* : the moments 1| E |∫_E v_h m_αβ d E, ∀ m_αβ∈ℳ_k-2(E) where ℳ_k-2(E) is the set of monomials defined as
ℳ_k-2{
m_αβ( x - x_Eh_E)^α( y - y_Eh_E)^β α,β∈ℕ , α + β≤ k - 2
}.
Thanks to these DoFs, it is possible to compute the following projections:
V_h(E) →_k(E), 0 V_h(E) →_k(E), 0P ∇ V_h(E) → [_k(E)]^2 .
Gluing together the local spaces, we define the global virtual element space as
V_h(Ω_h) = {v_h ∈ V s.t. v_h|_E ∈ V_h(E) for all E ∈Ω_h} ,
with the associated set of degrees of freedom:
* : the values of v_h at the vertices;
* : the values of v_h at k-1 points on each edge e;
* : the moments up to order k-2 for each element E∈Ω_h.
We finally recall from <cit.> the optimal approximation property for the space V_h(Ω_h).
Under the assumption (A1) for any v ∈ V ∩ H^s+1(Ω_h) there exists ∈ V_h(Ω_h) such that for all E ∈Ω_h it holds
v - _0,E + h_E ∇ (v - )_0,E≲ h_E^s+1 |v|_s+1,E ,
where 0 < s ≤ k.
§.§ Virtual Element Forms and the Discrete Problem
We start observing that the bilinear forms
a(·,·) ,
(·,·)
and
c(·,·), see (<ref>), (<ref>) and (<ref>),
can be obviously decomposed into local contributions
a(u, v) ∑_E ∈Ω_h a^E(u, v) ,
(u, v) ∑_E ∈Ω_h(u, v) ,
c(u, v) ∑_E ∈Ω_h c^E(u, v) .
Using the DoFs introduced in Section <ref>, we construct a computable counterpart of the above-mentioned forms, following the standard VEM procedure.
Hence, we define the bilinear form a_h^E(·, ·) V_h(E) × V_h(E) → as follows:
a_h^E(u_h, v_h) :=
∫_E 0P ∇ u_h ·0P ∇ v_h dE +
((I - ) u_h, (I - ) v_h) .
Above, the stabilizing bilinear form V_h(E) × V_h(E) → is required to be computable and to satisfy
α_*|v_h|_1,E^2 ≤(v_h, v_h) ≤α^* |v_h|_1,E^2 ,
for all v_h ∈ Ker() ,
for two positive uniform constants α_* and α^*.
In what follows, we choose the stabilization (cf. <cit.>, for instance), which is a common choice for VEM approach.
Following <cit.>, we replace the bilinear form b^E(·, ·) V_h(E) × V_h(E) →
with b_h^E(·, ·), defined as
b_h^E(u_h, v_h) ∫_E ·∇0 u_h 0 v_h dE +
∫_∂ E (·^E) (I - 0) u_h v_h ds .
In the numerical scheme, we will employ the skew-symmetrized form (cf. (<ref>)):
(u_h , v_h) = 1/2( b_h^E(u_h, v_h) - b_h^E(v_h, u_h) ) .
The reaction term is locally replaced by c_h(· , ·) V_h(E) × V_h(E) →, defined as
c_h^E(u_h, v_h) ∫_E 0 u_h 0 v_h dE +
|E| ((I - 0) u_h, (I - 0) v_h) .
Following <cit.>, we now introduce a VEM version of the local CIP-stabilization form,
defined as
J_h^E(u_h,v_h)
∑_e ⊂∂ Eγ2∫_e h_e^2 [∇ u_h] · [∇ v_h] ds
+
γ h_E ((I - ) u_h, (I - ) v_h) ,
where [∇·] denotes the gradient jump across e. If e is a boundary edge we set [∇· ] = 0. The parameter γ is defined as
γ (∂ E) ·^e _L^∞(∂ E) ,
where ^e is one of the two outward normal vectors to e. Since we will work with the assumption _[L^∞(Ω)]^2 = 1, we will treat γ as a constant.
Moreover, we impose the Dirichlet boundary conditions by using a Nitsche-type technique. To this aim, we define the local forms:
𝒩_h^E(u_h,v_h)
- ϵ⟨∇ u_h ·^E, v_h ⟩_Γ_E
+ ϵδ h_E⟨ u_h,v_h ⟩_Γ_E
+ 12⟨|·| u_h, v_h ⟩_Γ_E ,
where Γ_E = ∂ E ∩Γ, δ is a positive parameter to be chosen and ⟨· , ·⟩ is the L^2(Γ_E)-scalar product.
The standard definition of Nitsche's method also considers a term
- ϵ⟨ u_h, ∇ v_h ·^E⟩_Γ_E .
Since we are not interested in achieving symmetry and in order to simplify the analysis of the method, we drop this term. Another difference with the standard formulation of Nitsche's method is the convective term. Usually, it is locally defined as
- ⟨ (·^E) u_h, v_h ⟩_Γ_in∩Γ_E .
By integration by parts, in the definition of
(·,·), we should consider also
12⟨ (·^E) u_h,v_h ⟩_Γ_E .
Summing the last two terms, we recover our definition of 𝒩_h^E(·, ·).
Summing all of these contributions, we construct the discrete bilinear form ^E V_h(E) × V_h(E) → as
^E(u_h, v_h) = ϵ a_h^E(u_h , v_h) + (u_h , v_h) + σ c_h^E(u_h , v_h) + 𝒩_h^E(u_h,v_h) + J_h^E(u_h , v_h) ,
and summing over all the polygons we obtain the global versions of the bilinear forms
a_h(u_h, v_h) := ∑_E ∈Ω_h a_h^E(u_h, v_h) ,
(u_h, v_h) := ∑_E ∈Ω_h(u_h, v_h) ,
c_h(u_h, v_h) := ∑_E ∈Ω_h c_h^E(u_h, v_h) ,
J_h(u_h, v_h) := ∑_E ∈Ω_h J_h^E(u_h, v_h) ,
𝒩_h(u_h, v_h) := ∑_E ∈Ω_h𝒩_h^E(u_h, v_h) ,
and
(u_h, v_h) ∑_E ∈Ω_h (u_h, v_h) .
The discrete local and global load terms (here ℱ_h^E V_h(E) →) are
ℱ_h^E(v_h) ∫_E f 0 v_h ,
ℱ_h(v_h) := ∑_E ∈Ω_hℱ^E_h(v_h) .
Finally, the discrete problem reads as:
{ find u_h ∈ V_h(Ω_h) s.t.
(u_h, v_h) = ℱ_h(v_h) for all v_h ∈ V_h(Ω_h)..
§.§ Consistency of the method
Due to the polynomial projections entering in (<ref>), it is easily seen that, as usual for the VEMs, the solution u of the continuous problem (<ref>) does not solve the discrete scheme (<ref>) (thus, strong consistency does not hold). However, if u is more regular, say u∈ H^2(Ω)∩ H^1_0(Ω), then it holds:
(u , v_h) = ℱ̃(v_h) for all v_h ∈ V_h(Ω_h) .
where
(u,v_h)
∑_E ∈Ω_h (u, v_h) , ℱ̃(v_h) ∑_E ∈Ω_hℱ̃^E(v_h) ,
and the local forms are defined as follows.
∙ (u, v_h)
ϵ a^E(u, v_h)
+ (u, v_h)
+ σ c^E(u, v_h)
+𝒩̃_h^E (u, v_h)
+ J̃_h^E(u, v_h) ,
with
𝒩̃_h^E(u,v_h)
- ϵ⟨∇ u ·^E, v_h ⟩_Γ_E
+ ϵδ h_E⟨ u,v_h ⟩_Γ_E
+ 12⟨|·| u, v_h ⟩_Γ_E ,
where Γ_E = ∂ E ∩Γ, and
J̃_h^E(u, v_h)
12∑_e ⊂∂ Eγ∫_e h_e^2 [∇ u] · [∇ v_h] ds
=
12∑_e ⊂∂ Eγ∫_e h_e^2 [∇ u ·^e] [∇ v_h ·^e] ds ;
∙ ℱ̃^E(v_h) ∫_E f v_h .
§ STABILITY AND CONVERGENCE ANALYSIS
§.§ Preliminary results
Before proving the stability of the discrete problem, we mention some preliminary results that are useful for our purposes. The first one is a standard inverse estimate for the virtual element functions.
Under the assumption (A1), for any E ∈Ω_h, there exists a uniform positive constant such that
| v_h |_1,E≲ h_E^-1 v_h _0,E for all v_h ∈ V_h(Ω_h) .
We also recall, see <cit.>, the following inverse trace inequality.
Under the assumption (A1), for any E ∈Ω_h and for every v_h ∈ V_h(E) such that
Π^0,E_k-2 v_h ≡ 0,
it holds that
v_h _0,E≲
h_E^1/2 | v_h |_0, ∂ E .
We now construct a VEM version of the Oswald interpolation operator, see for instance <cit.> for the FEM framework.
We consider a point ν associated to a DoF in or and
we define E_ν⋃{ E ∈Ω_h s.t ν∈ E}, i.e. the union of the set of all the elements that contain the point ν.
The quasi-interpolation operator π for a sufficiently regular function v is defined as
π v= ∑_ν∈∪λ_ν (v) ϕ_ν + ∑_E ∈Ω_h∑_α +β≤ k-2μ_αβ^E(v) φ_αβ^E ,
where {φ_ν}_ν∈∪ are the canonical basis functions associated to the degree of freedom pointed at {ν}_ν∈∪ and the coefficients {λ_ν (v)} are defined as
λ_ν (v)1| E_ν|∑_ E ⊆ E_ν v^E(ν) |E| .
Above, and from now on in this section, a superscript E for a function denotes the restriction of that function to the element E.
Similarly, above {φ_αβ^E} denote the basis functions associated to the degrees of freedom , and {μ_αβ^E(v)} are the associated coefficients corresponding to v, defined as (cf. (<ref>)):
μ_αβ^E(v) = 1| E |∫_E v m_αβ d E .
We are ready to prove the following estimate concerning the interpolation error for piecewise polynomial functions. A FEM version of this result can be found in <cit.>.
Under assumption (A1), for every E ∈Ω_h it holds
(I - π) p _0,E≲ h^1/2∑_e ∈ℱ_E [ p ] _0,e for all p ∈_k(Ω_h) ,
where ℱ_E { e ∈ℰ s.t e ∩∂ E ∅} is the set of the edges with at least one endpoint which is a vertex of E.
We introduce the difference
δ (I - π ) p .
We restrict our attention to an element E∈Ω_h, and consider δ^E.
Since the DoFs in belong to one element only, we observe that for δ^E only the DoFs arising from and (i.e. the ones on the mesh skeleton), are involved.
Hence, noting that δ^E ∈ V_h(E) and Π^0,E_k-2δ^E =0 we can apply Lemma <ref>:
δ^E _0,E≲ h^1/2δ^E _0,∂ E≲ h δ^E _∞,∂ E .
Since the basis function associated to and are scaled in a way that their L^∞-norm is equal to 1, we have that
h δ^E _∞,∂ E≲ h
max_ν∈∪ | δ^E (ν) | .
Exploiting the definition of the Oswald interpolant, we observe that if ν∈ is not on the boundary, we have that
δ^E (ν) =
p^E (ν) - (π p)^E(ν) =
1| E ∪ E' |(
| E ∪ E' | p^E(ν) - | E | p^E(ν) - |E'| p^E'(ν)
)
=
c (p^E(ν) - p^E'(ν)) = c [p](ν) ,
where E' is the second element that shares the node ν.
Thanks to the mesh assumptions (A1), all the values
c
=
| E ∪ E' | - | E || E ∪ E' |
=
| E' || E ∪ E' |≈12 >0 .
are uniformly bounded from below and they do not depend on h; hence it holds
max_ν∈ | δ^E (ν) | ≲max_ν∈ | [p](ν) | .
If ν∈, a similiar computation allows to bound | δ^E (ν) | by means of the jumps of p at the nodes on the edges containing ν (this set is denoted by 𝒩_ν here below):
| δ^E (ν) | ≲max_ν' ∈𝒩_ν | [p](ν') | .
Combining (<ref>) and (<ref>), we get
h max_ν∈∪ | δ^E(ν) | ≲ h max_ν∈ e , e ∈ℱ_E | [p](ν) | ≲ h || [p] ||_∞,ℰ(E) ,
where ℰ(E) ⋃_e∈ℱ_E e. Since an inverse estimate gives
h || [p] ||_∞,ℰ(E)≲ h^1/2 || [p] ||_0,ℰ(E)≲
h^1/2∑_e ∈ℱ_E [p] _0,e ,
from (<ref>), (<ref>), (<ref>) and (<ref>) we obtain
(I - π) p _0,E = δ^E _0,E≲ h^1/2∑_e ∈ℱ_E [p] _0,e .
Under assumption (A1), for every E ∈Ω_h it holds
π p _0,E≲ p _0,𝒟(E) for all p ∈_k(Ω_h) ,
where 𝒟(E) ⋃{ K ∈Ω_h s.t. E̅∩K̅∅}.
Using triangular inequality, we obtain
π p _0,E≤ p _0,E + (I-π) p _0,E .
Thanks to Proposition <ref>, we control the second term with the jumps
(I-π) p _0,E≲
h^1/2∑_e ∈ℱ_E [ p ] _0,e
.
Thanks to the polynomial trace inequality, we conclude
(I-π) p _0,E≲ p _0,𝒟 (E) ,
hence
π p _0,E≤ p _0,E + (I-π) p _0,E≲ p _0,𝒟(E) .
§.§ Stability of the discrete problem
We start the theoretical analysis for the proposed method by introducing the local VEM-CIP norm
v_h^2_ , E
:=
ϵ ∇ v_h ^2_0,E +
h ·∇0 v_h ^2_0,E +
σ v_h ^2_0,E +
‖ξ (ϵ, ) v_h ‖^2_0,Γ_E +
J_h^E(v_h,v_h) ,
where
ξ (ϵ, )
(
ϵδ h +
12|·|)^1/2 ,
with global counterpart
v_h^2_∑_E ∈Ω_hv_h^2_, E .
The following two lemmas will be useful to prove the stability of the method.
Under assumptions (A1), given v_h∈ V_h(Ω_h), it holds
(v_h, v_h) ≳ϵ∇ v_h _0^2
+
J_h(v_h, v_h)
+
σ v_h _0^2
+
‖ξ (ϵ, ) v_h ‖^2_0,Γ .
We proceed locally, on each E∈Ω_h.
Thanks to the skew-symmetry property of
b_h^skew,E(·,·), testing the quadratic form
(·,·)
with v_h, we obtain
-ϵ⟨∇ v_h ·, v_h ⟩_Γ_E
+
ϵ∇ v_h _0,E^2
+
J^E_h(v_h, v_h)
+
σ v_h _0,E^2
+
‖ξ (ϵ, ) v_h ‖^2_0,Γ_E≲(v_h, v_h) .
We now handle the non-symmetric first term in (<ref>). Thanks to the Cauchy-Schwarz inequality and the Young's inequality for a positive constant α to be chosen, we have that
ϵ‖∇ v_h ‖^2_0,E
-
ϵ⟨∇ v_h ·, v_h ⟩_Γ_E
+
ϵδ h‖ v_h ‖_0,Γ_E^2
≥ϵ‖∇ v_h ‖^2_0,E
-
h ϵ2 α‖∇ v_h ·‖_0,Γ_E^2
+
( 1δ - α2) ϵh‖ v_h ‖^2_0,Γ_E .
Using the polynomial trace inequality, under the assumptions (A1), we have that
h ‖∇ v_h ·‖_0,Γ_E^2
≤
C_t ‖∇ v_h ‖_0,E^2
≤
C_t ‖∇ v_h ‖^2_0,E ,
for a uniform positive constant C_t. Hence, if we set α = C_t and 0 < δ < 2/C_t, we obtain
ϵ‖∇ v_h ‖^2_0,E
-
ϵ⟨∇ v_h ·, v_h ⟩_Γ_E
+
ϵδ h‖ v_h ‖_0,Γ_E^2
≳ϵ‖∇ v_h ‖^2_0,E +
ϵδ h‖ v_h ‖^2_0,Γ_E .
Inserting this in (<ref>), we obtain
ϵ∇ v_h _0,E^2
+
J^E_h(v_h, v_h)
+
σ v_h _0,E^2
+
‖ξ (ϵ, β) v_h ‖^2_0,Γ_E≲(v_h, v_h) .
Summing over all to elements E ∈Ω_h, we get the control of the symmetric terms in
·_:
ϵ∇ v_h _0^2
+
J_h(v_h, v_h)
+
σ v_h _0^2
+
‖ξ (ϵ, β) v_h ‖^2_0,Γ≲(v_h, v_h) .
Given v_h∈ V_h(Ω_h), let us set
w_h ,
where _h is the L^2-projection of onto the space of piecewise linear functions _1(Ω_h). Then, under assumptions (A1), if ϵ < h it holds
(v_h, w_h)
≥
C_1 h ·∇ v_h ^2_0,Ω
-C_2 (v_h, v_h) .
Thanks to Lemma <ref>, we first notice that
π (_h ·∇ v_h) _0,E≲_h ·∇ v_h _0,𝒟(E) ,
an estimate which will be frequently used in the sequel.
Recalling (<ref>), we locally have
(v_h, w_h)
=
ϵ a_h^E (v_h, )
+ J_h^E (v_h, )
+
σ c_h^E(v_h, )
+
𝒩^E_h(v_h, )
+
(v_h, )
= T_1 + T_2 + T_3 + T_4 + T_5 .
We consider each of the five terms in this equation.
Estimate for (𝐓_1).
Using Cauchy-Schwarz inequality, Lemma <ref>, estimate (<ref>) and recalling that ϵ < h, we get
ϵ a_h^E (v_h, )
≥
- ϵ a_h^E(v_h, v_h)^1/2 a_h^E(, )^1/2
≳
-ϵ^1/2∇ v_h _0,E ϵ^1/2||_1,E
≳
-ϵ^1/2∇ v_h _0,E ϵ^1/2 h^-1_0,E
≳
-ϵ^1/2∇ v_h _0,E h^1/2_h ·∇ v_h _0,𝒟(E) .
Estimate for (𝐓_2).
For the jump operator
J_h^E(·, ·),
we use again Cauchy-Schwarz inequality
J_h^E(v_h, )
≥
-J_h^E(v_h, v_h)^1/2 J_h^E(, )^1/2 .
Thanks to the trace inequality for polynomials, Lemma <ref> and estimate (<ref>), we obtain (w_h =):
J_h^E(w_h, w_h)
=
γ2∑_e ⊂∂ E∫_e h_e^2 [∇ w_h]^2 ds
+
γ h_E _J ( ( I - ) w_h, ( I - ) w_h )
≲
h ∇0 w_h _0,𝒟(E)^2
+ h_E ||_1,E^2
≲
h^-1 0 w_h _0,𝒟(E)^2 + h^-1_0,E^2
≲
h^-1_0,𝒟(E)^2
≲
h _h ·∇ v_h _0,𝒟(𝒟(E))^2 ,
where 𝒟(𝒟(E)) := ∪_E'⊆𝒟(E)𝒟(E').
Therefore, it holds
J_h^E(v_h, )
≳
-J_h^E(v_h, v_h)^1/2 h^1/2 _h ·∇ v_h _0,𝒟(𝒟(E)) .
Estimate for (𝐓_3).
Using a similar procedure, we control the bilinear form
c_h(·,·)
in this way
σ c_h^E(v_h, )
≳
- σ v_h _0,E _0,E
≳
- v_h _0,E h^1/2_h ·∇ v_h _0,𝒟(E) .
where we used h^1/2≲ 1 to simplify later developments.
Estimate for (𝐓_4).
For the Nitsche term, we have that
𝒩^E_h(v_h,w_h) =
- ϵ⟨∇ v_h ·^E, w_h ⟩_Γ_E + ϵδ h_E⟨ v_h, w_h ⟩_Γ_E
+ 12⟨|·| v_h, w_h ⟩_Γ_E .
We consider each of the three terms above. Using Cauchy-Schwarz inequality, trace inequality, ϵ<h and inverse estimate, the first term is estimated by
ϵ⟨∇ v_h ·^E, ⟩_Γ_E ≳
- ϵ h^-1/2‖∇ v_h ‖_0,E
h^-1/2‖‖_0,E
≳
-ϵ^1/2∇ v_h _0,E h^1/2_h ·∇ v_h _0,𝒟(E) .
For the second term we have
ϵδ h_E⟨ v_h, ⟩_Γ_E ≳
-ϵδ h‖ v_h ‖_0,Γ_E h^-1/2‖‖_0,E
≳
-ϵ^1/2δ h^1/2‖ v_h ‖_0,Γ_E h^1/2‖π ( _h ·∇ v_h) ‖_0,E
≳
- ‖ξ (ϵ, ) v_h ‖_0,Γ_E h^1/2_h ·∇ v_h _0,𝒟(E) .
For the last one, using the same estimates, we get
12⟨|·| v_h, ⟩_Γ_E ≳
- ‖ξ (ϵ, ) v_h ‖_0,Γ_E h^1/2_h ·∇ v_h _0,𝒟(E) .
Hence it holds
𝒩^E_h(v_h, w_h) ≳ - ( ϵ^1/2‖∇ v_h ‖_0,E + ‖ξ (ϵ, ) v_h ‖_0,Γ_E) h^1/2‖_h ·∇ v_h ‖_0,𝒟(E) .
Estimate for (𝐓_5). It is the most involved term.
The skew term (v_h, w_h) is composed by two parts
(v_h, w_h)
=
12 ( b_h^E(v_h, w_h) - b_h^E(w_h, v_h)) ,
and we consider each of these two terms separately.
The first term is defined as
b_h^E(v_h, w_h)
=
(·∇0 v_h, 0 w_h )_0,E +
((·^E) ( I - 0 ) v_h, 0 w_h)_0,∂ E .
We split the first term of (<ref>) as
( ·∇0 v_h, 0 w_h )_0,E =
( ·∇0 v_h, w_h )_0,E
+
( ·∇0 v_h, ( 0 - I ) w_h )_0,E
=
( ·∇0 v_h, h _h ·∇0 v_h )_0,E
+
( ·∇0 v_h, w_h - h _h ·∇0 v_h )_0,E
+
(·∇0 v_h, (0 - I) w_h)_0,E
η__1 + η__2 + η__3 .
We estimate each of these three quantities.
For the first term we have
η__1 =
(·∇0 v_h, )_0,E
=
h ·∇0 v_h ^2_0,E
+
(·∇0 v_h, h (_h - )·∇0 v_h)_0,E
≥
h ·∇0 v_h ^2_0,E
- C h^1/2·∇0 v_h _0,E h^1/2 | |_W^1,∞(E) h ∇0 v_h _0,E
≥
h ·∇0 v_h ^2_0,E
- C h^1/2·∇0 v_h _0,E h^1/2 | |_W^1,∞(E) v_h _0,E
Recalling (<ref>) and by Young's inequality we get:
η__2 = h ( ·∇0 v_h, (π - I) (_h ·∇0 v_h) )_0,E
≥
- h2·∇0 v_h ^2_0,E
- h2 (π - I) (_h ·∇ v_h) ^2_0,E .
Since _h is piecewise linear, for the second term we can use Proposition <ref> and obtain
h (π - I) (_h ·∇ v_h) _0,E^2
≲
h^2 ∑_e ⊂ℱ_E [_h ·∇ v_h] _0,e^2
≲
h^2 ∑_e ⊂ℱ_E [( _h - ) ·∇ v_h] _0,e^2
+
h^2 ∑_e ⊂ℱ_E [·∇ v_h] _0,e^2
≲
h^2 ∑_e ⊂ℱ_E [( _h - ) ·∇ v_h] _0,e^2
+
J_h^𝒟(E) (v_h, v_h)
On each e, we control the first term in the previous inequality as
h^2 [(_h -)·∇ v_h] _0,e^2
≲
h^4 | |^2_W^1,∞(E∪ E') h^-1∇ v_h _0,E∪ E'^2
≲
h | |^2_W^1,∞(E∪ E') v_h _0,E∪ E'^2
≲
h | |^2_W^1,∞(E∪ E') v_h _0,E∪ E'^2 ,
where E and E' are the two elements sharing the edge e.
Combining (<ref>) with (<ref>) and (<ref>), we obtain
η__2 ≥ -
h2·∇0 v_h ^2_0,E
- C(
h | |^2_W^1,∞(𝒟(E)) v_h _0,𝒟(E)^2
+J_h^𝒟(E)(v_h, v_h)
) .
It remains to control η__3. Since _h∈_1(E), it holds ( _h ·∇0 v_h, (0 - I) w_h )_0,E= 0.
Hence we have
η__3 = ( ( - _h) ·∇0 v_h, (0 - I) w_h )_0,E
≳
- ( - _h) ·∇0 v_h _0,E _0,E
≳
- | |_W^1,∞(E) h ∇0 v_h _0,E h _h ·∇ v_h _0,𝒟(E)
≳
- | |_W^1,∞(E) v_h _0.𝒟(E)^2
Collecting (<ref>), (<ref>) and (<ref>), from (<ref>) we get
( ·∇0 v_h, 0 w_h )_0,E≥h2 ·∇0 u_h ^2_0,E
- C(
J_h^𝒟(E)(v_h, v_h)
+ h^1/2·∇0 v_h _0,E h^1/2 | |_W^1,∞ v_h _0,E
+ h | |^2_W^1,∞(𝒟(E)) v_h _0,𝒟(E)^2
+ | |_W^1,∞(E) v_h _0,𝒟(E)^2
) .
Returning to (<ref>), we have to control the boundary term.
We first notice that, due to Agmon's inequality and Poincaré's inequality, it holds
( I - 0) v_h _0,∂ E≲ h^1/2| ( I - 0) v_h |_1,E .
Together with an inverse inequality for the polynomial 0 w_h, the definition of w_h (cf. (<ref>)), and Lemma <ref>, we thus get:
( (·^E) (I - 0) v_h, 0 w_h )_0, ∂ E ≳
- ( I - 0) v_h _0,∂ E 0 w_h _0,∂ E
≳
- h^1/2 | ( I - 0) v_h |_1,E h^-1/20 w_h _0,E
≳
- | ( I - 0) v_h |_1,E w_h _0,E
≳
- J_h^E(v_h, v_h)^1/2 h^1/2_h ·∇ v_h _0,𝒟(E) .
Above, we have also used the estimate, see (<ref>):
h | ( I - 0) v_h |_1,E^2 ≲ J_h^E(v_h, v_h) .
From (<ref>), (<ref>) and (<ref>) we get
b_h^E(v_h, w_h) ≥h/2 ·∇0 u_h ^2_0,E
- C(
J_h^𝒟(E)(v_h, v_h)
+ h^1/2·∇0 v_h _0,E h^1/2 | |_W^1,∞ v_h _0,E
+ h | |^2_W^1,∞(𝒟(E)) v_h _0,𝒟(E)^2
+ | |_W^1,∞(E) v_h _0,𝒟(E)^2
+ J_h^E(v_h, v_h)^1/2 h^1/2_h ·∇ v_h _0,𝒟(E)) .
Finally, we need to control
- b_h^E(w_h, v_h), see (<ref>).
Integrating by parts, we obtain
- b_h^E(w_h, v_h)
= - ( ·∇0 w_h, 0 v_h )_0,E
-
( (·^E) (I - 0) w_h, 0 v_h )_0,∂ E
=
( ·∇0 v_h, 0 w_h )_0,E
-
( (·^E) w_h, 0 v_h )_0,∂ E
=
( ·∇0 v_h, 0 w_h )_0,E
-
( (·^E) w_h,
(0 - I) v_h )_0,∂ E
-
( (·^E) w_h,
v_h )_0,∂ E .
The first two terms are similar to the case
b_h(v_h, w_h).
The last one vanishes on the interior edges when we sum over all E∈Ω_h. Hence, we need to consider the elements E sharing with ∂Ω at least an edge. Using Cauchy-Schwarz inequality, trace inequality, inverse estimates and the continuity of π, we obtain on these boundary edges
- ( (·^E) w_h,
v_h )_0,∂ E ≥
- ‖ξ (ϵ, ) v_h ‖_0,Γ_E‖ξ (ϵ, ) w_h ‖_0,Γ_E
≳
-
‖ξ (ϵ, ) v_h ‖_0,Γ_E
h^-1/2‖ w_h ‖_0,E
≳
-
‖ξ (ϵ, ) v_h ‖_0,Γ_E
h^1/2_h ·∇ v_h _0,𝒟(E) .
Therefore, from (<ref>), (<ref>), (<ref>) and (<ref>) we get
(v_h, w_h)
≥h/2 ·∇0 u_h ^2_0,E - C(
J_h^𝒟(E)(v_h, v_h)
+ h^1/2·∇0 v_h _0,E h^1/2 | |_W^1,∞ v_h _0,E
+ h | |^2_W^1,∞(𝒟(E)) v_h _0,𝒟(E)^2
+ | |_W^1,∞(E) v_h _0,𝒟(E)^2
+ ( J_h^E(v_h, v_h)^1/2 +
‖ξ (ϵ, ) v_h ‖_0,Γ_E) h^1/2_h ·∇ v_h _0,𝒟(E)) .
We now consider the five local estimates (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>). From (<ref>), summing over all the elements E∈Ω_h, we obtain
(v_h, w_h)
≥h/2·∇ v_h _0,Ω^2
- C (
∑_E ∈Ω_h( ϵ^1/2∇ v_h _0,E
+
J_h^E(v_h, v_h)^1/2 + v_h _0,E + ‖ξ (ϵ, β) v_h ‖_0,Γ_E) h^1/2_h ·∇0 v_h _0,E
+ J_h(v_h, v_h) +
∑_E ∈Ω_h(h | |^2_W^1,∞(E)+ | |_W^1,∞(E)) v_h _0,E^2
+
∑_E ∈Ω_h h^1/2_h ·∇0 v_h _0,E h^1/2 | |_W^1,∞(E) v_h _0,E) .
Above, we have also used the property that, due to assumption (A1), summing over the elements each polygon is counted only a uniformly bounded number of times, even when the terms involve norms on 𝒟(E) or 𝒟(𝒟(E)).
We now notice that the triangular inequality, standard approximation results and an inverse estimate give
h^1/2_h ·∇0 v_h _0,E≲
h^1/2( ·∇0 v_h _0,E
+
| |_W^1,∞(E) v_h _0,E).
Hence, from (<ref>), using also Young's inequality (with suitable constants) for the first and the last summations in the right-hand side, we get
(v_h, w_h)
≥ C_1 h ·∇ v_h _0,Ω^2 - C_2 ( ϵ∇ v_h _0^2 + J_h(v_h, v_h) + v_h _0^2
+ ‖ξ (ϵ, β) v_h ‖^2_0,Γ) .
From Lemma <ref>, we now obtain
(v_h, w_h)
≥
C_1 h ·∇ v_h ^2_0,Ω
-C_2 (v_h, v_h) .
With Lemmas <ref> and <ref> at our disposal, the inf-sup condition easily follows.
Under assumptions (A1), it holds:
v_h _≲sup_z_h ∈ V_h(Ω_h) (v_h, z_h) z_h _ for all v_h ∈ V_h(Ω_h).
We split the proof into two cases.
First case. We first consider ε < h. Given v_h ∈ V_h(Ω_h), we take z_h =w_h + κ v_h, where w_h is defined as in Lemma <ref>. From Lemmas <ref> and <ref>, for κ sufficiently large we have
(v_h, z_h) =
(v_h, w_h + κ v_h)
≳ v_h _^2 .
In order to conclude the proof of the inf-sup condition, we have to prove the estimate
w_h _≲ v_h _ ,
which obviously implies z_h _≲ v_h _. Recalling the norm definition (<ref>)-(<ref>) and that w_h :=, the above continuity estimate follows from Lemma <ref>, estimate (<ref>) and observing that
h ·∇0 w_h _0,E^2
≲
h^-10 w_h _0,E^2
≲
h _h ·∇0 v_h _0,𝒟(E)^2 ,
and
w_h _0,Γ_E^2 = _0,Γ_E^2
≲ h π( _h·∇Π^0_k v_h ) _0,E^2 +
h^3 | π(_h·∇Π^0_k v_h) |_1,E^2
≲
h π( _h·∇Π^0_k v_h) _0,E^2
≲ h _h·∇Π^0_k v_h _0,𝒟(E)^2
.
The above bounds (<ref>) and (<ref>) are to be combined with (<ref>).
Second case.
We now consider the case ε≥ h.
In such case the proof simply follows from Lemma <ref> and the observation that
h ·∇0 u_h ^2_0,E≲ε∇0 u_h ^2_0,E≲ε∇ u_h ^2_0,E ,
which allows to control also convection with (v_h, v_h).
§.§ Error estimates
We begin our error analysis, which follows the steps of <cit.>, with the following result.
Let u ∈ V and u_h ∈ V_h(Ω_h) be the solutions of problem (<ref>) and problem (<ref>), respectively. Furthermore, let us define
u - ,
where ∈ V_h(Ω_h)
is the interpolant function of u defined in Lemma <ref>.
Then under assumption (A1), it holds that
u - u_h _≲_
+
∑_E ∈Ω_h(
+ + + + + ) ,
where (cf. Section <ref>)
ℱ̃^E - ℱ_h^E _ ,
ϵ a^E(u, ·) - a_h^E(, ·) _ ,
(u, ·) - (, ·) _ ,
c^E(u, ·) - c_h^E(, ·) _ ,
𝒩_h^E(u, ·) - 𝒩_h^E(, ·) _ ,
J̃_h^E(u , ·) - J_h^E(, ·)_ = J_h^E(, ·)_ ,
where ·_ is the dual norm of ·_.
We first introduce the following quantities
u - Π^∇_k u ,
e_h u_h - .
Using triangular inequality, we have that
u - u_h _≤ u - u_ℐ_
+ u_ℐ - u_h _
=
e_ℐ_
+ e_h _ .
Thanks to the inf-sup condition, and recalling that u satisfies (<ref>), we have that
e_h _ ≲sup_v_h ∈ V_h(Ω_h)(e_h, v_h) v_h _
=
sup_v_h ∈ V_h(Ω_h)(u_h - u_ℐ, v_h) v_h _
=
sup_v_h ∈ V_h(Ω_h)ℱ_h(v_h) - (u_ℐ, v_h) v_h _
=
sup_v_h ∈ V_h(Ω_h)ℱ_h(v_h) - ℱ̃(v_h) + (u, v_h) - 𝒜_(u_ℐ, v_h) v_h _
=
sup_v_h ∈ V_h(Ω_h)∑_E ∈Ω_h( ℱ_h^E(v_h) - ℱ̃^E(v_h) + (u, v_h) - (u_ℐ, v_h) ) v_h _ .
Estimate (<ref>) easily follows by recalling the definitions of and , see (<ref>) and (<ref>)-(<ref>).
To properly bound all the terms in Proposition <ref> we make the following assumptions:
(A2) Data assumption. The solution u, the advective field and the load f in (<ref>) satisfy:
u ∈ H^+1(Ω_h) , f ∈ H^+1(Ω_h) , ∈ [W^+1_∞(Ω_h)]^2 ,
for some 0 < ≤ k.
Under assumptions (A1) and (A2),
the term ^2_ can be bounded as follows (for 0 < s ≤ k)
^2_,E≲ϵ h^2 | u |^2_+1,E + h^2+1 | u |^2_+1,E .
By definition of ·_, we have that
^2_,E
=
ϵ∇^2_0,E
+ h ·∇0 _0,E^2
+ σ_0,E^2
+ ξ ( ϵ, ) e_ℐ^2_Γ_E
+ J_h^E(e_ℐ, e_ℐ) .
Using lemma <ref>, we have that
ϵ∇^2_0,E
+ h ·∇0 _0,E^2
≲
(ϵ + h) ∇^2_0,E≲
(ϵ + h) h^2 | u |^2_+1,E ,
and
^2_0,E≲ h^2+2 | u |_+1,E^2 .
For the Nitsche term we have that
ξ ( ϵ, ) e_ℐ^2_Γ_E = ϵδ h⟨ e_ℐ, e_ℐ⟩_Γ_E + ⟨ | ·^E | e_ℐ, e_ℐ⟩_Γ_E .
Using trace inequality and interpolation estimate, we obtain
ϵδ h⟨ e_ℐ, e_ℐ⟩_Γ_E≲ϵδ h^2 e_ℐ_0,E^2
+
ϵδ | e_ℐ |_1,E^2
≲ϵ h^2s |u|_s+1,E^2 ,
and
⟨ | ·^E | e_ℐ, e_ℐ⟩_Γ_E≲ h^-1 e_ℐ_0,E^2
≲ h^2s+1 |u|_s+1,E^2 .
It remains to control the jump operator. We have
J_h^E(e_ℐ,e_ℐ)
=
γ2∑_e ⊂∂ E∫_e h_e^2 [∇ e_ℐ] · [∇ e_ℐ] ds
+
γ h_E _J ( (I - ) e_ℐ, (I - ) e_ℐ)
≲
h^2 ( h^-1∇ e_ℐ_0,𝒟(E)^2
+
h |∇ e_ℐ|_1,𝒟(E)^2)
+
h | (I - ) |^2_1,E
≲
h^2 (h^-1∇ e_ℐ_0,𝒟(E)^2 )
+
h | (I - ) |^2_1,E
≲
h ||^2_1,𝒟(E)≲ h^2+1 | u |_+1,𝒟(E)^2 .
Under the assumptions (A1) and (A2),
the term can be bounded as follows (for 0 < s ≤ k)
η_ℱ^E
≲
h^+1 | f |_+1,E .
It is sufficient to follow the same procedure of <cit.>. Using the orthogonality of 0, Cauchy-Schwarz inequality, Poincaré inequality and Lemma <ref>, we obtain
η_ℱ^E
=
ℱ̃^E(v_h) - ℱ^E_h(v_h)
=
(f, v_h - 0 v_h )_0, E
=
( (I-0) f, (I - 0) v_h )_0, E
≤ (I-0) f _0,E (I - 0) v_h _0, E
≲ (I-0) f _0,E v_h _0, E
≲
h^+1 | f |_+1,E v_h _,E .
Under the assumptions (A1) and (A2),
the term can be bounded as follows (for 0 < s ≤ k)
≲ϵ^1/2 h^ | u |_+1,𝒟(E) .
This result is proved following the line of Lemma 5.3 of <cit.>.
Adding and subtracting u, using Cauchy-Schwarz inequality, we obtain
=
ϵ ã^E_h(u, v_h) - ϵ a_h^E(u_ℐ, v_h)
=
ϵ ã^E_h(u - u,v_h) + ϵ a_h^E( u - u_ℐ,v_h)
≤ϵ (∇ e_π_0,E + (1 + α^*) ∇ ( u - u_ℐ) _0,E) ∇ v_h _0,E
≲ϵ (∇ e_π_0,E + ∇ e_ℐ_0,E) ∇ v_h _0,E≲ϵ^1/2 h ^ | u |_+1,E v_h _,E .
Under the assumptions (A1) and (A2),
the term can be bounded as follows (for 0 < s ≤ k)
≲
h^ + 1/2 u _+1
+
_[W^+1,∞]^2 h^+1 u _+2,E
+
∫_∂ E (· ^E) e_ℐ v_h ds .
Recalling the definition, we need to estimate
( ·∇ u, v_h )_0,E
-
( ·∇0 u_ℐ, 0 v_h )_0,E
-
∫_∂ E (· ^E) (I - 0) u_ℐ 0 v_h ds ,
( 0 u_ℐ, ·∇0 v_h )_0, E
-
( u, ·∇ v_h )_0,E
+
∫_∂ E (· ^E) (I - 0) v_h 0 u_ℐ ds .
By integration by parts, we have
η_b,A^E
=
( ·∇ u, (I - 0) v_h )_0,E
+
( ·∇ (u - 0 u_ℐ), 0 v_h )_0,E
-
∫_∂ E (·^E) (I - 0) u_ℐ 0 v_h ds
=
( ·∇ u, (I - 0) v_h )_0,E
-
( u - 0 u_ℐ, ·∇0 v_h )_0,E
+
∫_∂ E (·^E) (u - u_ℐ) 0 v_h ds
=
( (I - 0) ·∇ u, (I - 0) v_h )_0,E
+
( 0 u_ℐ - u, ·∇0 v_h )_0,E
+
∫_∂ E (·^E) e_ℐ 0 v_h ds
η_b,1^E + η_b,2^E + η_b,3^E ,
and
η_b,B^E
=
( 0 u_ℐ - u, ·∇0 v_h )_0,E
-
( u, ·∇ (I - 0) v_h )_0,E
+
∫_∂ E (·^E) (I - 0) v_h 0 u_ℐ ds
=
( 0 u_ℐ - u, ·∇0 v_h )_0,E
+
( ·∇ u, (I - 0) v_h )_0,E
+
∫_∂ E (·^E) (I - 0) v_h (0 u_ℐ - u) d s
=
(0 u_ℐ - u, ·∇0 v_h)_0,E
+
((I - 0)·∇ u,(I - 0)v_h)_0,E
+
∫_∂ E (·^E) (I - 0) v_h (0 u_ℐ - u) d s
η_b,2^E + η_b,1^E + η_b,4^E .
yielding the following expression for
η_b^E
2 η_b^E = 2η_b,1^E + 2η_b,2^E + η_b,3^E + η_b,4^E .
We now analyze each term in the sum above.
∙ η_b,1^E: using Cauchy-Schwarz, the continuity in 0 in L^2 and standard estimates, we obtain
η_b,1^E
=
( (I - 0) ·∇ u, (I - 0) v_h )_0,E
≤ (I-0) ·∇ u_0,E v_h _0,E
≤ (I - 0) ·∇ u _0,E v_h _,E
≲
h^+1|·∇ u |_+1,E v_h _,E
≲
h^+1 u _+1,E β_[W^+1_∞(E)]^2 v_h _,E .
∙ η_b,2^E: we have that
η_b,2^E
=
( 0 u_ℐ - u, ·∇0 v_h )_0,E
≤0 u_ℐ - u _0,E ·∇0 v_h _0,E
≤( (I - 0)u _0,E + e_ℐ_0,E)
·∇0 v_h _0,E
≲ h^ + 1/2 u _+1 v_h _,E .
∙ η^E_b,3 + η^E_b,4: we use a scaled trace inequality making use of the scaled norm
∀ v ∈ H^1(E), v _1,E^2 v _L^2(E)^2 + h^2_E | v |^2_H^1(E) .
We obtain
η^E_b,3 + η^E_b,4 =
∫_∂ E (·^E) e_ℐ 0 v_h d s
+
∫_∂ E (·^E) (I - 0) v_h (0 u_ℐ - u) d s
=
∫_∂ E (·^E) (0 - I) v_h (e_ℐ + u - 0 u_ℐ) d s
+
∫_∂ E (·^E) e_ℐ v_h d s
≲ (
e_ℐ_L^2(∂ E)
+
u - 0 u_ℐ_L^2(∂ E)
) (I - 0)v_h_L^2(∂ E)
+
∫_∂ E (·^E) e_ℐ v_h d s
≲ h_E^-1 (
e_ℐ_1,E
+
u - 0 u_ℐ_1,E
) (I - 0)v_h_0,E
+
∫_∂ E (·^E) e_ℐ v_h d s
≲
h^-1/2 (
e_ℐ_1,E
+
u - 0 u_ℐ_1,E
)
h^1/2∇(I - )v_h_0,E
+
∫_∂ E (·^E) e_ℐ v_h d s
≲ h^+1/2| u |_+1,E v_h _,E + ∫_∂ E (·^E) e_ℐ v_h d s ,
where in the last step we used the J_h(v_h,v_h) term in the definition of v_h _,E.
The thesis now follows gathering the last three inequalities in (<ref>).
Under the assumptions (A1) and (A2),
the term can be bounded as follows (for 0 < s ≤ k)
≲
h^ + 1 | u |_+1,E
Similarly to Lemma <ref>, we have that
=
c̃^E_h(u,v_h) - c_h^E(u_ℐ,v_h)
=
c̃^E_h(u - 0 u,v_h) + c_h^E(0 u - u_ℐ,v_h)
≤
( e_π_0,E + (1 + α^*) 0 u - u_ℐ_0,E) v_h _0,E
≲
( e_π_0,E + e_ℐ_0,E) v_h _0,E≲
h^ + 1 | u |_+1,E v_h _,E .
Under the assumptions (A1) and (A2),
the term can be bounded as follows (for 0 < s ≤ k)
≲
(ϵ^1/2 h^s + h^s+1/2 )| u |_s+1,E .
By definition of the two bilinear forms, we have that
=
- ϵ⟨∇ u · , v_h ⟩_Γ_E + ϵ⟨∇ u_ℐ· , v_h ⟩_Γ_E
+
ϵδ h_E⟨ u, v_h ⟩_Γ_E
- ϵδ h_E⟨ u_ℐ, v_h ⟩_Γ_E
- 12⟨ |·| u, v_h ⟩_Γ_E
+ 12⟨ |·| u_ℐ, v_h ⟩_Γ_E
+ + .
Now, we estimate each of the three terms. Using trace inequality, the first returns
=
- ϵ⟨∇ u · , v_h ⟩_Γ_E + ϵ⟨∇ u_ℐ· , v_h ⟩_Γ_E
=
- ϵ⟨∇ (u - ∇ u_ℐ) · , v_h ⟩_Γ_E
≲ϵ ( h^-1/2‖∇ u - ∇ u_ℐ‖_0,E + h^1/2 | ∇ u - ∇ u_ℐ |_1,E ) ‖ v_h ‖_Γ_E
≲ϵ^1/2 (‖∇ u - ∇ u_ℐ‖_0,E + h | ∇ u - ∇ u_ℐ |_1,E ) ‖ v_h ‖_cip,E .
Adding and subtracting u, using triangular inequality and Lemma <ref>, we obtain
≲ϵ^1/2 h^s | u |_s+1,E‖ v_h ‖_cip,E .
For the second term, using trace inequality and interpolation estimate, we have that
=
ϵδ h⟨ u, v_h ⟩_Γ_E - ϵδ h_E⟨ u_ℐ, v_h ⟩_Γ_E
≲ϵδ h‖ u - u_ℐ‖_Γ_E‖‖ v_h ‖_Γ_E≲( ϵδ h)^1/2‖ u - u_ℐ‖_Γ_E‖‖ v_h ‖_,E
≲ϵ^1/2 h^s | u |_s+1,E‖ v_h ‖_cip,E .
Finally, the last one is treated in a very similar way with respect to the previous one, it gives
=
-12⟨ |·| u, v_h ⟩_Γ_E - 12⟨ |·| u_ℐ, v_h ⟩_Γ_E
≲
h^-1/2‖ u - u_ℐ‖_0,E‖ v_h ‖_,E
≲ h^s+1/2 | u |_s+1,E‖ v_h ‖_,E .
Under the assumptions (A1) and (A2),
the term can be bounded as follows (for 0 < s ≤ k)
≲
h^s + 1/2 | u |_s+1,E .
Using Cauchy-Schwarz inequality, we have that
J^E_h(u_ℐ, v_h)
≤
J^E_h(u_ℐ, u_ℐ)^1/2 J^E_h(v_h, v_h)^1/2
≤
J^E_h(u_ℐ, u_ℐ)^1/2 v_h _,E .
Since the solution u is sufficiently smooth, we have that
J^E_h(u_ℐ, v_h)
=
∑_e ⊂∂ E∫_eγ2 h^2_e [∇ u_ℐ]·[∇ u_ℐ]
+
h_E γ 𝒮_j^E( ( I - ) u_ℐ,( I - ) u_ℐ)
=
∑_e ⊂∂ E∫_eγ2 h^2_e [∇ ( u_ℐ - u)]^2
+
h_E γ 𝒮_j^E( ( I - ) u_ℐ,( I - ) v_h)
≲∑_E' ∈𝒟(E) h^2 ∇ ( u_ℐ - u) _0,∂ E'^2
+ h | ( I - ) u_ℐ |_1,E^2 .
Using trace inequality, we obtain for the first term
∇ u_ℐ - ∇ u _0,∂ E' ≲( h^-1 ∇ u_ℐ - ∇ u ^2_0,E' + h |∇ u_ℐ - ∇ u |^2_1,E')^1/2 .
Adding and subtracting ∇0 u, using Lemma <ref> and interpolation estimate, we obtain
h^-1/2∇0 u_ℐ - ∇ u_0,E' ≲
h^-1/2∇0 u - ∇ u _0,E' + h^-1/2∇0 u_ℐ - ∇0 u _0,E'
≲
h^ - 1/2| u |_ + 1 ,E' ,
and similarly, we have that
h^1/2|∇0 u_ℐ - ∇ u|_1,E' ≲
h^ - 1/2| u |_ + 1 ,E' .
Using Lemma <ref>, we have that
h^1/2 | ( I - ) u_ℐ |_1,E ≲
h^1/2 ( ∇_0,E + ∇_0,E )
≲
h^ + 1/2 | u |_ + 1, E .
We conclude
J^E_h(u_ℐ, v_h)
≲
h^s + 1/2 | u |_s+1,𝒟(E) v_h _, 𝒟(E) .
We thus have the following proposition.
Under the assumptions (A1) and (A2), let u ∈ V be the
solution of equation (<ref>) and u_h ∈ V_h(Ω_h) be the solution of equation (<ref>).
Then it holds that
u - u_h^2_≲∑_E ∈Ω_hΘ^E
(
ϵ h^2
+
h^2+1
) ,
where the constant Θ^E depends on
u_s+2,E, f_s+1,E, β_[W^s+1_∞(E)]^2.
It it sufficient to use Proposition <ref> combined with Lemmas <ref>, <ref>, <ref>, <ref>, <ref>, <ref> <ref>, noting that
∑_E ∈Ω_h∫_∂ E∖∂Ω (·^E) e_ℐ v_h d s = 0 ,
and the contributions stemming from ∂Ω are controlled as in (<ref>).
§.§ A special case: advection-diffusion problem with ∈_1(Ω)
We consider problem (<ref>) in a particular situation: we assume an advection term ∈_1(Ω), i.e. globally linear, and we allow the reaction coefficient σ = 0. We do not make further assumptions on the diffusion coefficient ϵ and on the load term f. Thus, the advection-diffusion problem reads as (cf. (<ref>))
{ find u ∈ V such that:
ϵ a(u, v) + (u, v) = ℱ̃(v) for all v ∈ V..
In this case, even without the reaction term, we are able to prove robust estimates for the approximation of problem (<ref>).
Using the same approach as before, the discrete version of problem (<ref>) reads as
{ find u_h ∈ V_h(Ω_h) such that:
^ ad(u_h, v_h) = ℱ_h(v_h) for all v_h ∈ V_h(Ω_h),.
where
^ ad(u_h, v_h) ∑_E ∈Ω_h^ ad,E (u_h, v_h) ,
and
^ ad,E (u_h, v_h) ϵ a_h^E(u_h , v_h) + (u_h , v_h) + 𝒩_h(u_h, v_h) + J_h^E(u_h , v_h) .
The key observation is that a suitable inf-sup condition still holds true without the help of the L^2-type norm stemming from the reaction term. In fact, introducing the local norm
v_h^2_, ad, E
:=
ϵ ∇ v_h ^2_0,E +
h ·0P ∇ v_h ^2_0,E +
‖ξ (ϵ, ) v_h ‖^2_0,Γ_E +
J_h^E(v_h,v_h) ,
and its global counterpart
v_h^2_, E∑_E ∈Ω_hv_h^2_, ad, E ,
similarly to Proposition <ref>, we have the following result.
Under assumptions (A1), it holds that
v_h _, ad≲sup_z_h ∈ V_h(Ω_h)^ ad (v_h, z_h) z_h _ , ad for all v_h ∈ V_h(Ω_h),
for a constant that does not depend on h and ϵ.
The proof of (<ref>) is analogous to the one of Proposition <ref>, with the simplification that in Lemma <ref> it holds _h=. The main difference stands in the treatment of η__1, η__2 and η__3 in (𝐓_5), see (<ref>). These terms are the only ones requiring the help of the L^2 norm in the general case (apart, of course, the reaction term itself).
Regarding the term η__1, in our present case we immediately have the advective norm:
η__1 =
h ·∇0 v_h ^2_0,E
Since ∈_1(Ω), it follows that ·∇0 v_h∈_k(E), so that we can directly bound η__2 using Young's inequality and Proposition <ref>:
η__2 = h ( ·∇0 v_h, ( π - I ) (·∇0 v_h) )_0,E
≥ - h2·∇0 v_h ^2_0,E -
h2 (π - I) (·∇0 v_h) ^2_0,E
≥ - h2·∇0 v_h ^2_0,E - C J_h^𝒟(E)(v_w,v_h) ,
which is the counterpart of (<ref>).
Furthermore, again since ·∇0 v_h∈_k(E), it follows that
η__3 := (·∇0 v_h, (0 - I) w_h)_0,E = 0 .
Once the above stability result has been established, the next error estimate can be proved using the same arguments of Proposition <ref>.
The only difference is handling the terms η_ℱ^E and η_b,1^E which now must be bounded using diffusion (since reaction is not available) and therefore paying a price in terms of ε but with a better rate in terms of h. For the sake of conciseness we here omit the simple alternative derivations for such terms.
Under the assumptions (A1) and (A2), let u ∈ V be the
solution of equation (<ref>) and u_h ∈ V_h(Ω_h) be the solution of equation (<ref>).
Then it holds that
u - u_h^2_, ad ≲∑_E ∈Ω_hΘ^E
(
ϵ h^2
+
h^2+1
+
h^2( + 2)ϵ) ,
where the constant Θ^E depends on
u_s+1,E, f_s+1,E, β_[W^s+1_∞(E)]^2/β_E.
§ NUMERICAL EXPERIMENT
In this section, we investigate the actual computational behavior of the proposed method.
Model problem.
We consider a family of problems in the unit square Ω=(0, 1)^2. We choose the boundary conditions and the source term (which turns out to depend on ϵ, σ and ) in such a way that the analytical solution is always the function
u(x, y) := sin(π x)sin(π y) .
Different choices of the parameters σ, ϵ and of the advective term (x,y) will be selected. Since the pointwise values of the numerical solution u_h are unknown, the following error quantities will be considered:
* H^1-seminorm error
e_H^1 := √(∑_E∈𝒯_h∇(u-Π_k^∇ u_h)^2_0,E) ;
* L^2-norm error
e_L^2 := √(∑_E∈𝒯_h(u-0 u_h)^2_0,E) .
We will consider two different families of mesh:
* a mesh composed by highly distorted quadrilaterals obtained perturbing a mesh composed of structured squares;
* a centroidal Voronoi tessellation of the unit square.
These two families are represented in Figure <ref>.
Effects of the CIP stabilization. The first aspect we investigate is the benefits of inserting the CIP term in the variational formulation of the problem. We thus consider an advection-dominated regime and choose the parameters ϵ = 1e-9, σ = 0, along with a constant advection term
(x, y) := [[ 1; 0.5 ]] .
We consider a centroidal Voronoi tesselation of the domain Ω into 256 polygons.
The degree of the method is set to k=1.
In Figure <ref> we observe that by inserting the bilinear form J_h(·,·) in the variational formulation, we are able to accurately approximate the analytic solution u(x,y) of the model problem.
If we omit the CIP term, we obtain (as expected) a definitely unsatisfactory numerical solution, which exhibits nonphysical oscillations all over the computational domain. We also remark that these instabilities reach peaks of the order of 10^2, despite for the analytic solution we have u _L^∞(Ω)=1.
Convergence analysis We now investigate the convergence of the numerical method by means of the above-introduced norms, and choosing different consistency order, i.e. k=1,2,3.
The convective term is
(x, y) := [[ 1; 0.5 ]] .
We consider a diffusion-dominated case (ϵ = 1), and an advection-dominated one (ϵ = 1e-9).
Thus, we are in the framework of Section <ref>. Accordingly, we neglet the reaction term (hence, σ = 0) and the theoretical error bound of Proposition <ref> holds.
We compare the method with and without the jump term J_h(·,·). The results are obtained using the Voronoi mesh family.
In Figure <ref>, we observe that in the case ϵ = 1 the two methods behave in the same way. Instead, in the advection-dominated regime we observe that the optimal convergences are attained when inserting the stabilising jump term; without it, as expected, the method displays unsatisfactory results, especially for the low-order case.
Effect of the reaction term. We now consider an advection-dominated problem with a variable advection term not in _1(Ω).
In particular, we select
(x, y) := [[ -2 π sin(π (x+2 y)); π sin(π (x+2 y)) ]] .
We recall that for this case we are able to prove robust error bounds only with the aid of the reaction term, see Proposition <ref>.
The diffusive coefficient is set to ϵ = 1e-9. We consider two different families of mesh. The first one is made by the usual Voronoi polygons and the second one is composed of distorted squares.
We select two different values for the reaction term: σ = 1 and σ = 0.
Figure <ref> shows that there is no significant difference between the cases σ=1 and σ =0. As already mentioned, for this latter case Proposition <ref> does not apply, and no satisfactory theoretical analysis is available, yet. However, the numerical outcomes seems to suggest that it could be possible to drop the reaction term even if the advection term is not globally linear. We note also that we achieve a good convergence also in the case that the mesh is composed of unstructured quadrilaterals.
tocsection
plain
§.§ CIP stabilizing form
Io e Carlo siamo rimasti d'accordo per smembrare questa sezione e ridefinirla nella sezione sulla discretizzazione. Dovremmo quindi rimuoverla, al moment l'ho copia incollata nei vari pezzi nelle session successive We start observing that the bilinear forms
a(·,·) ,
(·,·)
and
c(·,·)
can be decomposed into local contributions
a(u, v) ∑_E ∈Ω_h a^E(u, v) ,
(u, v) ∑_E ∈Ω_h(u, v) ,
c(u, v) ∑_E ∈Ω_h c^E(u, v) .
It is also possible to consider a decomposition of the bilinear form 𝒩(·,·)
𝒩(u, v) ∑𝒩^E(u, v) ,
where the sum is taken over all the elements E ∈Ω_h such that ∂ E∩Γ has positive measure.
Following <cit.>, we introduce the local CIP-stabilization form
J̃_h^E H^q(E) × H^q(E)→, with q > 3/2,
defined as
J̃_h^E(u, v)
12∑_e ⊂∂ E∫_e γ h_e^2 [∇ u] · [∇ v] ds
=
12∑_e ⊂∂ E∫_e γ h_e^2 [∇ u ·^e] [∇ v ·^e] ds ,
where [∇ u] denotes the jump of ∇ u across e, γ is a parameter that on each edge is defined as
γ (e)·^e _L^∞(e) ,
and ^e is one of the two outward normal vectors to e.
If e is a boundary edge we set [∇ u ] = 0. Since we are in the setting _[L^∞(Ω)]^2 = 1
We define the local bilinear form
(·,·) for sufficiently smooth functions
as
(u, v)
ϵ a^E(u, v)
+ (u, v)
+ σ c^E(u, v)
+𝒩^E (u,v)
+ J̃_h^E(u, v)
for all u, v ∈ V.
The global bilinear form
(·, ·)
is obtained by summing the contributions from all the polygons
(u,v)
∑_E ∈Ω_h (u, v) .
Io a questo punto rimuoverei questa parte e nel momento di fare l'analisi dell'errore scriverei che se u è regolare allora J(u,v) è zero... Then, the problem we consider is the following:
{ find u ∈ V s.t.
(u, v) = ℱ̃(v) for all v ∈ V..
If
u ∈ H^2(Ω)
we have that ∇ u is well defined for every edge e.
Than, for every polygon E and for all v ∈ V
J̃_h^E(u,v) = 0 . ] |
http://arxiv.org/abs/2307.04691v1 | 20230710164441 | Metastable cosmic strings | [
"Wilfried Buchmuller",
"Valerie Domcke",
"Kai Schmitz"
] | hep-ph | [
"hep-ph",
"astro-ph.CO",
"gr-qc"
] |
0.4
CERN-TH-2023-118
MS-TP-23-37
0.4
July 2023
2.5cm
Metastable cosmic strings
1cm
Wilfried Buchmüller^a, Valerie Domcke^b, Kai Schmitz^c
^a Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany
^b Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
^c Institute for Theoretical Physics, University of Münster, 48149 Münster, Germany
2cm
Many symmetry breaking patterns in grand unified theories (GUTs) give rise to cosmic strings that eventually decay when pairs of GUT monopoles spontaneously nucleate along the string cores. These strings are known as metastable cosmic strings and have intriguing implications for particle physics and cosmology. In this article, we discuss the current status of metastable cosmic strings, with a focus on possible GUT embeddings and connections to inflation, neutrinos, and gravitational waves (GWs). The GW signal emitted by a network of metastable cosmic strings in the early universe differs, in particular, from the signal emitted by topologically stable strings by a suppression at low frequencies. Therefore, if the underlying symmetry breaking scale is close to the GUT scale, the resulting GW spectrum can be accessible at current ground-based interferometers as well as at future space-based interferometers, such as LISA, and at the same time account for the signal in the most recent pulsar timing data sets. Metastable cosmic strings thus nourish the hope that future GW observations might shed light on fundamental physics close to the GUT scale.
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§ INTRODUCTION
The formation of topological defects is a generic feature of cosmological phase transitions <cit.>.
Such defects are tied to spontaneous symmetry breaking in extensions of the Standard Model (SM), in particular in grand
unified theories (GUTs). They include Nielsen–Olesen strings <cit.>, 't Hooft–Polyakov monopoles <cit.>, unstable “dumbbells” or “X-strings” connecting a monopole–antimonopole pair <cit.>, and other composite defects <cit.>.
Monopoles would overclose the universe and must therefore be avoided or diluted by inflation. Domain walls will reach a scaling regime, but will still lead to an overclosure problem. On the contrary, cosmic strings evolve towards a scaling regime where their fraction of the total energy density remains constant. Together with characteristic signatures in the cosmic microwave background and in gravitational lensing, the stochastic gravitational-wave background (SGWB) from cosmic strings is a potentially very interesting messenger from the early universe (for reviews and references, see, e.g., Refs. <cit.>).
For a large class of supersymmetric GUTs with symmetry breaking chains avoiding the monopole problem, cosmic-string formation is unavoidable <cit.>. Making use of supersymmetric hybrid inflation <cit.>, the string scale is close to the GUT scale, a prominent example being the breaking of B-L, the difference between baryon and lepton number <cit.>. String scales below the GUT scale are also possible and may be related to intermediate-mass right-handed neutrinos, which could render the SGWB a probe of thermal leptogenesis <cit.>.
Pulsar timing array (PTA) observations <cit.> can probe the string tension of stable cosmic strings down to Gμ≲ 10^-10 <cit.>, where G denotes Newton's constant and μ is the energy per unit length of the string. These observations have now entered a new phase with evidence for a common-spectrum process at nanohertz frequencies first reported in Refs. <cit.>, followed by evidence for Hellings–Downs angular correlation, the smoking-gun signal of a SGWB, reported by PTA collaborations across the world in Refs. <cit.>.
Beyond the astrophysical interpretation in terms of inspiraling supermassive black-hole binaries <cit.>, possible cosmological interpretations include stable and metastable cosmic strings <cit.> (see, e.g., Refs. <cit.> for an overview of possible cosmological signals). However, the originally favoured GUT-scale strings with a tension in the range Gμ≃ 10^-(8⋯6) are firmly excluded by these results, as they would lead to too large an SGWB signal in the PTA band.
However, in theories where strings couple to monopoles, strings can decay by quantum tunneling into string segments connecting monopole–antimonopole pairs <cit.>. In the semiclassical approximation, the decay rate per string unit length is given by <cit.>
Γ_d = μ/2 πexp( - πκ) with κ = m_M^2/μ ,
where m_M is the monopole mass. Given the exponential dependence of the decay rate on the parameter κ, and considering monopole masses larger than the string scale, metastable strings have generally been assumed to be effectively stable (see, e.g., Refs. <cit.>). However, a particularly interesting phenomenology is obtained for metastable cosmic strings with √(κ)∼ 8. Such values can indeed be obtained for SO(10) models with B-L strings <cit.>. In this case, the cosmic-string network survives for about an hour (redshift z ∼ 10^7) until monopole production becomes efficient, which implies that at high frequencies the resulting GW spectrum resembles that of stable cosmic strings, whereas at lower frequencies, corresponding to GWs sourced at later times, the spectrum is strongly suppressed <cit.>. As a result, metastable cosmic strings can not only provide a good fit to the PTA signal for GUT-scale string tensions <cit.>, but moreover, they can also easily evade any bounds at PTA scales while still yielding a strong signal at higher frequencies <cit.>, i.e., in the frequency bands relevant for LISA <cit.> and ground-based interferometers <cit.>.
In this article, we will discuss the current status of metastable cosmic strings. Section <ref> deals with a minimal but representative example model: the breaking of SU(2)_R×U(1)_B-L down to U(1)_Y by an SU(2)_R Higgs triplet and two SU(2)_R Higgs doublets with quantum numbers suitable for an embedding in SO(10). The computation of the SGWB signal is presented in Section <ref>, with an emphasis on the theoretical prediction for the spectral tilt of the GW spectrum. Some aspects of stable and quasi-stable strings are reviewed in Section <ref>, and the role of inflation is described in Section <ref>. We conclude in Section <ref>.
§ METASTABLE STRINGS
Metastable strings are a characteristic prediction of GUTs that lead, via several steps of spontaneous symmetry breaking, to the SM gauge group G_SM = SU(3)_C×SU(2)_L×U(1)_Y. Strings with tensions above the electroweak scale result from the spontaneous breaking of a U(1) group that commutes with G_SM. Similarly, monopoles arise once a non-Abelian gauge group is broken to a subgroup containing a U(1) factor. Then, if the U(1) symmetry involved in the production of monopoles partially overlaps or coincides with the U(1) symmetry responsible for string formation, the strings become metastable, i.e., pairs of monopoles and antimonopoles spontaneously nucleate along the strings by quantum tunneling.
SM extensions giving rise to strings must feature a gauge group of at least rank 5. Starting from an exceptional Lie group at high energies, we can, e.g., consider the following symmetry breaking chain,
G_SM⊂SU(5)×U(1)_X⊂SO(10) ⊂E(6) ⊂… ,
where SU(5) refers to the Georgi–Glashow
SU(5) GUT group or to the flipped SU(5)
model. Another possibility is to consider a sequence
featuring an extended electroweak sector,
G_SM
⊂SU(3)_C×SU(2)_L×SU(2)_R×U(1)_B-L⊂G_PS⊂SO(10) ⊂E(6) ⊂… ,
where G_PS = SU(4)×SU(2)_L×SU(2)_R denotes the Pati–Salam group.
If a symmetry group G is broken to a subgroup
H, the quotient ℳ = G/H
corresponds to the manifold of degenerate vacuum states.
The types of defects that may be formed in the symmetry breaking are
governed by the topology of ℳ, which is encoded in the
homotopy groups π_n(ℳ). Topologically stable
strings can form if the first homotopy group is nontrivial,
π_1(ℳ) ≠ I, i.e., there are loops in ℳ that cannot be
contracted to a point. Similarly, topologically stable magnetic monopoles can arise if the
second homotopy group is nontrivial, π_2(ℳ) ≠ I, so
that there exist non-contractable two-dimensional surfaces in ℳ.
We shall be particularly interested in two-step symmetry breakings
G→H→K, where the
homotopy group G/K is trivial, but the homotopy groups
of the individual steps, G/H and
H/K, are nontrivial. In this case, metastable defects can form.
A simple example is the breaking of SO(10) to the Standard
Model group via SU(5). The result crucially depends on the
chosen Higgs representation <cit.>. The breaking chain
SO(10) 45→SU(5) ×U(1) 45⊕126→G_SM×ℤ_2
yields stable monopoles and, in the second step, also stable strings.
On the contrary, for the closely related symmetry
breaking with a 16-plet,
SO(10) 45→SU(5) ×U(1) 45⊕16→G_SM ,
the homotopy group of ℳ = SO(10)/G_SM is trivial, π_1(ℳ) = I, and there
are no topologically stable strings. However,
cosmologically interesting metastable strings can now form.
Metastable strings can break apart into segments in consequence of
quantum tunneling events leading to the spontaneous nucleation of
monopole–antimonopole pairs. Eventually, string decay leads to a
population of short string segments where each segment has a monopole
on one end and an antimonopole on the other. In the example in Eq. (<ref>),
monopoles are formed both in the first and second breaking step,
where the latter also determines the string energy scale.
Besides, there are also other composite topological defects, which can be created in other symmetry breaking chains. One example are ℤ_2-strings, also known as “necklaces”, which correspond to one-dimensional string–monopole–string configurations, i.e., configurations where two strings are attached to each monopole <cit.>. More details and references on composite topological defects can be found in Ref. <cit.>.
Realistic GUTs require large Higgs representations in order to break the GUT gauge group down to the SM, which complicates their analysis. On top, nonsupersymmetric models are sensitive to large radiative corrections and hence suffer from a severe naturalness problem. This observation triggered the investigation of even more complicated models in the literature: supersymmetric GUTs with even more complicated Higgs sectors. In view of this situation, it is important not to loose sight of the fact that conventional spontaneous symmetry breaking is not the only way in which a fundamental GUT gauge group can be reduced to the SM gauge group. Higher-dimensional theories such as orbifold GUTs or string theory represent intriguing alternatives that deserve consideration (a review and references can, e.g., be found in Ref. <cit.>). In these constructions, the fundamental GUT gauge group is first partially broken in a geometric way, namely, by the the compactification of extra dimensions, and only the remnant subgroup remaining after this first step is further reduced to the SM group via conventional spontaneous symmetry breaking. For these reasons, we will restrict ourselves to the simplest possible case leading to metastable strings in the following, the first embedding in Eq. (<ref>), which may mark the end of a long symmetry breaking chain, which we, however, do not specify in detail,
G_SM⊂SU(3)_C×SU(2)_L×U(2) ,
U(2) = SU(2)_R×U(1)_B-L /ℤ_2 .
In order to break this group down to G_SM, we consider a Higgs triplet U ∼ (3,0) of SU(2)_R alongside a pair of Higgs doublets of SU(2)_R, S ∼ (2,q) and S_c ∼ (2̅,-q), that carry charges ± q under U(1)_B-L.
The breaking of SU(2)_R leads to monopoles while the breaking of U(1)_B-L implies strings, yielding the necessary ingredients for metastable strings.
Also, note that we divide out a ℤ_2 factor in Eq. (<ref>), which is necessary to avoid double counting of the center of SU(2)_R, which consists of the identity element and its negative, {I,-I}, and which is also contained in U(1)_B-L. For earlier discussions of defects in U(2) models with triplet and doublet Higgs fields but without supersymmetry, see Refs. <cit.>.
§.§ Strings from supersymmetric B-L breaking
The prospects to explain the recent PTA signal in terms of a SGWB from metastable cosmic strings motivate us to consider large string tensions. We are specifically interested in symmetry breaking scale far above the electroweak scale, at least of the order of v_s ∼ 10^13 GeV. It is reasonable to expect unbroken supersymmetry at such high energies, which is why we will focus on supersymmetric models of symmetry breaking in the rest of this paper, following the analysis presented in Ref. <cit.>.
Our starting point is a supersymmetric Abelian Higgs model with two chiral superfields S and S_c and a gauge singlet ϕ that gives rise to spontaneous B-L breaking. The fields S and S_c carry charge q and -q under U(1)_B-L, respectively, and the Kähler potential and superpotential of the model (we use the same conventions as in Ref. <cit.>) are given by
K = S^† e^2gqV S + S_c^† e^-2gqVS_c + ϕ^†ϕ ,
P = 1/4 W W + λϕ(v_s^2 - SS_c) .
Here, V is a vector superfield, W is the supersymmetric field strength, and v_s is the scale of spontaneous symmetry breaking, which we can choose to be real and positive.
From the auxiliary fields of the vector and chiral superfields, we can derive the scalar potential,
𝒱 = 1/2 D^2 + |F_S|^2 + |F_S_c|^2 + |F_ϕ|^2 ,
where the F and D terms follow from solving the associated equations of motion,
D = - gq (|S|^2 - |S_c|^2) ,
F_S^* = λϕ S_c ,
F_S_c^* = λϕ S ,
F_ϕ^* = -λ(v_s^2 - SS_c) .
The scalar potential and the kinetic terms for the scalar and vector fields constitute the bosonic part of the Lagrangian,
ℒ_b = -1/4 F_μνF^μν - (D_μ S)^*(D^μ S)
- (D_μ S_c)^*(D^μ S_c) - ∂_μϕ^* ∂^μϕ -
𝒱 ,
with covariant derivatives D_μ = (∂_μ + igq A_μ)S and D_μ S_c = (∂_μ - igq A_μ)S_c, and where A_μ is the vector component in the vector multiplet V, F_μν is the corresponding field strength, and where chiral superfields and their scalar components are denoted by the same symbols.
The vacuum manifold of the model is described by a D-flat direction, |S|^2 = |S_c|^2, which represents a flat moduli space of vacuum states with unbroken supersymmetry and spontaneously broken U(1)_B-L symmetry,
S = v_s e^iα ,
S_c = S^* , ϕ = 0 .
The particle excitations around the true vacuum are best described if we expand the fields S and S_c around their vacuum expectation values (VEVs),
S = v_s e^iα + S' , S_c = v_s e^-iα + S'_c .
In this field basis, we then find that the Goldstone multiplet (S'-S'_c)/√(2) is “eaten” by the massless vector multiplet V, which results in a massive vector multiplet with mass m_V = √(2)gqv_s. Similarly, the orthogonal linear combination (S'+S'_c)/√(2) and the singlet field ϕ fuse in a massive chiral multiplet with mass m_S = λ v_s.
As in the nonsupersymmetric case, the vacuum manifold ℳ is the circle S^1, which has a nontrivial first homotopy group, π_1(ℳ) = ℤ. The model thus admits exited states in the form of topologically stable strings. On the supersymmetric moduli space, i.e., along the D-flat direction, these strings are described by the Nielsen–Olesen string solutions <cit.>. Static strings along the z axis and with winding number n correspond to field configurations of the form
S = v_s f(ρ) e^niφ = S_c^* ,
A_0 = 0 , A_i = -n/gρ h(ρ) ∂_iφ ,
where we work in cylindrical coordinates (ρ,φ,z) and with boundary conditions
f(0) = h(0) = 0 , f(∞) = h(∞) = 1 .
Strings described by these field configurations exhibit a total magnetic flux along the string of 2n π/g with n ∈ℤ∖{0} and a string tension (i.e., energy per unit length) of
μ = 2π v_s^2 B(β) = π m_V^2/(gq)^2 B(β) with β = m_S^2/m_V^2 = λ^2/2g^2 .
Here, the parameter β measures the ratio of the Higgs and vector boson masses; and B is a slowly varying function of β, normalized such that B→1 in the Bogomol'nyi limit β→ 1 <cit.>. If β < 1, the strings will have larger Higgs-field cores than gauge-field cores, m_S^-1 > m_V^-1, which gives rise to type-I strings, in analogy to similar condensed-matter systems. Type-I strings are stable and attracted towards each other.
The model defined in Eq. (<ref>) is intriguing, as it features a singlet field ϕ that can be identified as the inflaton in supersymmetric hybrid inflation <cit.>. It is, moreover, straightforward to extend it by adding supersymmetry breaking terms and by coupling it to a set of chiral right-handed-neutrino superfields. Extensions of the model along these lines can account for leptogenesis, dark matter, and a stage of cosmic inflation in accord with recent bounds on the primordial scalar spectral index in observations of the cosmic microwave background <cit.>. Scenarios of this type also give rise to stable cosmic strings with tension Gμ∼ 10^-7, where the specific value of the string tension is dictated by the other phenomenological aspects of the model, in particular the energy scale of inflation. For stable strings, such large string tensions have, however, been known to be in conflict with PTA measurements for many years <cit.>, which renders such scenarios unviable.
§.§ Monopoles from supersymmetric SU(2) breaking
Next, we turn to topologically stable 't Hooft–Polyakov monopoles <cit.> produced by the spontaneous breaking of SU(2) to U(1). To implement this breaking, we consider an SU(2) triplet U^a (a = 1,..,3) and work with the following Kähler potential and superpotential,
K = U^† e^2gV U ,
P = 1/8 tr[W W] .
Here, V = V^a T^a denotes the SU(2) vector superfield and the (T^a)_bc = -iϵ_abc are the SU(2) generators in the adjoint representation. The bosonic Lagrangian of the theory is given by
ℒ_b = -1/4 F^a_μνF^aμν - (D_μ U^a)^*(D^μ U^a)
- ig ϵ_abc D^a U^b * U^c + 1/2 D^aD^a +
F^a *_U F^a_U ,
with gauge-covariant derivative (D_μ U)^a = ∂_μ U^a - g ϵ_abc A^b_μ U^c and non-Abelian field strength tensor F^a_μν = ∂_μ A^a_ν - ∂_ν A^a_μ -g ϵ_abc A^b_μ A^c_ν.
In passing, we mention that the Lagrangian in Eq. (<ref>) corresponds to the bosonic Lagrangian of SU(2) Super-Yang–Mills theory with 𝒩=2 supersymmetry. Supersymmetric UV completions of the SM of this type can occur in certain orbifold compactifications. Starting with a supersymmetric Pati–Salam or SO(10) theory in five or six dimensions, orbifold constructions can lead to an 𝒩 = 2 sector with gauge group SU(2)_R in four dimensions. The total gauge group containing SU(2)_R as a subgroup is then spontaneously broken down to the SM in subsequent symmetry breaking steps.
The classical theory defined by the potentials in Eq. (<ref>) features again a moduli space spanned by the flat direction U^a = u/√(2) δ_a3 modulo gauge transformations. Thanks to the large symmetry of the theory, this flat direction is preserved at the quantum level as well as when nonperturbative corrections are taken into account. The flat direction is therefore also present in the full theory, where it interpolates between two phases: a confinement phase with monopole condensation at small values of u and a perturbative Higgs phase at large values of u <cit.>. In the following, we will be concerned with the Higgs phase of the model, which corresponds to field values u much larger than the confinement scale Λ.
As in the previous case of the Abelian Higgs model, the vacuum degeneracy along the flat moduli space can be lifted by coupling the superfield U to a new gauge singlet superfield ϕ' via a superpotential term of the form
P = 1/8 tr[W W] + λ'/2ϕ' (v_u^2/2 - U^T U) ,
where v_u is a mass scale that we can choose to be real and positive and which sets the scale of spontaneous SU(2) breaking. The new superpotential in Eq. (<ref>) now only exhibits 𝒩=1 supersymmetry, as the new term breaks 𝒩=2 supersymmetry. Next, we derive the equations of motions for the auxiliary fields contained in V, U, and ϕ',
D^a = ig ϵ_abc U^b* U^c ,
F_U^a* = λ' ϕ' U^a ,
F_ϕ'^* = -λ'/2(v_u^2/2 - U^T U) .
Instead of a supersymmetric moduli space, we now find a supersymmetric vacuum at
U^a = v_u/√(2) δ_a3 , ϕ' = 0 ,
where the value of U^TU is now fixed and U^a is determined up to an SU(2) rotation.
Meanwhile, one rotation of the fields U^a still represents an unbroken symmetry, despite the fact that U^TU has nonvanishing expectation value, which means that a U(1) subgroup of SU(2) survives in the new ground state after symmetry breaking. The particle spectrum of the theory now consists of: a massless vector multiplet, V^3; a charged vector multiplet with mass m_V = gv_u that has “eaten” the Goldstone multiplets U^1,2; and a massive chiral multiplet with mass m_U = λ' v_u/√(2) composed of the multiplets U'^3=U^3-v_u and ϕ'.
After symmetry breaking, the theory contains excited states in the form of topologically stable monopoles. To see this, note that the vacuum manifold ℳ is a 2-sphere S^2 spanned by the SU(2) rotations acting on the fields U^a in the ground state, just like in the nonsupersymmetric case. The vacuum manifold thus has nontrivial homotopy group π_2(ℳ) = ℤ, which indicates the existence of monopole solutions. The simplest monopole configuration is the “hedgehog” solution <cit.>, corresponding to radial field profiles of the form
U^a = v_u/√(2) f(r) x^a/r , A^a_0 = 0 ,
A^a_i = h(r) ϵ_aij x^j/gr^2 .
Now, r denotes the radial coordinate in spherical coordinates rather than cylindrical coordinates, r=(x^i x^i)^1/2, and the functions f and h are subject to the boundary conditions
f(0) = h(0) = 0 , f(∞) = h(∞) = 1 .
From Eq. (<ref>), we read off that the scalar field profile points into the radial direction, U^a ∝ϕ̂^̂â≡ x^a/r. The same is therefore true for the unbroken symmetry generator. Similarly, one obtains the following gauge-invariant magnetic field strength at large distances,
B_i = -1/2 ϕ̂^a ϵ_ijk F^a_jk = x^i/gr^3 ,
which allows us to identity the magnetic charge of the monopole, 4π/g.
In general, the monopole mass is 2n π/g with n ∈ℕ, i.e., the 't Hooft–Polyakov monopole
corresponds to n=2.
The mass of the monopole is subject to the Bogomol'nyi bound <cit.>,
m_M ≥4π m_V/g^2 = 4π v_u/g ,
where the equal sign holds in the Prasad–Sommerfield limit λ'/g → 0 <cit.>. For nonzero values of the ratio of coupling constants, λ'/g, there exist no analytical expressions for the functions f and h in Eq. (<ref>). One therefore has to resort to numerical solutions of the field equations, which show that m_M is a monotonically increasing function of the Higgs mass. In the limit λ'/g →∞, one finds in particular an upper bound m^max_M ≃ 4π m_V/g^2 × 1.79 <cit.>.
§.§ Monopoles and metastable strings
Let us now combine the constructions in Secs. <ref> and <ref> and discuss metastable B-L strings decaying into short string segments with monopoles and antimonopoles on their ends. Defects of this type form if we embed the electroweak part of the SM gauge group, G_ EW = SU(2)_L×U(1)_Y, in the group G_221 = SU(2)_L×SU(2)_R ×U(1)_B-L/ℤ_2 and spontaneously break G_221 down to G_ EW in two steps. Hypercharge Y then follows from the linear combination of the neutral SU(2)_R and U(1)_B-L generators, Y = T^3_R + (B-L)/2. The two symmetry-breaking steps in this model break SU(2)_R×U(1)_B-L/ℤ_2 to U(1)_Y and end on the vacuum manifold ℳ = U(2)/U(1) = S^3. This manifold contains the union of the vacuum manifolds of stable strings and monopoles, S^1 ∪ S^2, and has trivial homotopy groups π_1(ℳ) and π_2(ℳ). The model thus neither features topologically stable monopoles nor strings. Instead, we will see that it can give rise to metastable strings or unstable dumbbells.
In order to break U(2) = SU(2)_R×U(1)_B-L/ℤ_2 down to U(1)_Y, we shall work with similar Higgs representations as in Secs. <ref> and <ref>. Specifically, we introduce a B-L-neutral SU(2)_R triplet U as well as two oppositely B-L-charged SU(2)_R doublets S, S_c,
U ∼(3,0) , S ∼(2,q) , S_c ∼(2̅,-q) ,
under SU(2)_R×U(1)_B-L. Defects in nonsupersymmetric U(2) models with triplet and doublet Higgs representations were previously discussed, e.g., in Refs. <cit.>. In the following, we are, however, interested in the supersymmetric version of the model, which we construct by choosing the Kähler potential K and superpotential P as a combination of Eqs. (<ref>), (<ref>), and (<ref>), supplemented by an additional mass term in P,
[Our model is different from standard left-right-symmetric models, as we work with neutral triplets. In left-right-symmetric models, the triplets typically carry U(1) charge and thus occur in pairs <cit.>.]
K = U^† e^2gV U + S^† e^2(gṼ + g'qV')S + S_c^† e^-2(gṼ+g'qV')S_c + ϕ^†ϕ +
ϕ'^†ϕ' ,
P = 1/8 tr[W W] + 1/4 W'W' + 2 h S^T_c Ũ S
+ λ'/2ϕ' (v_u^2/2 - U^T U) + λϕ(v_s^2 - S^T_cS) - h v_u S^T_c S .
Here, U = (U^1,..,U^3)^T is the triplet field written as a vector in the triplet representation; Ũ = U^a τ^a/2 is the triplet field written as a matrix in the doublet representation; V = V^a T^a is the SU(2)_R vector field in the triplet representation; Ṽ = V^a τ^a/2 ≡ T_R is the SU(2)_R vector field in the doublet representation; V' is the U(1)_B-L vector field; and W and W' are the supersymmetric SU(2)_R and U(1)_B-L field strengths.
The covariant derivative in the bosonic sector, induced by the Kähler potential, is given by D_μ S = ∂_μ S + i(gṼ + g'qV')S.
The terms involving the Yukawa coupling h are introduced for the following reason: First, the trilinear coupling, coupling the fundamental triplet U^a to the composite triplet S^T_cτ^a S ensures that a U(1) subgroup survives after symmetry breaking. Without this term, the initial U(2) group would in general be broken completely, which in our case would mean that no hypercharge gauge group U(1)_Y would remain in the electroweak sector. Second, the mass term for the pair of doublet fields, i.e., the last term in P in Eq. (<ref>), ensures that U(2) symmetry breaking results in a supersymmetric vacuum with ⟨ P⟩ = 0. This serves the purpose to separate the energy scales of SU(2)_R and U(1)_B-L breaking from the energy scale of supersymmetry breaking. Without this mass term, we would generically expect a contribution to the gravitino mass from the U(2) sector of the order of ⟨ P⟩/M^2_P∼ hv_uv_s^2/M^2_P. However, if ⟨ P⟩ = 0 after U(2) symmetry breaking, we retain the possibility that a separate supersymmetry-breaking sector results in a hierarchically smaller gravitino mass.
As in Secs. <ref> and <ref>, the model exhibits again D-flat directions that are lifted by the coupling to the singlets ϕ and ϕ'. The supersymmetric true vacuum then corresponds to
U^a = v_u/√(2) δ_a3 , S = S_c = v_s [ 1; 0 ] , ϕ' = √(2)hv_s^2/λ' v_u , ϕ = 0 .
The fundamental triplet U^a and the composite triplet S^T_cτ^aS are parallel in this vacuum configuration, as desired, thanks to the Yukawa coupling h ≠ 0 in Eq. (<ref>). As mentioned above, this is necessary to keep an unbroken U(1) symmetry in the vacuum. Without the Yukawa coupling h, the relative orientation of U^a and S^T_cτ^aS would not be fixed.
Next, let us discuss the mass spectrum of the model. In order to identify the mass eigenstates, we must shift the chiral multiplets around their vacuum expectation values,
U^3 = v_u/√(2) + U^3' , S = [ v_s + S^0'; S^- ] , S_c = [ v_s + S^0'_c; S^+ ] , ϕ' = √(2)hv_s^2/λ'v_u + ϕ̂ .
Then, by inspecting the terms linear in the vector fields V, Ṽ, and V' in the Kähler potential in Eq. (<ref>), we can identify the Goldstone multiplets,
Π^∓ = 1/√(v_u^2 + v_s^2)(v_u U^∓ + v_s S^∓) , Π^0 = 1/√(2)(S^0' - S^0'_c) ,
where U^± = (U^1 ∓ i U^2)/√(2), which are respectively “eaten” by the vector multiplets
V^± = 1/√(2)(V^1 ∓ iV^2) ,
V_X = (cosΘ V^3 + sinΘ V') ,
with tanΘ = 2g'q/g. The vector multiplet orthogonal to these two fields,
V_Y = -sinΘ V^3 + cosΘ V' ,
remains massless, while the vector multiplets V^± and V_X acquire masses
m^2_V = g^2 (v_u^2 + v_s^2) ,
m^2_X = g^2/2cos^2Θ v_s^2 .
In addition to the Goldstone multiplets Π^± and Π^0 and vector multiplets V^±, V_X, and V_Y, we are left with six chiral multiplets, Σ^±, Σ^0, U^3', ϕ, and ϕ̂. The mass matrix of these fields follows from the quadratic part of the superpotential,
P_m = - 2√(2)h(v_u^2+v_s^2/v_u) Σ^-Σ^+
- λ' v_u/√(2)ϕ̂ U^3' - √(2)v_s(λϕ - h U^3') Σ^0
- h v_s^2/v_u U^3'U^3' ,
where the linear combinations
Σ^± = 1/√(v_u^2 + v_s^2)(-v_s U^± + v_u S^±) , Σ^0 = 1/√(2)(S^0' + S^0'_c) ,
are orthogonal to the Goldstone multiples Π^± and Π^0, respectively. We emphasize again the role of the Yukawa coupling: in absence of the h-dependent terms in Eq. (<ref>), the superpotential P_m simply corresponds to the mass terms discussed in Secs. <ref> and <ref>, i.e., the mass terms for SU(2)_R
and U(1)_B-L breaking in isolation.
For suitably chosen parameter values, the model discussed in this section allows us to break SU(2)_R ×U(1)_B-L down to U(1)_Y in two subsequent steps, each of which corresponding to a cosmological phase transition in the early universe. In the first step, a nonvanishing triplet expectation value ⟨ U^a⟩ breaks SU(2)_R to U(1)_R; and then in a second step, nonvanishing doublet expectation values ⟨ S⟩ and ⟨ S_c⟩ break U(1)_R×U(1)_B-L down to U(1)_Y. Note that analogous symmetries are present in the electroweak sector, where SU(2)_L contains the subgroup U(1)_L and where U(1)_L ×U(1)_Y contain in turn the electromagnetic subgroup U(1)_Q; even though electroweak symmetry breaking does not involve any Higgs triplets.
Up to now, we treated the U(1)_B-L gauge coupling times the charge of the doublet fields, qg', as a free parameter. This is no longer possible as soon as one begins to consider embeddings of our model in either of the symmetry breaking chains in Eqs. (<ref>) and (<ref>).
In Pati–Salam or SO(10) GUT extensions of the SM, the Higgs doublets S and S_c are embedded into Pati–Salam (4,1,2) ∼χ_L and (4̅,1,2̅) ∼χ_R^c representations, or into 16, 16 representations of SO(10), respectively [see Eq. (<ref>)]. Here, the Higgs doublets S,
S_c are identified as the “lepton doublets” in χ_L, χ^c_R.
For the Pati–Salam embedding, the
covariant derivative in the bosonic sector reads
D_μχ_L = ∂_μχ_L + i(g T^a_RV^a + g'1/2(B-L) V')χ_L. The normalization
condition g^' 2/4 tr[(B-L)^2] = g^2tr[(T^3_R)^2]
then implies g'√(2/3) = g, which corresponds to the mixing angle
tanΘ = -√(3/2). The covariant derivative with the two U(1) factors
U(1)_R and U(1)_B-L then reads
D_μχ_L = ∂_μχ_L + ig (T^3_R V^3+ √(3/2) 1/2(B-L) V') χ_L.
Note that the field S carries charge ± 1/2 with respect to the generators T^3_R and 1/2(B - L),
respectively.
Embedding the doublets S, S_c in 16-, 16-plets
Φ, Φ^c of SO(10), as in Eq. (<ref>),
implies that heavy Majorana neutrino masses must
be generated by the nonrenormalizable operator
ℒ_n = 1/M_* h_ij S^T L^c_i S^T L^c_j ⊂1/M_* h_ij Φ^c ψ_i Φ^cψ_j .
Here, the fields L^c_i = (n^c_i,e^c_i)^T,
i=1,..,3, denote the SU(2)_R doublets of right-handed neutral and charged
leptons that are contained in the SO(10) 16 representations
ψ_i of matter,
and h_ij are Yukawa couplings. Alternatively, one can follow
Eq. (<ref>) and break SO(10) with 126-,
126-plets Φ̃, Φ̃^c containing
the SU(5) singlets S̃, S̃_̃c̃. Heavy
neutrino masses are now generated by the renormalizable couplings
ℒ_n = h_ijS̃ L^c_i L^c_j ⊂
h_ijΦ̃ψ_i ψ_j ,
as assumed, e.g., in Ref. <cit.>.
The VEVs of S̃, S̃_̃c̃ leave a ℤ_2 discrete
symmetry unbroken, which leads to topologically stable strings.
The cosmological realization of the two symmetry-breaking stages in our model (in the form of cosmological phase transitions) leads to the formation of defects: monopoles in the first step and strings in the second step. For a monopole–string–antimonopole
configuration, the magnetic fluxes of the string and the (anti)monopole have to match[Note that, in the U(2) model, this is only possible if g'q and g are integer multiples of each other. This is guaranteed by the Pati–Salam embedding.] (see, e.g., Ref. <cit.>). The string solution with lowest energy has winding number n=1.
As the symmetry breaking field S has charge 1/2, it carries magnetic
flux 4π/g. This can be matched by a n=2 monopole with mass m_M ∼
4π v_u/g. Together with the string tension μ≃ 2π v_s^2, we then
obtain for the parameter κ, which controls the metastability of
cosmic strings,
κ = m_M^2/μ∼8π/g^2v_u^2/v_s^2 .
In supersymmetric theories, one expects g^2 ∼ 1/2 at the unification
scale. This implies √(κ)∼ 7 v_u/v_s.
As we shall see in the following section, metastable strings can be
relevant for GWs in the PTA band for √(κ)≳ 8, which
corresponds to v_u ≳ v_s.
Note that the model predicts confined as well as unconfined magnetic flux
for the monopole, which is estimated as 4π/g sin^2Θ (for a discussion,
see Ref. <cit.>).
An important open question concerns the range of validity of the
relation in Eq. (<ref>). The size of the magnetic cores of the monopole and the string are given by
m_V^-1 and m_X^-1, respectively. The string decay rate will also
be affected by
the false vacuum cores, whose size is given by
the Higgs masses m_U and m_S for monopole and string, respectively.
Moreover, Eq. (<ref>) uses estimates for the mass and tension of
an isolated monopole
as well as an isolated string, respectively. So far, no calculations have carried out for
spatially extended composite defects.
As v_u approaches zero,
the semiclassical approximation used in the derivation of
Eq. (<ref>) breaks down, and metastable strings turn into
dumbbells that decay immediately.
In the case where SU(2)_R×U(1)_B-L is broken
to U(1)_Y by VEVs of the doublets S and S_c only,
dumbbells or
X-strings form, which are completely analogous to the Z-strings
of the SM <cit.>.
For |tanΘ| = √(3/2),
X-strings are known to be unstable
<cit.>.
Metastable strings are a generic feature of GUTs. The supersymmetric breaking of U(2) = SU(2)_R×U(1)_B-L /ℤ_2 is the simplest example (albeit a representative one) of a much richer structure that occurs in realistic GUTs. It is an intriguing prospect that the metastability of cosmic strings might be tested with gravitational waves, which would provide direct information about the energy scales of GUT symmetry breaking stages.
§ STOCHASTIC GRAVITATIONAL-WAVE BACKGROUND
The SGWB sourced by a metastable cosmic-string network was recently computed in Ref. <cit.> (see also Refs. <cit.>). As for stable cosmic strings, a population of string loops with number density ∘n(ℓ, t') radiates GWs with the power density per frequency <cit.>
P_gw(t', f') = G μ^2 ∑_k = 1^k_maxℓ/f' ∘n(ℓ, t') P_k ,
where f' = 2 k/ℓ indicates the GW frequency, emitted by a loop of length ℓ oscillating in its kth harmonic excitation; t' is the time of GW emission and P_k = Γ/(k^4/3ζ[4/3]) with Γ≃ 50 is the power emitted by a single loop (assuming the emission is dominated by the contribution from cusps). Integrating over t', we obtain the spectral energy density in GWs today normalized by the critical energy density,
Ω_gw(t_0, f) = 16 π (Gμ)^2/3 H_0^2 f∑_k k P_k ∫_0^z_idz'/H(z') (1 + z')^6 ∘n(2 k/f', t(z')) .
Here H(z) is the Hubble parameter, we have switched the time-variable to redshift z, and the argument of the loop number density ensures that we are accounting for all GWs emitted at frequency f' such that after red-shifting, they are observed at frequency f today.
The remaining challenge is to determine the loop number density ∘n(ℓ, t'). In the limit of stable cosmic strings, we adopt the velocity-dependent one-scale (VOS) model <cit.> within the Nambu–Goto framework, in which one-dimensional string loops are formed at a fixed fraction α of the horizon and then shrink due to GW emission, ℓ(t) = α t' - Γ G μ (t - t'). In this case, the loop number density can be determined analytically by solving the corresponding kinetic equation, up to integration constants which can be extracted from simulations <cit.>. For example, in a radiation-dominated background, this yields
∘n^rad_∞(ℓ, t) = B/t^3/2 (ℓ + Γ G μ t)^5/2 Θ(α t - ℓ) ,
where B = 0.18 and α = 0.1 are obtained from a fit to numerical simulations. The subscript ∞ (for κ→∞) refers to stable cosmic strings.
For metastable cosmic strings, the kinetic equations are modified to take into account the decay of string loops to segments through the formation of a monopole–antimonopole pair, as well as the formation of segments from longer segments and super-horizon strings <cit.>. If the monopoles carry no unconfined flux, the segments themselves can have cosmological lifetimes and contribute to the SGWB <cit.>. However, as demonstrated in Ref. <cit.>, the GW spectrum generated by cosmic string loops alone provides a good approximation to the full spectrum in most of the parameter space, even when there is a contribution from segments.
Moreover, the example in Sec. <ref> has unconfined flux. We therefore focus
on the GW spectrum from string loops, in which case the key change compared to stable cosmic strings is an additional decay term in the kinetic equation for the loop number density accounting for the monopole–antimonopole formation on the loops. Matching the number density to Eq. (<ref>) at early times, t ≪ t_s = 1/Γ_d^1/2, then yields for the loop number density of the metastable cosmic string network at t > 1/Γ_d^1/2,
∘n^rad(ℓ, t) = B/t^3/2 (ℓ + Γ G μ t)^5/2 e^- Γ_d [ ℓ (t - t_s) + 12Γ G μ (t - t_s)^2] Θ(α t_s - ℓ - Γμ ( t - t_s)) .
Here, the exponential factor accounts for the decay of the loops at t > t_s through the generation of monopoles, and the Heaviside function ensures that loop formation only occurs at t < t_s. For expressions for the loop number densities involving evolution during the matter-dominated era as well as expressions for the number densities of super-horizon strings and segments, see Ref. <cit.>.
Fig. <ref> shows the GW spectrum obtained by inserting Eq. (<ref>) (and corresponding expressions for the matter-dominated era) into Eq. (<ref>). The dotted black curves show the limit of stable cosmic strings, κ→∞, whereas the colored curves show the prediction for the spectrum for two different values of the ratio of the symmetry breaking scales κ and the string tension μ. Large frequencies correspond to GWs produced at early times, and hence the spectrum produced by stable and metastable strings is identical, featuring a plateau at
Ω_gw^plateau≃128 π/9 B Ω_r (G μ/Γ)^1/2 ,
where Ω_r h^2 = 4.15 · 10^-5 is the density parameter of radiation today.
At lower frequencies, the earlier decay of the metastable cosmic
string loops suppresses the GW signal, leading to a drop in the
spectrum proportional to f^2. This drop sets in at a frequency <cit.>
f_low∼ 3· 10^-9 Hz(50/Γ)^3/4(10^-8/Gμ)^1/2exp(-π(κ/4-16)) .
GWs from metastable strings can be observed if their decay happens
sufficiently late such that their redshifted frequencies are not much below
f_low. On the observational side, a lower cutoff on the
measurable frequencies at PTAs is given by the observation times, i.e.,
f^PTA∼ (10 yr)^-1≃ 3 nHz.
The cutoff f_low depends exponentially on κ, and
from Eq. (<ref>) one reads off that the condition for a potential
discovery of GWs at PTAs, f_low≲ f^PTA,
requires √(κ)≳ 8.
Note that the predicted f^2 spectrum is remarkably different from the f^3 scaling expected for causal GW sources (including, e.g., GWs from domain walls) and is due to the fact that the cosmic-string network acts as a GW source over time scales much larger than a Hubble time.
PTA collaborations across the world have recently reported the observation of a stochastic common-spectrum process<cit.> showing evidence for Hellings–Downs spatial correlations <cit.>, the hallmark signature of a SGWB. While the most plausible source remains supermassive black-hole binaries, the observed tension with common astrophysical population models <cit.> motivates a thorough investigation of a possible cosmological contribution. Upcoming data will improve our understanding of the spectral tilt, the isotropy and the presence of resolvable individual sources in this SGWB, which will all help to distinguish an astrophysical from a cosmological origin.
With these promises and caveats in mind, we now focus in more detail
on the GW signal of metastable cosmic strings in the PTA frequency
band. As shown in Refs. <cit.>, this
could explain the observed GW signal for 10^-11≲ G μ≲
10^-7 and √(κ)≳ 8. For models of hybrid and
tribrid inflation compatible with these values, see,
e.g., Refs. <cit.>. From Eq. (<ref>),
one reads off that √(κ)≃ 8 can be achieved if the two symmetry breaking scales v_u
and v_s are close to each other.
Metastable cosmic strings thus lead to observable effects at PTAs for v_u ≳
v_s. If the astrophysical origin of the currently observed signal
should be confirmed, which corresponds to interpreting the current PTA data as an upper
bound on a cosmological signal, this would shift the interest to the region v_u ≲ v_s (or to smaller values of Gμ),
which remains compatible with such a constraint while still allowing for a large SGWB signal in the LISA and LIGO bands.
[The described connection between PTA observation times
and GUT-scale parameters may appear surprising, but an analogous case
is known from neutrino physics. Neutrino mass differences
√(Δ m^2)∼ 0.05 eV could only be discovered in atmospheric neutrino oscillations
because the oscillation length is L ∼ 10^4 km (earth diameter) <cit.>. Smaller mass differences,
e.g. √(Δ m^2)∼ 0.005 eV, would have remained unobserved.
]
Within the approximations mentioned above, the GW spectrum from metastable cosmic strings only depends on two parameters, the string monopole mass m_M and the string tension μ, or, dropping the logarithmic dependence on the Yukawa couplings, the two symmetry breaking scales v_u and v_s. Interpreting the PTA data as a SGWB, the current data <cit.> indicate an amplitude of 10^-10≲Ω_gw^PTA h^2 ≲ 10^-9 at the PTA peak sensitivity f_PTA = 3 nHz. The spectral index n_t = dlnΩ_gw/dln f is less constrained and varies more significantly across the different data sets, 0 ≲ n_t ≲ 3, with the PPTA data set preferring slightly smaller values, the EPTA 10.5 year data set preferring larger values and the NANOGrav data lying in the middle. Upcoming data and analysis will significantly improve the measurement of the spectral tilt, allowing a distinction between different SGWB sources. Remarkably, within the framework of metastable cosmic strings, the two observables Ω_gw^PTA and n_t allow to determine the two model parameters, v_u and v_s.
This is shown in Fig. <ref>, where we fix the amplitude of the GW spectrum at f = 3 nHz to two distinct reference values, Ω_gw^PTA h^2 = 5 · 10^-10 (left) and Ω_gw^PTA h^2 =10^- 9 (right), and show how a future improved measurement of n_t (x-axis) would determine the GUT-scale symmetry breaking parameters (on the two vertical axes).
[Since the cosmic string signal is not a perfect power law over the frequency range of PTAs, the precise value of the tilt n_t (both in the model prediction and signal reconstruction) depends on the underlying assumptions. Here, for concreteness, we determine n_t by linearly interpolating between the signal predictions at 2 and 4 nHz. Note that n_t is related to the often quoted tilt α of the dimensionless characteristic strain and to the spectral index (-γ) of the timing-residual power spectral density as n_t = 2α + 2 = 5 - γ.]
The limit of stable cosmic strings requires Gμ≃ 4 · 10^-11 (7 · 10^-11) yielding n_t ≃ 0.7 (0.6) to reproduce this SGWB amplitude for Ω_gw^PTA h^2 = 5 · 10^-10 (10^-9).
As the cosmic-string lifetime and hence κ is reduced, the string tension μ needs to be increased to maintain the same SGWB amplitude at 3 nHz. For quasi-stable strings, an increase in Gμ comes with a decrease in n_t, until with a further decrease of κ the f^2 part of the spectrum enters the PTA band and the spectral index starts increasing again. For large string tensions (large v_s), the desired SGWB amplitude can only be achieved by significantly reducing the string lifetime (reducing κ), recovering the asymptotic f^2 scaling. Of course in this case, unless the reheating temperature is very low or a non-standard cosmological history is invoked <cit.> (see, e.g., Ref. <cit.>), the SGWB will exceed the bound Ω_gw≤ 5.8 · 10^-9 (1.7 · 10^-8 for a linear prior) set by the LIGO–Virgo–KAGRA (LVK) collaboration in the 100 Hz range <cit.>. If the current preference for a spectral index larger than n_t ≃ 1 persists, this moreover disfavours the limit of stable strings resulting in a sweet spot with v_s = few × 10^14 GeV and √(κ)≃ 8.3 (see also Ref. <cit.>).
We conclude this section by drawing attention to some theoretical uncertainties and open questions in the calculation of the GW spectrum. Our calculations here are based on the Nambu–Goto action, taking cosmic strings to be infinitely thin, and moreover we focus on the GW emission by cusps on the cosmic string loops. Alternatively, cosmic strings can be modeled using lattice simulations of classical field theory (Abelian Higgs model) and one may consider GW emission from kinks as well as GW bursts. For reviews and more detailed discussions, see Refs. <cit.>. In summary, a robust understanding of the substructure on string loops remains challenging but is crucial to accurately estimate the GW spectrum.
§ STABLE AND QUASI-STABLE STRINGS
The GW spectrum of metastable strings in the PTA frequency band and above
agrees with the one of stable strings for monopole-mass–string-tension ratios
√(κ)≳ 9 <cit.>.
These strings are
usually referred to as quasi-stable.
They can explain the PTA results for Gμ = 10^-(11⋯10), corresponding to a spectral tilt of 0 ≲ n_t ≲ 0.8 <cit.>, where the upper bound is relatively sensitive to the details of the modelling of the cosmic-string network <cit.>.
Many studies have explored possible connections between an
intermediate string scale v_s ∼ 10^13 GeV
and other predictions of
supersymmetric and nonsupersymmetric GUT models. In
nonsupersymmetric SO(10) models, requiring gauge coupling
unification together with an intermediate string scale significantly
restricts the allowed symmetry breaking chains. Moreover, the
unification scale has to be large enough such that the proton decay
rate is smaller than current experimental bounds <cit.>.
The situation is similar in supersymmetric SO(10) models
<cit.>. Since SO(10) GUTs with a large string
scale around 10^13 GeV generically contain heavy Majorana neutrinos,
weakly interacting neutrinos with sub-eV Majorana masses, as
well as leptogenesis are naturally incorporated. The discussion of
SO(10) models can be extended to E_6 models where
new types of monopoles and strings appear <cit.>.
One may also consider extensions of SO(10) models with a
Peccei–Quinn symmetry whose breaking yields an axion
<cit.>. The model predicts two types of monopoles,
related to the GUT scale and an intermediate scale, and in addition
topologically stable strings produced at an intermediate scale below
10^13 GeV. The related SGWB is too weak to be observed by
PTAs or LVK, but can be detected at SKA, LISA and ET/CE. On the other hand, for baryogenesis after
primordial-back-hole evaporation as discussed in Ref. <cit.>, the predicted scale of the B-L strings is too large to be consistent with
PTA data. These are some examples of the discriminating power of SGWB signals in the PTA band.
So far, we have assumed that a fundamental GUT is broken to the
SM by a sequence of symmetry breakings that are all
realized by the Higgs mechanism. One can then expect a plethora of
topological defects produced in cosmological phase
transitions as sources of a SGWB. However, it is far from obvious that
all symmetry breakings are realized by the Higgs mechanism.
Nonsupersymmetric GUTs
suffer from severe fine-tuning problems, and supersymmetric GUTs with
realistic fermion mass matrices require large Higgs representations,
which make them almost intractable. Attractive alternatives are GUTs
in higher dimensions and in string theories. Compactification to four
dimensions will then reduce the GUT group to a subgroup whose
further breaking to the SM group could then proceed via the
Higgs mechanism (for a review and references, see, e.g., Ref. <cit.>).
This still leaves room for some topological
defects. Clearly, the discovery of monopoles or evidence for strings
in a SGWB would be extremely valuable as a guide to a grand unified theory beyond the SM.
§ COSMOLOGICAL DEFECTS AND INFLATION
Metastable cosmic strings require two steps of spontaneous symmetry
breaking. In the first step, an SU(2) group, which may be
embedded in some GUT group, is broken to U(1),
leading to monopoles as topological defects. In the second step, a
U(1) group is spontaneously broken leading to strings
as topological defects.
This U(1) group must not be orthogonal
to the U(1) contained in SU(2) in order to allow the
string to split into segments having monopoles and antimonopoles at
the ends.
Between the SU(2) and U(1) phase transitions an
inflationary period must have taken place in order to dilute the
produced monopoles but to keep the cosmic strings. In fact, one
of the motivations for cosmic inflation has been the “monopole
problem” of GUTs in standard cosmology (see, e.g., Ref. <cit.>).
One may worry that a mass ratio of √(κ)∼ 8 between the topological defects sourced by these phase transitions does not leave enough space for an inflationary period. However, while the chronological order of the symmetry breaking stages in hybrid inflation is set by the Higgs masses, these are linked to the corresponding symmetry breaking scales (or Higgs vacuum expectation values) through Yukawa couplings. This leaves enough freedom to implement hybrid inflation <cit.>.
In the supersymmetric SU(2)_R×U(1)_B-L model
discussed in Sec. <ref>, inflation is naturally
realized by means of F-term hybrid inflation
<cit.>. The two singlets, needed in the
superpotential to ensure SU(2)_R and U(1)_B-L breaking,
play the role of inflatons. In combination with the supersymmetric
SM and right-handed neutrinos, a consistent picture of
inflation, leptogenesis and dark matter is obtained for a large scale
of B-L breaking, v_s ∼ 10^15 GeV
<cit.>.
Alternatively, one can consider sneutrino tribid inflation in a gauged U(1)_B-L
extension of the supersymmetric SM. Metastable strings are
again obtained by embedding the model in SO(10) <cit.>.
Depending on the pattern of supersymmetry breaking, one obtains
gravitino dark matter <cit.>. Note that in all SO(10)
models the precise connection between GUT masses and couplings and the
monopole-mass–string-tension ratio
√(κ) is an open question that remains to be investigated.
Monopoles and strings have also been considered in nonsupersymmetric
SO(10) models with an intermediate string scale
v_s ∼ 10^13 GeV. The inflaton is introduced as a
GUT-singlet scalar field whose potential is generated by radiative corrections.
Monopoles and strings may be present today at an observable level, and
stochastic GWs may respect PTA and LVK bounds and only become visible
at LISA, SKA, BBO and ET/CE <cit.>. The formation of
monopole–antimonopole–string configurations may lead to a suppression
of the GW spectrum at PTA frequencies <cit.>.
In GUT models with monopoles and metastable strings the incorporation of
inflation is of crucial importance. One is then faced with the
challenging problem to treat gauge coupling unification, fermion
masses, proton decay, baryogenesis, (potentially) supersymmetry
breaking and dark matter, together with inflation and the formation of
a cosmic string network in a quantitatively consistent way. In the
examples mentioned above, some progress has been made, but there
is much room for improvement. Evidence for (metastable) strings
from a SGWB would be a key element to guide us toward grand unified theories.
§ CONCLUSIONS AND OUTLOOK
The evidence for a gravitational-wave background at nanohertz frequencies recently reported by PTAs around the globe <cit.> opens a new window to study the evolution of our universe. The observed signal at nanohertz frequencies, ten orders of magnitude below the LIGO–Virgo–KAGRA band, may have an astrophysical origin — inspiralling supermassive black-hole binaries — but it might also be a remnant of events in the early universe.
One possible cosmological interpretation of the observed signal are metastable cosmic strings, which have a strong theoretical motivation in the framework of GUTs. The corresponding GW spectrum is characterized by two parameters: the string tension μ = 2π v^2_s, where v_s is the associated symmetry breaking scale, and the ratio between monopole mass squared and string tension, κ = m^2_M/μ, which determines the lifetime of the string network.
The signal in the most recent PTA data sets <cit.> is well described by a power law with a characteristic amplitude of the order of 10^-10≲Ω_gw h^2 ≲ 10^-9 and a positive spectral tilt, 0 ≲ n_t ≲ 3, around the current PTA peak sensitivity of roughly 3 nHz. In terms of the parameters of metastable cosmic strings, this implies a string tension 10^-11≲ Gμ≲ 10^-7 and a decay parameter √(κ)≳ 8, where the upper (lower) bound on Gμ (√(κ)) arises from the constraints set by ground-based interferometers on the amplitude of the SGWB. As illustrated in Fig. <ref>, a value of the spectral tilt n_t ≳ 1, as preferred by the most recent PTA data, favours values v_s ≥ few × 10^14 GeV, close to the GUT scale. Such large values of the string tension will be conclusively tested once the LIGO and Virgo ground-based interferometers reach design sensitivity in the coming years <cit.>.
In order to distinguish metastable cosmic strings from other interpretations of the SGWB signal at PTA frequencies, a more precise
determination of the spectral tilt will be important. Moreover, like most other cosmological signals, the SGWB from metastable cosmic strings is largely isotropic, as opposed to the significant anisotropies, and the possible presence of resolvable sources, which are expected for a GW signal from supermassive black-hole binaries <cit.>. Future pulsar observations and combinations of existing PTA data sets will shed light on these questions in the near future. In addition, GW observations in other frequency bands are an extremely powerful probe of the cosmic-string hypothesis, as the predicted signal spans many orders of magnitude in frequency: smaller frequencies would be valuable to test the characteristic f^2 behaviour of the spectrum, current and future ground-based detectors will be able to distinguish GUT-scale metastable strings from intermediate-scale stable strings, and the space-based interferometer LISA will probe string tensions down to values well below the current reach of PTAs.
If upcoming observations point to an astrophysical origin of the current PTA signal, the results presented here can be interpreted as upper bounds on Gμ and κ, demonstrating the potential of ground- and space-based interferometers to probe the remaining parameter space of GUT-scale cosmic strings.
On the theoretical side, the calculation of the GW spectrum has to be improved in several ways. Most importantly, a precise and robust understanding of the substructure of string loops is crucial for the estimation of the SGWB. In addition, the large value of v_s suggested by the recent PTA data calls for further explicit studies of metastable strings in GUT models. As explained in Section <ref>, the value √(κ)≃ 8.3 hinted at by the data requires the energy scales v_u and v_s of the symmetry breakings leading to monopoles and strings, respectively, to be close to each other. This is a strong constraint on GUT model building that remains to be investigated, with consequences for neutrinos, leptogenesis and inflation.
Acknowledgments
The work of K. Sc. is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group, GRK 2149: Strong and Weak Interactions — from Hadrons to Dark Matter.
JHEP
|
http://arxiv.org/abs/2307.03891v3 | 20230708035823 | MARBLER: An Open Platform for Standarized Evaluation of Multi-Robot Reinforcement Learning Algorithms | [
"Reza Torbati",
"Shubham Lohiya",
"Shivika Singh",
"Meher Shashwat Nigam",
"Harish Ravichandar"
] | cs.RO | [
"cs.RO",
"cs.MA"
] |
Feature selection simultaneously preserving both class and cluster structures
Suchismita Dasmycorrespondingauthor and Nikhil R. Pal
August 12, 2023
=============================================================================
Multi-agent reinforcement learning (MARL) has enjoyed significant recent progress, thanks to deep learning. This is naturally starting to benefit multi-robot systems (MRS) in the form of multi-robot RL (MRRL).
However, existing infrastructure to train and evaluate policies predominantly focus on challenges in coordinating virtual agents, and ignore characteristics important to robotic systems. Few platforms support realistic robot dynamics, and fewer still can evaluate Sim2Real performance of learned behavior.
To address these issues, we contribute MARBLER: Multi-Agent RL Benchmark and Learning Environment for the Robotarium. MARBLER offers a robust and comprehensive evaluation platform for MRRL by marrying Georgia Tech's Robotarium (which enables rapid prototyping on physical MRS) and OpenAI's Gym framework (which facilitates standardized use of modern learning algorithms).
MARBLER offers a highly controllable environment with realistic dynamics, including barrier certificate-based obstacle avoidance. It allows anyone across the world to train and deploy MRRL algorithms on a physical testbed with reproducibility.
Further, we introduce five novel scenarios inspired by common challenges in MRS and provide support for new custom scenarios.
Finally, we use MARBLER to evaluate popular MARL algorithms and provide insights into their suitability for MRRL.
In summary, MARBLER can be a valuable tool to the MRS research community by facilitating comprehensive and standardized evaluation of learning algorithms on realistic simulations and physical hardware.
Links to our open-source framework and the videos of real-world experiments can be found at <https://shubhlohiya.github.io/MARBLER/>.
§ INTRODUCTION
With increasing demand for robotics to operate in complex real-world environments, coordination of multiple robots is becoming paramount. However, the complexity of exact solutions to important problems (e.g., coverage control <cit.>, path-planning <cit.>, and task allocation <cit.>) grows exponentially as the number of robots increase <cit.>. Consequently, Multi-Robot Reinforcement Learning (MRRL) <cit.> is emerging as a promising alternative paradigm to address this challenge.
MRRL has proven useful for delivery robots <cit.>, coordinated robotic exploration <cit.>, multi-robot communication <cit.>, multi-robot path planning <cit.>, multi-robot target localization <cit.> and more <cit.>. However, despite being developed for robotics, learning algorithms are rarely evaluated in the real-world, with a few notable exceptions <cit.>. However, even the exceptions were tested on smaller teams (2, 2, 3, and 4 robots, respectively) and on ad-hoc platforms, rending reproducibility time-consuming and difficult.
In contrast, Multi-Agent Reinforcement Learning (MARL) algorithms can be evaluated in a systematic way in many standardized simulated environments, such as the Multi-Agent Particle Environment (MPE) <cit.> and the StarCraft Multi-Agent Challenge (SMAC) <cit.>. While it might possible use existing MARL environments to evaluate algorithms developed for MRS, they lack realistic robot dynamics and likely have a large sim2real gap. Further, they do not directly allow for evaluation and benchmarking on physical robots.
In this work, we develop an integrated and holistic platform that can enable seamless training of MRRL policies and their evaluation on physical robots. Specifically, we contribute Multi-Agent RL Benchmark and Learning Environment for the Robotarium (MARBLER). MARBLER is a bridge between the MARL community and the physical robots in the Robotarium <cit.> that makes it easy to evaluate MRRL algorithms and design novel scenarios. The Robotarium is a remotely-accessible, publicly-available, and free-to-use testbed for MRS that allows for up to 20 robots at once in a highly-customizable environment.
As such, MARBLER enables machine learning researchers to develop and test algorithms for physical robots, and control theorists to experiment with state-of-the-art (SOTA) learning algorithms.
Our MARBLER platform has the following key benefits:
* The simulated robots in MARBLER exhibit dynamics similar to that of physical robots as it is built on top of the Robotarium's simulator. Further, MARBLER includes support for barrier certificates to prevent collisions, forcing algorithms to learn in realistic settings.
* MARBLER inherits the open-access benefits of the Robotarium, enabling anyone across the world to train coordination algorithms and systematically deploy on a physical multi-robot testbed with reproducibility.
* MARBLER is compatible with any learning algorithm that can be used with the OpenAI Gym interface.
* MARBLER currently has 5 novel scenarios inspired by common and challenging problems in MRS.
* MARBLER is open-source and allows users to easily add new scenarios or modify existing ones.
By creating an interface between MARL algorithms and the Robotarium, MARBLER is the first publicly-available environment that can evaluate Sim2Real capability in MRRL. Further, MARBLER can serve as a benchmark to evaluate learning algorithms in simulation with real-world constraints and readily deploy them on physical robots.
In addition, we conducted detailed evaluations of existing MARL algorithms by leveraging Extended PyMARL (EPyMARL) <cit.> within MARBLER. Our experiments reveal insights into how different characteristics of existing algorithms (e.g., policy gradient vs. valued-based, parameter sharing, etc.) impact performance in both simulated and physical multi-robot systems.
§ RELATED WORK
§.§ MARL and MRRL Platforms
The Multi-Agent Particle Environment (MPE) <cit.> is a popular framework for evaluating MARL algorithms, consisting of cooperative and adversarial 2D tasks.
In MPE, agents apply forces to particles which can interact with landmarks and other agents. This is a popular setup in MARL environments and has been extended by platforms such as VMAS <cit.>: a vectorized version of MPE that is supported by GPUs to allow for more complex scenarios and faster training. However, particle simulators have very different dynamics than real robots making them poor choices for MRRL benchmarking.
Another popular MARL environment is StarCraft Multi-Agent Challenge (SMAC) <cit.> which is considerably more complex, requiring agents to handle partial observability over long horizons.
However, the agent dynamics in SMAC is still considerably different from real world robots, again making it a poor choice to evaluate MRRL algorithms.
There are few frameworks that are designed to benchmark MRRL algorithms and fewer that are able to evaluate Sim2Real performance of algorithms. SMART <cit.> is one such evironment. However, SMART is limited to scenarios involving autonomous driving, it only supports up to four robots, and neither their evaluation test bed nor their source code is publicly available.
The other MRRL environment that allows for Sim2Real testing is MultiRoboLearn <cit.>: an open-source framework that provides an OpenAI Gym interface for easier integration. However it also only supports a maximum of 4 robots, and, like SMART, it does not have a publicly available testbed. Additionally, creating new scenarios in MultiRoboLearn requires creating custom environments in Gazebo <cit.>, introducing significant overhead.
In contrast to existing environments, MARBLER's simulator closely mimics the constraints of physical robots and allows researchers to evaluate Sim2Real capabilities in a standardized and reproducible way. Therefore, MARBLER is the first MRRL benchmark that has both a realistic simulator and a physical testbed that anyone can use.
§.§ MARL Algorithms
A variety of MARL algorithms have been proposed that perform very well in simulated environments. PPO <cit.> is an effective actor-critic policy gradient method for single agent RL. MAPPO <cit.> is the multi-agent extension of PPO where a single centralized critic is conditioned on all agent's observations to learn a joint state value function and a separate actor for each agent tries to learn the best action to take conditioned only on the agent's individual observations.
In contrast to MAPPO, QMIX <cit.> and VDN <cit.> are value-based methods that decompose the joint state-action value function into individual state-action value functions. VDN learns to decompose the team value function agent-wise while QMIX learns agent-specific Q networks and combines them monotonically via hypernetworks.
In SMAC and MPE, MAPPO, QMIX, and VDN have been shown to be three of the best performing MARL algorithms <cit.>.
However, while these algorithms have performed very well in simulation, there is limited testing of their real world performance. <cit.> evaluated VDN's and QMIX's performance on robots and <cit.> and <cit.> evaluate different versions of multi-agent PPO based algorithms on real robots. However, these are some of the only works to do real-world evaluations and the experiments only used at most four robots and are not easily reproducible.
Another important design problem in MRRL is if robots should share parameters. When robots share parameters, their networks all learn together which greatly reduces the number of parameters to be trained. However, this leads to robots all learning the same behavior. To combat this, robots have unique IDs appended to their observations but this approach still only allows robots to learn policies with limited heterogeneity <cit.>. Alternatively, each robot can learn its own set of network parameters which allows robots to learn truly heterogeneous behavior but greatly increases the number of environment interactions needed for robots to learn, which can be expesive in realistic settings.
§.§ The Robotarium
The Robotarium<cit.> is a remotely accessible multi-robot laboratory developed by Georgia Tech. It features a 12ft x 14ft testbed, 8 Vicon motion-capture cameras and allows up to 20 GRITSBots <cit.> to operate at once.
The Robotarium has inbuilt control barrier certificates (CBF) <cit.> which provide a provable guarantee of online collision avoidance for the robots, by ensuring a minimum inter-robot distance.
Control commands that don't satisfy constraints are updated with minimum possible deviation before execution, by a quadratic-program based controller.
Hence, the policies learned in environments utilizing CBFs will have to adapt to these actuator constraints which makes the platform more realistic and allows policies to be run on real robots.
The Robotarium also provides a Python simulator that closely resembles how the robots will act in the real Robotarium. Once programs are working in simulation, the Robotarium has a publicly accessible website where anyone in the world can upload their programs for them to then be run in the real Robotarium on real robots.
§ THE MARBLER PLATFORM
Historically, evaluating MRRL algorithms using the Robotarium's simulator has been a challenging task. The lack of a standardized framework for MRRL in the Robotarium means that researchers have to create scenarios from scratch, design the low level control algorithms to control the robots after they select an action, control how the graphics are displayed, and more. As a result, to the best of our knowledge, only <cit.> has evaluated deep reinforcement learning algorithms with the Robotarium, despite its open accessibility to researchers. Addressing this limitation, MARBLER establishes a cohesive and user-friendly API tailored specifically for MRRL experiments. Researchers can design novel environments or employ the pre-existing default environments to execute their algorithms, thereby allowing reproducibility across studies.
Moreover, owing to its integration with the Robotarium's simulator, MARBLER streamlines the process of transitioning trained robots from simulation to real-world deployment. Through the execution of a single script, users can generate the files necessary for submitting their policies to the physical Robotarium. Because the Robotarium is accessible to all users free of charge, MARBLER is the first platform that allows for the deployment of MRRL algorithms on real robots in a highly reproducible manner.
§.§ Core Components
MARBLER is comprised of four core components that form the foundation of the platform:
Core: The Core component serves as the fundamental building block of MARBLER, leveraging the Robotarium's python simulator. It encompasses critical functionalities necessary for the environment, such as environment resetting and discrete time step advancement. By utilizing the capabilities of the Robotarium's simulator and CBFs, MARBLER incorporates realistic dynamics that emulate the constraints encountered by real robots.
Scenarios: The scenarios module defines the environments the robots interact in and the specific tasks they must accomplish.
Gym Interface: Each scenario within MARBLER is registered as a Gym environment, which allows for direct compatibility with the algorithms and tools that support the Gym interface.
Test Pipeline: The Test Pipeline
provides a streamlined process for importing trained robots into the simulation environment, giving researchers a way to visualize robots' performance and collect test data. Subsequently, researchers can execute a script to prepare their files for submission to the Robotarium, which can then be uploaded to the real Robotarium, enabling evaluation in a real-world setting.
§.§ Scenarios
§.§.§ Existing Scenarios
To facilitate immediate testing and evaluation using MARBLER, we introduce five scenarios inspired by diverse MRRL problems. These scenarios are designed to offer researchers a starting point for experimentation and can be easily customized by modifying the scenario's associated configuration file. Parameters such as the number of robots, communication methods, scenario difficulty, and more, can be adjusted as needed.
A complete overview of these scenarios is available in the supplementary material[Supplementary material can be found
https://shubhlohiya.github.io/MARBLER/assets/supplementary.pdfhere].
but we include brief descriptions here:
Simple Navigation (Fig. <ref>):
Robots navigate towards a known destination point. This scenario is an easy starting point for algorithms to learn in.
Predator Capture Prey (PCP) (Fig. <ref>):
Sensing robots and capture robots must work together to capture the prey. Sensing robots know the location of prey within their sensing radius and must communicate this to the blind capture robots. Inspired by the Predator Capture Prey scenario in <cit.>.
Warehouse (Fig. <ref>):
Robots must navigate to their color zone on the right to receive a load and then unload in their color zone on the left while avoiding collisions; a Multi-Robot Path Finding environment <cit.>.
Material Transport (MT) (Fig. <ref>):
Robots with varying speeds and capacities must collaborate to efficiently unload two zones: one nearby with a large amount of material and one further away with a small amount of material. This is a task allocation problem <cit.> where the robots must collaborate to unload the zones within a time limit.
Arctic Transport (AT) (Fig. <ref>):
Drones can move fast over any tile and have a large sensing radius. Ice and water robots have a limited sensing radius and move fast over some tiles but slow over other tiles. Robots are rewarded based on how far the ice/water robots are from the goal zone so the drones must guide the ice/water robots. This is a Multi-Robot Path Planning scenario <cit.> where the drones must find a path to the goal zone and communicate it to the ice/water robots.
§.§.§ Creating New Scenarios
MARBLER provides a user-friendly approach to create new scenarios, similar to MPE and VMAS. Researchers can customize the action space, observation space, visualizations, and other relevant parameters without needing to interact with the underlying Robotarium code, allowing researchers to develop tailored scenarios that align with their specific use cases. Our GitHub includes comprehensive documentation to create new scenarios.
§ EXPERIMENTS
§.§ Experiment Setup
For all our experiments, we used the EPyMARL framework to train our robots. Because the scenarios in MARBLER have been registered as Gym environments, they are directly compatible with EPyMARL. This allowed us to train policies using the various learning algorithms available in EPyMARL with no modifications.
Baselines: We compared MAPPO <cit.>, QMIX <cit.>, and VDN <cit.> with parameter sharing. To investigate the effects of parameter sharing, we also evaluated QMIX without parameter sharing (QMIX_NS).
§.§ Evaluation Protocol
We evaluated all algorithms in the PCP, Warehouse, MT, and AT scenarios with 4, 6, 4, and 4 robots respectively. Before training each algorithm, we ran a hyperparameter search in the Simple Navigation environment in a manner similar to <cit.>. Exact details on the hyperparameter search along with the hyperparameters we used for each algorithm can be found in the supplementary material[Supplementary material can be found
https://shubhlohiya.github.io/MARBLER/assets/supplementary.pdfhere].
We trained VDN and QMIX for a total of 5 million time steps in each scenario. Given the conflicting evidence about off-policy algorithms being more sample efficient than on-policy algorithms due to their use of a replay buffer <cit.>, we trained MAPPO for a total of 25 million time steps. We trained five seeds for each algorithm.
Because the Robotarium immediately stops a run when robots collide or go outside the acceptable boundaries, we used strict CBFs so that, if the robots attempt to get within 20cm from each other, their movement slows to the point to where they almost stop. We also penalize the robots and end the episode if robots collide or drive outside the boundaries of the environment. By doing this, the robots are able to successfully run in the Robotarium after training.
In all scenarios, robots had full communication and in all scenarios except MT, robots had unlimited bandwidth in their communications. Exact details about how the environments were configured for these evaluations are included in the supplementary material.
§.§ Computational Requirements
We trained all models using CPUs; primarily with a Dual Intel(R) Xeon(R) Gold 6226 <cit.> and an Intel(R) Core(TM) i7-12700KF. It took 16084 CPU hours to train all models (excluding hyperparameter searches).
§ RESULTS
To compare baselines, first we look at training evaluation returns to evaluate sample efficiency and how much of an impact different seeds make which can be seen in Fig. <ref>. Then, we compared the best performing models for each algorithm in each scenario. To do this, we took the model that achieved the highest reward for each algorithm and evaluated the model in simulation and on real robots to compare performances. In simulation, we ran each model for 100 episodes and on the real robots, we ran each model for 10 episodes. The results can be seen in table <ref>.
§.§ Value Based vs. Policy Gradient
For the first 5 million timsteps, VDN is the best performing algorithm in every scenario. After 25 million steps, MAPPO's best performing seeds approaches that of VDN's in MT and AT and surpasses it in Warehouse. However, all seeds in MAPPO converge to lower performance in PCP than in any of the value based methods.
Additionally, MAPPO's performance is much more influenced by its seed than in any value-based method. This is contradictory to the findings in <cit.> but it seems that VDN generally outperforms MAPPO in MARBLER suggesting that value based methods, particularly VDN, may be more applicable to physical robots than policy gradients.
§.§ Effects of Parameter Sharing
The performance of models trained with parameter sharing vs. without parameter sharing depends on the heterogeneity of the environment. In the Warehouse scenario, where robots are homogeneous except for their loading zone locations, QMIX outperformed QMIX_NS significantly. In MT, the robots need to learn slightly different policies to ensure that all zones are unloaded within the time limit, but the optimal policies are similar. In AT, drones and ice/water robots had fundamentally different optimal policies, yet neither QMIX nor QMIX_NS utilized the drones' enhanced sensing radius, resulting in similar policies for all robots. In AT and MT, with limited heterogeneity, QMIX showed a significant performance advantage over QMIX_NS but much less significant than in Warehouse. However, in the PCP scenario, where very different policies were learned for the Predator and the Capture robots, QMIX and QMIX_NS performed similarly. Thus, as heterogeneity increases, the gap between policies trained with and without parameter sharing shrinks, consistent with the findings from <cit.>. This suggests that in scenarios with more diverse heterogeneity, models trained without parameter sharing may outperform those trained with it.
Additionally, robots trained with QMIX_NS went out of bounds a total of 10 times in simulation and 6 times on real robots. In contrast, robots trained with all parameter sharing methods only went out of bounds once in simulation and once on real robots. When a single robot goes out of bounds, all robots are given a large negative penalty and the episode ends.
This suggests it is much more difficult for robots to learn how to handle events where a single robot can cause all other robots to suffer a penalty without parameter sharing.
§.§ Sim2Real Gap
As shown in table <ref>, there are few significant differences between the algorithms' performance in simulation and in the real Robotarium. This gives strong evidence that the simulator is very similar to real robots. However, there is one key difference between the real experiments and the simulated experiments: the robots never collide in simulation and robots go out of bounds more than 6x more often on average on real robots. The only time an algorithms' metrics were significantly worse on real robots vs. in simulation was when the real robots collided or went out of bounds.
To further evaluate this, we retrained VDN in PCP using less safe CBFs that are only effective at 17cm and do not slow the robots as much when their within the safety radii. In addition, we did not stop the episode or penalize the robots for driving out of bounds or colliding. This is how the Robotarium's safety mechanisms are setup by default. Other than these two modifications, we trained these models the same way as the original VDN models.
As seen in table <ref>, the differences between the test performance of the robots with the default CBFs compared to the safe CBFs in simulation is not significant. However, when we ran these robots in the Robotarium, they collided 3/10 episodes, despite using the recommended method of preventing collisions, the robots never colliding in the 100 simulated episodes, and the robots with the safe CBFs never colliding. This gives more evidence that, when it comes to safety, there is a significant Sim2Real gap which highlights the second major benefit of using MARBLER: even if robots seem to learn safe policies in simulation, those policies may not run safely in the real world. This makes MARBLER the first open platform created that can be used to evaluate how safe learned MRRL policies are.
§ CONCLUSION
We introduce MARBLER, the first open platform with Sim2Real capabilities, realistic robot dynamics, and the ability to evaluate how safe MRRL algorithms are. MARBLER environments are fully compatible with OpenAI Gym, providing an easy interface with modern learning algorithms.
To demonstrate the utility of MARBLER, we developed five MRRL scenarios and utilized the EPyMARL framework to benchmark popular MARL algorithms, both in simulation and in the real-world. We believe MARBLER will help researchers benchmark Sim2Real transfer capabilities of MRRL algorithms in a systematic and reproducible way, making it an invaluable tool for the research community.
IEEEtran
|
http://arxiv.org/abs/2307.04092v1 | 20230709042603 | Coupled-channel $D^\ast K^\ast -D_s^\ast ρ$ interactions and the origin of $T_{c\bar{s}0}(2900)$ | [
"Man-Yu Duan",
"Meng-Lin Du",
"Zhi-Hui Guo",
"En Wang",
"Dian-Yong Chen"
] | hep-ph | [
"hep-ph",
"hep-ex"
] |
[addref]
|
http://arxiv.org/abs/2307.04783v1 | 20230710180000 | Confinement in $(1+1)$ dimensions: a holographic perspective from I-branes | [
"Carlos Nunez",
"Marcelo Oyarzo",
"Ricardo Stuardo"
] | hep-th | [
"hep-th"
] |
.
Confinement in (1+1) dimensions: a holographic perspective from I-branes
Carlos Nunez^a[[email protected]],
Marcelo Oyarzo^b[[email protected]] and Ricardo Stuardo^a[[email protected]]
^aDepartment of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
^bDepartamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Abstract
In this paper we holographically study the strongly coupled dynamics of the field theory on I-branes (D5 branes intersecting on a line). In this regime, the field theory becomes (2 + 1) dimensional with 16 supercharges. The holographic dual background has an IR singularity. We solve this singularity by compactifiying the theory on a circle, preserving 4 supercharges. We study various aspects of the QFT: confinement, symmetry breaking, Entanglement Entropy, etc. We also discuss a black hole solution and make some comments on the string σ-model on our backgrounds.
Dedicated to the memory of Roman Jackiw.
tocempty
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§ INTRODUCTION
Shortly after the conjectured duality between super conformal field theories and string/M theory on spaces with an AdS factor <cit.> -<cit.>, these ideas were extended to non-conformal situations <cit.>-<cit.>.
In this work, we holographically study one particular two dimensional field theory that at weak coupling is defined on the intersection of two stacks of D5 branes (these are called I-branes). Dynamical features of these theories imply that, as the coupling is increased the field theory turns (2+1)-dimensional and doubles the amount of SUSY preserved. This dynamics is very well explained in <cit.>.
The string background, holographic dual to this strongly coupled QFT is well understood at large values of the radial coordinate, where it can be written as an intersection of NS five branes. Good control over the string σ-model in such background has been developed <cit.>, <cit.>. Nevertheless, this background is singular in the IR (at small values of a suitable radial coordinate). In this paper we propose a 'completion' of this dual background, making it trustable at low energies. The new solution is very explicit and simple. It preserves four supercharges and can be thought of as the dual to a (2+1)-dimensional field theory that is compactified to (1+1)-dimensions. The QFT is a two nodes quiver with Chern-Simons terms, connected by bifundamental matter.
In the bulk of this paper, we study holographically various aspects of the strongly coupled dynamics of this QFT. We define a suitable gauge coupling, that suggest a low energy confining behaviour. Theta angles and the breaking of U(1)-R symmetries are discussed, together with an estimate of a density of degrees of freedom as a function of the energy. Maldacena-Wilson loops are calculated (again indicating confinement). Also, 't Hooft loops and Entanglement Entropy on a strip are calculated, discussing how the non-local UV dynamics of the system impacts on these observables.
We also briefly touch upon two aspects that will be further developed in future publications: we present a black hole solution, obtained as analytic continuation of our new background and calculate some characteristic quantities.
Also, we shortly discuss some aspects of the string σ-model on our backgrounds.
The paper is organised as follows. In Section <ref> we present the supergravity backgrounds studied in the rest of the paper. In Section <ref> we propose the QFT dual to these backgrounds, with various characteristic observables calculated. This proposal is sharpened in Section <ref>, where the IR confining behaviour is determined and the influence of the high energy LST dynamics on observables like 't Hooft loops and Entanglement Entropy is discussed. Some aspects of the string σ-model are discussed in Section <ref>. The conclusions and future lines of research suggested by this paper are written in Section <ref>. Various appendices are included, these discuss in great detail the very many interesting technical aspects needed in the main body of this work.
§ THE SUPERGRAVITY BACKGROUNDS
In this section we write the supergravity backgrounds studied. The first background already appears in the bibliography <cit.>, the second background is new. We refer the reader to Appendices <ref> and <ref> for detailed derivations.
The associated charges are studied. A black hole solution is obtained as a bonus, by performing analytic continuations. Some characteristic observables of the black hole are calculated.
§.§ Background I
To describe the backgrounds we
use the coordinates ( t,x, φ, r ,θ_A,ϕ_A,ψ_A
,θ_B,ϕ_B,ψ_B). We set α'=g_s=1 and define two sets of left-invariant forms of SU(2),
ω̂_1= cosψ_A dθ_A +sinψ_Asinθ_A dϕ_A, ω̃_1= cosψ_B d θ_B +sinψ_Bsinθ_B dϕ_B,
ω̂_2= -sinψ_A dθ_A +cosψ_Asinθ_A dϕ_A, ω̃_2= -sinψ_B dθ_B +cosψ_Bsinθ_B d ϕ_B,
ω̂_3= dψ_A +cosθ_Adϕ_A, ω̃_3= dψ_B +cosθ_Bd ϕ_B.
In terms of these we present the first background. The string frame metric, the Ramond three form F_3, the potential C_2 and the dilaton Φ read,
ds^2_st = r{ -dt^2+dx^2+ (e_A^2+e_B^2) dφ^2/2+ 8 dr^2/r^2 (e_A^2+e_B^2)
+2/e_A^2[ ω̂_1^2+ω̂_2^2+ ω̂_3^2]
+2/e_B^2[ ω̃_1^2+ω̃
_2^2+ ω̃_3^2]} ,
F_3 = dC_2= -2/e_A^2ω̂_1∧ω̂
_2∧ω̂_3 -2/e_B^2ω̃_1∧ω̃_2∧ω̃_3 ,
C_2 =
-2/e_A^2ψ_A sinθ_Adθ_A
∧ dϕ_A - 2/e_B^2ψ_B sinθ_Bdθ_B
∧ dϕ_B .
Φ = log r .
Here (e_A,e_B) are parameters which are fixed when imposing charge quantisation. The coordinate φ is non-compact and could be rescaled to absorb the prefactor (e_A^2+e_B^2) /2.
There are two three-spheres labelled by Ŝ^3 and S̃^3 respectively parametrised by the Euler angles [θ_A, ϕ_A, ψ_A] and [θ_B,ϕ_B,ψ_B]. The range for these angles are θ
_A,B∈ 0,π ψ _A,B∈[0,4π], and ϕ _A,B∈[
0,2π].
The Ricci scalar for the metric in eq.(<ref>) is
R= -3( e_A^2 +e_B^2)/2r,
indicating a singularity at r=0. This is the singular behaviour found close to a stack of D5 branes–see equation (3.38) in the paper <cit.>. In fact, the background can be understood as the backreaction of two stacks of D5 branes that intersect along the non-compact coordinates (t, x) and extend respectively over (y_1,y_2,y_3,y_4) and (w_1,w_2,w_3,w_4), which can be written as radial coordinates and three spheres. After backreaction (at strong coupling), the two stacks share the directions [t,x,φ] wrap the spheres Ŝ^3[θ_A,ϕ_A,ψ_A] and S̃^3[θ_B,ϕ_B,ψ_B], as we find in the background (<ref>). See details in Appendix <ref>.
The singular behaviour at r∼ 0 indicates the need of a description in terms of other variables. On the other hand for large values of the radial coordinate r, the growth of the dilaton and the string coupling (g_s∼ e^Φ) requires an S-duality and the description of the system is in terms of an intersection of NS five branes. The system is then dual to two Little String Theories (LST) that intersect along (t,x,φ) each one wrapping the spheres Ŝ^3 and S̃^3.
We further elaborate on this background in Section <ref> and Appendix <ref>.
We are interested in resolving the singular behaviour at r=0 of the background in eq.(<ref>), making the solution in terms of D5 branes trustable in the ”IR-regime”. The goal is to write a trustable dual description for a strongly coupled QFT (that is UV-completed by a LST). We are also interested in preserving some amount of SUSY for stability purposes.
§.§ Background II
We write below a solution to the equations of motion of Type IIB supergravity that resolves the singular behaviour by compactifying the coordinate φ, with a precise period. A fibration between the spheres and the coordinate φ is also needed. This solution reads,
ds^2_st = r{ -dt^2+dx^2+f_s( r) dφ ^2+4/r^2f_s( r)
dr^2+2/e_A^2[ ω̂_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ
(r)dφ) ^2] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3-e_BQ_Bζ( r)
dφ) ^2] } ,
F_3 = dC_2= 2 ζ'(r)dr∧ dφ∧(
Q_A/e_Aω̂_3+Q_B/e_Bω̃
_3) +2/e_A^2ω̂_1∧ω̂
_2∧( e_AQ_Aζ (r)dφ -ω̂_3)
+2/e_B^2ω̃_1∧ω̃_2∧( e_BQ_Bζ (r)dφ -ω̃_3) ,
C_2 = ψ_A( 2Q_A/e_Aζ ^'(
r) dr∧ dφ -2/e_A^2sinθ_Adθ_A
∧ dϕ_A) +2/e_Acosθ_AQ_Aζ( r) dφ∧ dϕ_A
+ψ_B( 2Q_B/e_Bζ ^'( r)
dr∧ dφ -2/e_B^2sinθ_Bdθ_B
∧ dϕ_B) +2/e_Bcosθ_BQ_Bζ( r) dφ∧ dϕ_B .
Φ = log r .
Here (e_A, Q_A, e_B, Q_B) are parameters. The functions f_s(r), ζ(r) are given by
f_s(r) = e_A^2+e_B^2/2-m/r^2-2(
Q_A^2+Q_B^2) /r^4≡e_A^2+e_B^2/2r^4
(r^2-r_+^2)(r^2-r_-^2) ,
ζ( r) = 1/r^2-1/r_+^2 ,
r_±^2=m±√(
m^2+4(Q_A^2+Q_B^2)(e_A^2+e_B^2))/e_A^2+e_B^2.
If the parameter m=0 and e_A Q_B=± e_B Q_A, the background preserves four supercharges. For the SUSY study and the details of the construction of the background in eqs.(<ref>)-(<ref>), see Appendices <ref> and <ref>.
Note that the circle parametrised by the angle φ shrinks smoothly at r=r_+ if we choose its periodicity to be
φ∼φ +L_φ,
L_φ =8 π/r_+f^'_s(r_+) =4 π/e_A^2+e_B^2( 1+m/√(m^2+4 (e_A^2+e_B^2)(Q_A^2+Q_B^2) )) .
In the BPS limit, the Ricci scalar associated with the geometry in eq.(<ref>) is
R=-(e_A^2+e_B^2)/2 e_A^2 r^5(4 Q_A^2 + 3 e_A^2 r^4 ).
That is bounded for all the range of the radial coordinate [r_+,∞).
Notice that in the case Q_A=Q_B=m=0 (conversely, in the strict limit r→∞), the background in eq.(<ref>) becomes that in eq.(<ref>). In some of the observables we discuss below, we can perform a regularisation that takes away the effects on the observable that come from the background (<ref>) from the same observable computed in the background (<ref>). Also, in what follows, for any object, ξ _A,B, (like a D-brane) extended along the spheres we use the notation ξ_A≡ξ̂ and ξ _B≡ξ̃.
§.§ Conserved charges
To calculate the D5 brane charges associated with the Ramond Field F_3 in eq.(<ref>), we define the three-cycles,
ℳ_A = (ψ_A,θ_A,ϕ_A) , ℳ_B = (ψ_B,θ_B,ϕ_B).
Let us call the A-stack of branes to be the one extended along the coordinates [t,x,φ, θ_A,ϕ_A,ψ_A] and analogously for the B-stack.
To calculate the number of branes in the A-stack, we need to integrate F_3 over the three cycle ℳ_B–as this is orthogonal to the A-brane stack. Analogously, the number of branes in the B-stack will be obtained by integrating F_3 over ℳ_A.
Setting α'=g_s=1, the quantisation condition for Dp-branes is
(2π)^7-p g_s α'^7-p/2 N_Dp= ∫_Σ_8-p F_8-p, leads to N^i_D5 = 1/(2π)^2∫_ℳ_i F_3.
After choosing a convenient orientation for the three-cycles (equivalently, changing the sign of F_3) we find for the D5 charges,
N_A= 8/e^2_B ,
N_B = 8/e^2_A .
This implies a quantisation condition for the parameters (e_A, e_B). We could have chosen different three cycles, leading to the same conditions. The result is the same for the either of the backgrounds in eqs.(<ref>), (<ref>).
§.§ Bonus: a black hole solution
Let us consider our new background in eq.(<ref>)[The material in this section arose in discussion with Juan Maldacena, whom we gratefully acknowledge.]. We perform a double Wick rotation
φ→ it , Q_A,B→ -iQ_A,B , t → iy,
we find the black hole configuration, which we present in Einstein frame
ds_E^2 = √(r){ dy^2+dx^2-f_bh( r) dt^2+4/
r^2f_bh( r) dr^2+2/e_A^2[ ω̂
_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ(
r) dt) ^2] .
+. 2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3-e_BQ_Bζ( r) dt)
^2] } ,
F_3 = dC_2=2ζ ^'( r) dr∧ dt∧(
Q_A/e_Aω̂_3+Q_B/e_Bω̃
_3) +2/e_A^2ω̂_1∧ω̂
_2∧( e_AQ_Aζ( r) dt-ω̂_3)
+2/e_B^2ω̃_1∧ω̃_2∧( e_BQ_Bζ( r) dt-ω̃_3) ,
Φ = log( r) .
where
f_bh( r) = e_A^2+e_B^2/2-m/r^2+
2( Q_A^2+Q_B^2) /r^4≡e_A^2+e_B^2/
2r^4( r^2-r_+^2) ( r^2-r_-^2) ,
ζ( r) = 1/r^2-1/r_+^2 , r_±^2=m±√(m^2-4( Q_A^2+Q_B^2) (
e_A^2+e_B^2) )/e_A^2+e_B^2 .
This configuration F_3 has both "electric" and magnetic
parts. Nevertheless, the integrating of the magnetic part of F_7=⋆ F_3 does not lead to charge of D1 brane since the 7-cycle in which one integrates is not closed, thus is not possible to apply Gauss's law.
In general f_bh( r) has two real roots r_±.
The extremal black hole is obtained when r_+=r_- that is,
m^2=4( Q_A^2+Q_B^2) ( e_A^2+e_B^2), r_+^2=r_-^2=2√(Q_A^2+Q_B^2/e_A^2+e_B^2),
f_bh( r) =e_A^2+e_B^2/2r^4(
r^2-r_+^2) ^2=
e_A^2+e_B^2/2r^4( r^2-2√((
Q_A^2+Q_B^2) /( e_A^2+e_B^2) ))
^2 ,
The preservation of SUSY imposes extremality and
e_AQ_A± e_BQ_B=0 .
The BPS background with Q_B=e_A/e_BQ_A reads,
ds_E^2 =√(r){ dy^2+dx^2+4/r^2e_A^2+e_B^2
/2r_+^4ζ( r) ^2dr^2+2/e_A^2[
ω̂_1^2+ω̂_2^2+ω̂_3^2-2ω̂
_3e_AQ_Aζ dt] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+ω̃_3^2-2ω̃_3e_AQ_Aζ dt]
} .
F_3 =2d[ ζ( r) dt∧( Q_A/e_A
ω̂_3+Q_B/e_Bω̃_3) ] -2/e_A^2ω̂_1∧ω̂_2∧ω̂_3-2/e_B^2ω̃_1∧ω̃
_2∧ω̃_3,
Φ= log(r).
Note that g_tt=0, thus the vector ∂ _t is null.
In what follows we consider the non-BPS black hole background (<ref>). This configuration is rotating along the directions ∂_ψ_A and ∂_ψ_B. Since the fibrations do not decay at infinity, the coordinate system used in (<ref>) correspond to a rotating frame, i.e. an observer at infinity is rotating with the system. We can move to a non-rotating frame via large gauge transformation which cancels the constant term of ζ(r) in (<ref>). Also for simplicity in the computation of the charges we shift the Dilaton by a constant and the F_3 by a factor
ζ(r)=1/r^2 , Φ→Φ-2log(e_A^2+e_B^2/2) , F_3 →e_A^2+e_B^2/2F_3.
The geometry is asymptotically locally flat and has the same causal structure as the Reissner-Nordström spacetime. Nevertheless,
the spacetime is asymptotically conformal to ℝ^1,3× S^3 × S^3.
We compute the conserved charges associated to the spacetime by using the Noether-Wald method <cit.>. The expression for the charges are derived in detail in Appendix <ref>. The energy (mass), angular momentum associated to ∂_ψ_A and ∂_ψ_B, the temperature and the entropy for this configuration are given by [The asymptotic form of the metric leaves an ambiguity in the normalisation of the time-like killing vector at infinity that appears in the computation of the energy and the temperature. Therefore, the temperature in ten dimensions is the same as in four dimensions up to a numerical factor.]
E = 2m/e_A^3e_B^3r_+^2κ ^2( 16π
^2) ^2L_xL_y ,
J_A = 8Q_A/e_A^4e_B^3κ ^2( 16π
^2) ^2L_xL_y ,
J_B =8Q_B/e_B^4e_A^3κ ^2( 16π
^2) ^2L_xL_y ,
T = e_A^2+e_B^2/16π-4(
Q_A^2+Q_B^2) /16π r_+^4 ,
S =2r_+^2/e_A^3e_B^3G_10( 16π)
^2L_xL_y .
According to our normalisation κ^2=8π G_10 where G_10 is the Newton constant in ten dimensions. These quantities satisfy the first law of thermodynamics as expected
dE=TdS+Ω _AdJ_A+Ω _BdJ_B ,
where the angular velocities are
Ω _A= e_A Q_A/r_+^2 , Ω _B= e_B Q_B/r_+^2 .
This background can be understood as the lift to Type IIB of the four dimensional planar black hole with electric charges, found in <cit.>. The presence of the electric charges in four dimensions corresponds to rotations of the branes in ten dimensions.
We leave this black hole background here, as it is not the focus of the rest of this work. We move into the study of the dual Field Theories to the backgrounds I and II in eqs.(<ref>) and (<ref>)
§ A PROPOSAL FOR THE DUAL FIELD THEORY AND ITS OBSERVABLES
Here we present a proposal for the field theory dual to our new background in eq.(<ref>). It is convenient to first discuss the field theory dual to
the D5-D5 intersection and the background in eq.(<ref>).
§.§ The holographic dual to the Background I
We start discussing the field theory on I-branes.
The result in eq.(<ref>) indicates the presence of two stacks of D5 branes, with N_A and N_B being the number of branes on each stack.
When taken at weak coupling these stacks intersect over two dimensions. In <cit.> it was shown that when two stacks of branes intersect along (4k+2), being the transverse dimensions a multiple of four (in our case k=0, D5 stacks intersect in two dimensions and have eight transverse directions) the massless spectrum contains chiral fermions, arising from the open strings connecting the branes. These fermions give rise to gauge (and gravitational) anomalies on the intersection. The anomalies are cancelled by anomaly inflow from the 'bulk of the brane'. This implies that the D-branes world-volume action must contain a Chern-Simons term.
When studied at weak coupling the D5-D5 system preserve chiral supercharges. We have two gauge groups SU(N_A)× SU(N_B) with chiral fermions transforming in the ( N_A, N̅_B) representation, the system has SO(1,1) Poincare symmetry. The anomaly is cured by inflow from the bulk of the D5 branes. In other words, the dynamics of the intersection is not decoupled from the brane dynamics. The system preserves eight SUSYs <cit.>. The weakly coupled field theory is summarised by the quiver in Figure <ref>.
As the couplings grow large, the above description breaks down and is replaced by a description in terms of the background in eq.(<ref>). This is carefully described in <cit.>. In the strong coupling regime, the system preserves SO(2,1) with SO(4)× SO(4) R-symmetry and sixteen SUSYs. The three dimensional field theory has gauge groups with Chern Simons terms SU(N_A)_N_B× SU(N_B)_N_A <cit.>, <cit.>.
At strong coupling, these stacks intersect in the coordinates (t,x,φ). One of the stacks extends along (θ_A,ϕ_A,ψ_A) whilst the other does it over (θ_B,ϕ_B,ψ_B).
The papers <cit.>, <cit.>, argue that the field theory is gapped.
Notice that the background in eq. (<ref>) is not trustable for all the range of the radial coordinate. In fact, for large values of r, the dilaton becomes large the type IIB system is better described by performing an S-duality and describing the dynamics in terms of the Little String Theory on the two stacks of NS five branes.
That is, the field theory above described has a non-field theoretical UV completion.
On the other hand, for r→ 0 the Ricci scalar in eq.(<ref>) diverges and the background is not trustable. We then need to replace the description by the one given by the configuration in eq.(<ref>).
§.§ The field theory dual to the Background II.
Here, we analyse the background in eq.(<ref>). The supergravity solution is smooth, hence the dual QFT is strongly coupled.
In fact, the Ricci scalar in eq.(<ref>) is finite for r>r_+ and the string coupling (proportional to e^Φ) is bounded below some value r< r_*. The value of r_* is determined by observing that the dilaton in eq.(<ref>) can be changed by Φ=Φ_0+ log r at the cost of rescaling the Ramond form F_3→ e^-Φ_0 F_3. These scalings make the string coupling g_s=e^Φ= r e^Φ_0. The value r_*∼ e^-Φ_0 (for which g_s∼ 1), can be made arbitrarily large by suitably choosing the parameter Φ_0. Notice that a chosen large and negative Φ_0, makes the charges of the D5 branes larger.
We then conclude that the background is trustable in a large region of the radial coordinate [r_+,r_*). Consequently, in a large regime of energies the dual field theory is strongly coupled. At very high energies, when the string coupling becomes large we should S-dualise arriving to a configuration of intersecting and wrapped NS-five branes. The high energy behaviour of the field theory is UV-completed in terms of a Little String Theory.
We now discuss the flowing to lower energies. In this case, the coordinate φ is compactified, shrinks to zero size and one ends with a QFT with less SUSY, smaller R-symmetry and effectively in (1+1) dimensions.
Whilst for large values of the radial coordinate, the backgrounds I and II coincide at leading order, a relevant operator is deforming the field theory dual to background II. This deformation is associated with the subleading terms, proportional to the parameters Q_A, Q_B. As we lower the energy (still at strong coupling)
a Kaluza-Klein spectrum of massive modes arises due to the compactification of the branes on φ.
At energies around the scale set by r_+, the QFT dual to the background in eq.(<ref>) should be a (1+1) dimensional QFT. This should be the reduction of the Yang-Mills-Chern-Simons SU(N_A)_N_B× SU(N_B)_N_A to (1+1), preserving four supercharges. The QFT is expected to be gapped, confine and break part of the R-symmetry.
In what follows, we start the study of this interesting field theory. We do so by calculating observables of the two-dimensional QFT using probes of the background in eqs. (<ref>)-(<ref>).
These probes inform us about gauge couplings, theta-angles, symmetry breaking, confinement, etc. We present a quantity that indicates the number of degrees of freedom (density of states in terms of the energy).
§.§ Gauge coupling
To study the background using D-branes probes, it is first useful to set our conventions for the Dirac-Born-Infeld-Wess-Zumino (DBIWZ) action, describing the dynamics of branes in our background.
Consider a ten dimensional manifold M_10 equipped with a metric tensor G_μν, Neveu-Schwarz two form B_μν, dilaton Φ and Ramond potentials encoded in the poly-form C. In this space, there is an embedded manifold Σ of dimension (p+1) with (9-p)
space-like normal vectors. This embedded manifold hosts a Dp-brane. We denote the coordinates on the
Dp-brane as X^M with M=0,1,… ,p. and the induced metric g_MN.
The action of a single Dp-brane is the Dirac-Born-Infeld-Wess-Zumino action
given by
S_Dp, DBI[ g_MN, F_MN] =T_p∫ d^p+1xe^-Φ√(-( g_MN+ F_MN) ) .
S_Dp, WZ[ C, F_MN]= - T_p∫_Σ C∧ e^- F_MN
Here F_MN= B_MN +2πα' F_MN, where B_MN is the pull-back of the background Neveu-Schwarz two-form on Σ, and F_MN is an Abelian gauge field strength defined on the brane. The tension of the Dp brane is T_p=1/(2π)^7-p, in our chosen units. For the case of our backgrounds, we have B_MN=0 and the poly-form C= C_2 given in eq.(<ref>), and/or its electric dual C_6, given in Appendix <ref>.
With our choice of units (g_s=α'=1) we can perform a small field F_MN expansion of the action in eq.(<ref>), equivalent to a small-α' expansion. We obtain an effective action for
the Dp-brane,
S_Dp, DBI=T_p∫ d^p+1xe^-Φ√(- g_MN)[ 1-1/4
( 2π) ^2F^BCF_BC+𝒪(F^3) ]
where F^BC=g^BMg^CNF_MN. The Wess-Zumino part of the action in eq.(<ref>) contains a finite number of terms.
Specialising for our backgrounds with B_MN=0 we have for any D_p brane probe,
S_Dp,WZ=
- T_p∫_Σ_p+1 C_p+1 - 2π C_p-1∧ F_2 +(2π )^2/2 C_p-3∧ F_2^2 -(2π)^3/6 C_p-5∧ F_2^3+(2π)^4/24C_p-7∧ F_2^4.
In what follows we study the backgrounds in eqs. (<ref>), (<ref>)-(<ref>) with various probe branes in Type IIB.
The first probe is a D5 brane that extends on the directions [t,x,φ, θ_A,ϕ_A,ψ_A]. This is like a probe that extends where the A-stack originally was. We will switch on an electric field on its worldvolume.
Calculation in the Background I
Let us start by performing the probe calculation in the background of eq.(<ref>). The dual QFT is (2+1)-dimensional, as the coordinate φ is not compact. By expanding the Born-Infeld action, we find a Maxwell term, with coupling (the details of this calculation are spelled out below)
1/g_YM,A^2= 16π^4/e_A^3T_5 √(e_A^2+ 2 e_B^2).
Similarly, the Wess-Zumino term gives,
S_D5,WZ= -2π^2 T_5∫ C_2∧ F_2∧ F_2=- ∫_S^3_A F_3 ∫_t,x,φ A_1∧ F_2= -N_B ∫_t,x,φ A_1∧ F_2.
We have performed an integration by parts, used the quantisation condition in eq.(<ref>) and set T_5=1/(2π)^2 (in our units).
There is a similarly symmetric calculation for a D5 probe along [t,x,φ, θ_B,ϕ_B,ψ_B]. In agreement with the field theory picture discussed above, we find two gauge groups with Yang-Mills Chern-Simons dynamics, SU(N_A)_N_B× SU(N_B)_N_A, with fixed gauge couplings. This is exactly in agreement with the field theory expectations <cit.>, that we summarised in the previous section.
Calculation in the Background II
Let us now study the case for which the (2+1) QFT has been compactified along the φ-direction and we are dealing with a (1+1) dimensional QFT.
We work with the background in eqs.(<ref>)-(<ref>), and follow the calculation above, by first writing the induced metric on the probe D5 brane,
ds_D̂_5^2=r{ -dt^2+dx^2+[ f_s( r)
+2Q_B^2ζ( r) ^2] dφ ^2+2/e_A^2
[ ω̂_1^2+ω̂_2^2+( ω̂
_3-e_AQ_Aζ (r)dφ) ^2] } .
From here we calculate
e^-Φ√(- g_MN^( D̂_5) )=r^2( 2/
e_A^2) ^3/2sinθ_A√(f( r)
+2Q_B^2ζ( r) ^2).
The effective action
for the brane in eq. (<ref>) reads,
S_D̂_5, BI = -T_D_5∫ dt dx dφ dθ_Adϕ_Adψ_A( 2/e_A^2) ^3/2r^2sinθ_A√(f( r) +2Q_B^2ζ( r) ^2)(1-(2π)^2/4 F^BCF_BC) ,
=
T_D_5L_φ(
4π) ^2( 2/e_A^2) ^3/2r^2√(f(
r) +2Q_B^2ζ( r) ^2)∫ dtdx(1 -(2π)^2/4
F^BCF_BC) .
We turn on F_tx so that
F^BCF_BC=2F_txF_txg^ttg^xx=21/r^2F_txF_txη
^ttη ^xx=1/r^2F_μν^2 .
From here we identify the Yang-Mills coupling for this probe D5,
1/g_YM,A^2=( 2π)^4T_D̂_5 L_φ( 2/e_A^2) ^3/2√(f( r) +2Q_B^2ζ( r) ^2).
If the D5 brane probes the SUSY preserving background, we impose Q_B=e_B/e_AQ_A and m =0 on the parameters appearing in eqs.(<ref>)-(<ref>). This implies r_±^2=±2Q_A/e_A and L_φ=4π
/(e_A^2+e_B^2). Together with eq.(<ref>) and the quantisation condition (<ref>), the gauge coupling reduces to[For large values of the radial coordinate r and decompactifying φ, this result reduces to that in eq.(<ref>). ]
1/ĝ_YM,A^2= 16π^4/e_A^3 T_D̂_5L_φ√(( 1-2Q_A/
e_Ar^2) ( 2e_AQ_A/r^2+e_A^2+2e_B^2) ) ,
with limiting values
1/ĝ_YM,A^2={[ 16 π^4/e_A^3T_D̂_5L_φ√(e_A^2+2e_B^2) , r→∞; 0 , r→ r_+ ]. .
In other words, the gauge coupling grows very large at low energies and asymptotes to a constant value for high energies. As discussed, at very high energies the field theory is best described in terms of a Little String Theory (LST). This calculation above refers to the gauge coupling g_YM,A. Other interactions in the QFT may become large at high energies, in such a way that the field theory is strongly coupled in the UV. This is in agreement with the background in eq.(<ref>) being weakly curved for all values of the radial coordinate [r_+,∞).
Had we studied a D5 probe extended along [t,x,φ, θ_B,ϕ_B,ψ_B], with an electric field F_tx switched on the brane, the result would be,
1/g_YM,B^2=( 2π)^4T_D_5 L_φ( 2/e_B^2) ^3/2√(f( r) +2Q_A^2ζ( r) ^2).
§.§ Theta angle
From the viewpoint of the (2+1) dimensional QFT, represented by Background I in eq.(<ref>), we can consider dimensionally reducing the Chern-Simons term obtained in eq.(<ref>). We then obtain a theta-term proportional to N_B ∮_φ A_φ for the QFT on the A-stack.
For the (1+1) viewpoint, additional probes calculate the Θ-angle of each gauge group. Let us use a D3 probe, extended along
[t,x,θ_A,ϕ_A] with an electric field F_tx switched on. We study the Wess-Zumino term following eq.(<ref>) and using the two-form potential pulled-back on this D3 probe
C_2|_D_3=-2/e_A^2ψ_Asinθ_Adθ_A∧ dϕ_A .
Replacing this C_2 in eq.(<ref>) and using that F_2= F_tx dt∧ dx we find the Wess-Zumino term for this probe is (note that C_4=0 in the background),
S_WZ,D3=-T_3 2π∫ C_2∧ F_2 = T_3 16 π^2 /e_A^2ψ_A∫ F_txdt ∧ dx .
The Θ-angle associated with the gauge group should be identified according to ,
S_WZ,D3= Θ_A/4π^2∫ dt dx F_tx, ⟶ Θ_A =T_D_364π^4 /e_A^2ψ_A= ψ_A N_B/2 .
We have used 16 π^4 T_D3=1 (in our units) and the quantisation condition in eq.(<ref>). Notice that the periodic identification Θ_A∼Θ_A + 2 k π implies that the angle ψ_A gets quantised to the values
Δψ_A= 4 k π/N_B, with k=0,1,2...., N_B-1.
For k=N_B we have Δψ_A=4π, covering the full circle.
Had we considered the D3 probe extended along [t,x, θ_B,ϕ_B] with all other coordinates fixed, we would have found (the calculation is identical),
Θ_B = ψ_B N_A/2.
Another way of understanding the R-symmetry breaking would be to consider euclidean D1 branes wrapping [θ_A,ϕ_A].
The Wess-Zumino action contributes to the partition function via
Z_D1∼ e^i/2π∫ C_2.
This contribution should not depend on a (large) gauge transformation parameter ϵ_A, that appears as we change ψ_A→ψ_A+ϵ_A. We enforce
(i/2π) (- 2ϵ_A/e_A^2)∫sinθ_A dθ_A dϕ_A= 2 i k π , leads to ϵ_A= 4 k π/N_B.
These results indicate that the background's continuous isometries transforming ψ_A,B→ψ_A,B + ϵ_A,B, with (ϵ_A,B being constants), are actually broken. In fact, the allowed changes are Δψ_A,B= 4 k π/N_B,A, with k=0,1,2...., N_B,A-1, which should be interpreted as the breaking of the two field theory global symmetries U(1)_A,B into discrete subgroups.
The argument used to derive eqs.(<ref>)-(<ref>) is not airtight. It uses a two manifold that has a boundary and at the same time a gauge choice is made for the potential C_2 in eq.(<ref>). It would be better to have an argument for R-symmetry breaking that is explicitly gauge invariant. In the coming section we present a different holographic perspective on the U(1)_A,B breaking with this property.
§.§ U(1)-A,B symmetry breaking pattern
Some supersymmetric field theories exhibit a classical U(1) R-symmetry that is quantum mechanically broken to a discrete subgroup. The symmetry breaking can be understood diagrammatically or in terms of instantons. The supergravity dual to the given field theory should encode this, but the mechanism should not involve instantons (as that are very suppressed in supergravity). The fact that the Ramond potentials are not gauge invariant under the U(1) R-symmetry is key. The breaking of the global R-symmetry in the field theory manifest as spontaneous breaking in supergravity. The vector field in the bulk, dual of the R-symmetry current acquires a mass. We find this below for our background of eq.(<ref>). The argument that follows is gauge invariant at all steps. All along this section, we set m=0 and focus only in the BPS case.
The U(1) symmetry of the metric is perturbed, the Lagrangian for this fluctuation is described by the usual F_μν^2-term. For the perturbation to be consistent, the Ramond fields must be also perturbed, this contributes to the mass term for the fluctuation. The massive gauge field is understood as symmetry breaking in supergravity.
In the previous section, we hinted at a breaking of the isometries represented by the Killing vectors ∂_ψ_A,B. To better understand the breaking of the field theory global U(1) symmetries associated with the translations in ψ_A,B we proceed as explained in <cit.>, <cit.>. We give full details in Appendix <ref>. In the holographic background, we gauge the isometry by replacing, both in the metric and in the Ramond potential of eq.(<ref>)
dψ_A,B→ dψ_A,B + A_A,B, ψ_A,B→ψ_A,B + ϵ_A,B.
Such that a change ψ_j →ψ_j + ϵ_j is compensated by A_j→ A_j + d ϵ_j.
We then study the Lagrangian for the gauge fields A_A,B, by replacing these changes in the string frame Lagrangian.
The metric perturbation changes the Ricci scalar to
e^-2Φ R → e^-2Φ(R -1/42r/e^2_AF^2_A
-1/42r/e^2_BF^2_B).
The kinetic term for F_3 changes as,
1/12F^μνλF_μνλ→ 1/12F^μνλF_μνλ
+ 1/24Q^2_A +e^2_A r^4/e^2_Ar^6(A_A-dϵ_A)^2
+ 1/24Q^2_B +e^2_B r^4/e^2_Br^6(A_B-dϵ_B)^2
+4Q_AQ_B/e_Ae_Br^6(A_A-dϵ_A)·(A_B-dϵ_B).
These perturbations around an isometry direction are usually known to be consistent. It is interesting to note that the kinetic term for the gauge field A_(A,B) comes only from the Ricci scalar. The information of ϵ_A,B on the other hand, comes only from F^2_3. Defining
W_(A,B) = A_A,B - dϵ_A,B,
and imposing the BPS condition e_AQ_B= e_BQ_A, the Lagrangian for the perturbation reads,
ℒ = -1/42/r e^2_AF^2_A
-1/42/r e^2_BF^2_B
+1/2r^2 (W^(A)_μW^(A)μ+W^(B)_μW^(B)μ)
+1/24Q^2_A/e^2_Ar^6(W^(A)_μ+ W^(B)_μ)^2.
We interpret this result as follows. There are two U(1) global symmetries in the QFT, holographically they are represented by the invariance of the metric under changes in ψ_A and ψ_B. These global symmetries are broken to discrete groups ℤ_N_B and ℤ_N_A, as indicated by eq.(<ref>). This breaking is an effect of the lack of invariance of the gauge potential C_2. The breaking of the global symmetries is addressed in this section without appeal to the Ramond potentials, by observing that gauging the metric isometries ∂_ψ_A,B leads to a breaking of the gauge symmetry, by a mass term. These mass terms are dependent on the radial coordinate.
Contrary to what happens for the duals to N=1 SYM, the metric in eq.(<ref>) does not break these discrete isometries to ℤ_2. In other words, there is not a radial-regime in the metric that explicitly breaks the isometry ∂_ψ_A,B. We interpret this result as the (presumably anomalous) breaking of the two U(1)_A× U(1)_B→ℤ_N_B×ℤ_N_A in the QFT not being followed by a further spontaneous breaking. One might argue that VEVs are not allowed in a two dimensional QFT, hence no further breaking can take place by the formation of a condensate. This argument is not completely rigorous, as our QFT is two dimensional in the far IR, but get UV completed around the confining scale to a higher dimensional QFT.
We study now a different observable that gives an approximate idea of the number of degrees of freedom as a function of the energy (a density of states).
§.§ Holographic central charge
Consider a generic holographic background dual to a QFT in (d + 1) spacetime dimensions, with metric and dilaton given by
ds^2 = a(r,y^i) [ dx^2_1,d+b(r) dr^2]+ g_ij(r,y^i)dy^i dy^j, Φ(r,y^i).
Following <cit.>, we define quantities V_int, H according to
V_int= ∫ dy^i √(e^-4 Φ a(r, y^i)^d [g_ij]), H= V_int^2.
From these we define the holographic central charge (or free energy),
c_hol = d^db(r)^d/2 H^2d+1/2/G^(10)_N (H')^d,
where G^(10)_N = 8π^6 is (in our conventions), the ten-dimensional Newton constant.
The holographic central charge of eq.(<ref>) makes perfect sense for backgrounds with an AdS-factor. In those cases the quantity in eq.(<ref>) is a number depending on the parameters of the background and it was successfully matched with the free energy of the CFT. In contrast for our case, without an AdS-factor we use eq.(<ref>) to give an indication of the number of degrees of freedom of the QFT.
Let us first compute the quantities in eqs.(<ref>)-(<ref>)
for the solution in eq.(<ref>), dual to a (2+1) dimensional QFT. We find,
d=2, a(r, y^i)=r, b(r)= 8/(e_A^2+e_B^2) r^2, V_int= N r^3, N= 8 (4π)^4/e_A^3e_B^3,
H= N^2 r^6, c_hol= 8 N/9 G_N (e_A^2+e_B^2) r^3.
This quantity diverges at large energies, hinting at a UV completion in terms of a system in higher dimensions (a LST). Also, it vanishes for r=0, indicating a gapped system.
Notice nevertheless that the calculation should not be trusted close to r=0, as the background (<ref>) is singular there.
Let us now calculate for the background in eq.(<ref>), as a dual to a (1+1)-dim QFT. We find
d=1, a(r,y^i)= r, b(r)= 4/r^2 f_s(r),
V_int= N r^2√(f_s(r)), Ĥ= N^2 r^4 f_s(r), N=(4π)^48/e_A^3 e_B^3 L_φ,
c_hol= N/2 G_Nf_s(r) r^2/( f_s(r) +r/4 f'_s(r) ).
At high energies
the number of degrees of freedom grows unbounded (as r^2), signalling the UV completion in terms of a decompatified QFT in higher dimensions. At very low energies the number of degrees of freedom vanish, as f_s(r_+)=0. In this case, the calculation is trustable, hence the gapped character of the system is clear.
A related calculation can be done that encodes the fact that we can think our field theory as a three dimensional QFT with anisotropies (or a QFT with a flow across dimensions). We follow the treatment described in section 8.2 of the paper <cit.>, see also <cit.>. Using the notation of <cit.> we have,
d=2, α_0=α_1=r, α_2=r f_s(r), β= 4/r^2 f_s^3/2, H= N^2 r^4 f_s(r), N=(2/e_A e_B)^3 (4π)^4,
c_flow= N/G_Nr^2 f_s/(f_s(r)+ r/4 f_s(r)')^2
.
The result in eq.(<ref>) has a similar interpretation. At low energies we have no degrees of freedom, at high energies an unbounded growth in the degrees of freedom indicates the UV completion. Note that the growth at high energies for the flow-central-charge–eq.(<ref>), is slower than it is for the case in which we consider the QFT to be two dimensional, see eq. (<ref>). This same feature occurs when considering flows between conformal points in different dimensions.
These results suggest that our QFT generates a mass gap at low energies. The behaviour of the gauge couplings at low energies, see eq.(<ref>) also suggest that the QFT is confining. To ascertain the confining behaviour we calculate Maldacena-Wilson and 't Hooft loops, that provide order parameters for confinement. We also investigate the Entanglement Entropy (EE), which gives information about the interplay between a confining IR and the non-local UV dynamics of the QFT.
§ MALDACENA-WILSON, 'T HOOFT LOOPS AND ENTANGLEMENT ENTROPY
In this section we calculate different observables to learn more about the proposed field theory.
We start calculating the Maldacena-Wilson loops <cit.>, in order to test the above proposal that at low energies the QFT presents a mass gap and confines. We start with a summary of the formalism to compute Maldacena-Wilson loops. This same formalism is then adapted for the study of 't Hooft loops and Entanglement Entropy.
§.§ General comments on Maldacena-Wilson loops and similar probes
We start summarising general results pertaining holographic Wilson loops. We follow the treatment of <cit.>, <cit.>. This generic treatment is also useful for the study of other probes that reduce to an 'effective string' in the background. Hence it will apply to 't Hooft loops, Entanglement Entropy, as we discuss below.
Consider a generic holographic background of the form
ds^2=-g_ttdt^2+g_xxdx⃗^2+g_rr
dr^2+g_ijdθ ^idθ ^j .
We assume that g_tt, g_xx, g_rr depend only on the radial coordinate r. As usual, we propose a string embedding (parametrised in terms of (τ,σ) the worldsheet coordinates), which leads to a Nambu-Goto action for the F1-string of the form,
t=τ, x=x(σ), r=r(σ).
S_NG= T_F1∫ dτ dσ√(g_tt(r) g_xx(r)x'^2 + g_tt(r)g_rr(r) r'^2).
From this action, the equations for the string moving in the generic background reduce to (see <cit.> for a detailed derivation)
dr /dσ=±dx/dσV_eff( r ) .
We defined the effective potential
V_eff( r ) =F( r ) /CG( r
) √(F^2( r ) -C^2) , F^2( r ) =g_ttg_xx, G^2( r )=g_ttg_rr.
The constant C=F^2x'/√(F^2 x'^2+ G^2 r'^2) is obtained from one of the equations of motion. In the simpler case in which we take x(σ)=σ we find eq.(<ref>) from the conserved Hamiltonian. In that case C= F(r_0), being r_0 the turning point of the string satisfying r'(σ)=0 (these are called U-shaped embeddings). We set C=F(r_0) in what follows. We enumerate below a set of properties of the U-shaped embeddings.
* This formalism applies to an open string whose end points are at r →∞, where we add a D-brane. Dirichlet boundary conditions for the string at r →∞ require that V_eff|_r →∞∼∞.
* We compute the separation between the two ends of the string on the D-brane, which can be thought as the separation between a quark-antiquark pair. The energy of the pair of quarks calculated from the Nambu-Goto action needs regularisation, implemented by subtracting the mass of two non-dynamical strings extended along the whole range of the radial coordinate [r_+, ∞).
The separation and energy are given as functions of r_0 (the distance from the origin of the radial coordinate, r_+, to the position of the turning point of the string). The expressions for these quantities are,
L_QQ( r_0) =2∫_r_0^+∞dz/V_eff(z) ,
E_QQ( r_0) =F( r_0) L_QQ( r_0)
+2∫_r_0^+∞dzG( z) /F( z) √(F( z) ^2-F( r_0) ^2)-2∫_r_+^+∞dz G( z) .
*
To obtain a finite contribution coming from the upper limit of the QQ̅ pair separation in (<ref>), a further restriction on the behaviour of the effective potential at infinity is needed. See <cit.> for a derivation,
V_eff|_r → +∞∼ r^β , with β >1 .
*
When expanded close to the end of the space, which is at r=r_+ in the case of the background in eq.(<ref>), we find V_eff∼ (r-r_+)^γ. If 1 ≤γ, the separation between the pair becomes infinite–see <cit.>. Otherwise (if γ<1) we have screening behaviour.
*
There is an analytic relation between E_QQ and L_QQ which is
dE_QQ/dr_0=F(r_0) dL_QQ/d r_0 ⟶ dE_QQ/dL_QQ=F(r_0).
We have inverted the relation (<ref>) as r_0=r_0(L_QQ).
*
For generic backgrounds, the evaluation of the integral in eq.(<ref>) need not be simple nor have an expression in terms of elementary functions. Nevertheless, the quantity
L̂_QQ(r_0)= πG/F'|_r_0,
provides a reasonable approximation to eq.(<ref>).
We check this approximate expression for different observables studied below.
*
Following <cit.> we define
Z(r_0)= d/dr_0L̂_QQ(r_0)= πd/dr_0( G(r_0)/F'(r_0)).
The stability of the U-shaped string embedding in eq.(<ref>) is guaranteed if Z(r_0)<0 <cit.>. This is valid for any observable that can be reduced to an effective string action of the form (<ref>)[
It was shown in <cit.>,<cit.> that embeddings for which Z(r_0)≥ 0 do not satisfy two physically well motivated criteria <cit.>:
The force between the quark and the antiquark is always attractive and positive dE_QQ/dL_QQ>0. It is also a non-increasing function of the separation d^2 E_QQ/d L_QQ^2≤ 0.
The proposal in <cit.> is that the two criteria above are equivalent to the stability of the U-shaped embedding or conversely Z(r_0)<0.]
*
In calculations like those in eqs.(<ref>)-(<ref>) we can introduce a cutoff r_UV to regulate divergences coming from the upper limit in the integrals. We
define a quantity analog to (<ref>),
L_QQ(r_0,r_UV)= 2∫_r_0^r_UVdz/V_eff(z).
Following <cit.> we calculate
L_a= lim_r_0→∞lim_r_UV→∞ L_QQ(r_0, r_UV) and L_b= lim_r_UV→∞lim_r_0→ r_UV L_QQ(r_0, r_UV).
Whilst L_b=0 by definition, it is sometimes the case that L_a is nonzero.
If in this case the U-shaped configuration is unstable (Z(r_0)>0), there exist 'short configurations' (that appear very close to the cutoff). These short configurations are energetically favoured and induce a phase transition in the observable calculated <cit.>, <cit.>.
The existence of these short configurations that appear when introducing the UV-cutoff is not an artefact of the cutoff. They indicate that the string embedding we proposed in eq.(<ref>) is not capturing the dynamically favoured configuration.
In the study of the 't Hooft loop and the Entanglement Entropy we encounter these short configurations, that cure the problem of the instability of the embedding and introduce a phase transition in the observable. We take the instability of the configuration (cured by the introduction of a cutoff) as an indication that the dynamics of the LST is driving the observable.
Let us now apply this general treatment to our background in eq.(<ref>).
§.§ Maldacena-Wilson loops in our background
Let us apply the expressions in eqs.(<ref>)-(<ref>) to our background in eq.(<ref>).
The relevant functions are
F(r)=r, G(r)=2/√(f_s(r)), V_eff(r)= √(e_A^2+e_B^2/8)( 1/r_0 r) √((r^2-r_0^2)(r^2-r_+^2)(r^2-r_-^2)).
The condition in eq.(<ref>) to have Dirichlet boundary conditions for the string at r→∞, is satisfied. When expanded close to r_0=r_+, V_eff∼ (r-r_+), indicating that L_QQ diverges when r_0 approaches the end of the space r_+. This points to confining behaviour.
Another quick way of determining a confining behaviour is to check the function F(r_+), that intuitively represents the tension of a QCD-string at low energies in the QFT (close to the end of the space of the geometry). In this case we find F(r_+)=r_+, pointing to confining behaviour. In fact, a finite QCD-string tension leads to an energy growing with the separation of the quark pair.
Note also that the approximate formula for the separation of the quark pair eq.(<ref>), gives for the functions in eq.(<ref>)
L̂_QQ(r_0)=2 π/√(f_s(r_0)).
For r_0→∞ this gives L̂_QQ= π√(8/e_A^2+e_B^2), which refers to a characteristic length of the UV completion (the Little String Theory scale). On the other hand, for r_0∼ r_+, the approximate length diverges and 1/√(r_0-r_+). This indicates that the quark-anti-quark pair can be infinitely separated. According to eq.(<ref>), this gives an energy that scales linearly with the separation, another signal of a confining behaviour.
The quantity Z(r_0) in eq.(<ref>) reads,
Z(r_0)= 2√(2)π r_0/√(e_A^2+e_B^2)( -r_0^2(r_+^2+ r_-^2) +2 r_+^2r_-^2)/[ (r_0^2-r_+^2)(r_0^2-r_-^2)]^3/2.
By inspection, one find that Z(r_0)<0 in all the range. Hence the U-shaped embeddings are stable and no phase transition is expected for the Maldacena-Wilson loops.
The rigorous way of determining the low energy behaviour of the QFT (either confining or screening) is to analyse the expressions for the distance and the energy of the quarks pair.
The distance between the quark-antiquark pair and its energy are written from eqs.(<ref>)-(<ref>)
L_QQ( r_0) = √(32 /e_A^2+ e_B^2 ) r_0 ∫_r_0^∞z
√(1/( z^2-r_+^2) ( z^2-r_-^2)
( z^2-r_0^2) )dz ,
E_QQ( r_0) =r_0 L_QQ( r_0) +√(32/e_A^2+ e_B^2 )[
∫_r_0^∞ dz √(z^2( z^2-r_0^2) /( z^2-r_+^2) ( z^2-r_-^2) ) -
∫_r_+^∞ dz z^2/√(( z^2-r_+^2) ( z^2-r_-^2) )] .
The integrals can be performed analytically and expressed in terms of in terms of Elliptic integrals of the first kind. It is interesting to study this in the BPS limit, when m=0 and r_+^2=- r_-^2. See Appendix <ref> for a detailed study. Let us quote the explicit results for the separation and energy of the quark pair in eqs.(<ref>)-(<ref>). Defining
the elliptic integrals,
K(x)=∫_0^π/2dθ/√(1-x sin^2θ), E(x)= ∫_0^π/2√(1 -x sin^2θ) dθ,
we can write the explicit expressions for the separation and energy of the quark-antiquark pair, which read
L_QQ( r_0) =2√(8/e_A^2+e_B^2)
r_0/r_+/√(r_0^2/r_+^2-1)𝐊(
1-r_-^2/r_+^2/1-r_0^2/r_+^2) ,
E_QQ( r_0) = 2r_+√(8/e_A^2+e_B^2)
[ r_0^2/r_+^2/√(r_0^2/r_+^2-1)𝐊
( 1-r_-^2/r_+^2/1-r_0^2/r_+^2) .
. -𝐄( 1-r_-^2/r_+^2/
1-r_0^2/r_+^2) √(r_0^2/r_+^2-1)+C(
r_-/r_+) ] ,
where
C( r_-/r_+) =𝐄( r_-^2/r_+^2)
+λ _-𝐊( 1-r_-^2/r_+^2) +iλ _-
𝐊( r_-^2/r_+^2) -( 1-r_-^2/r_+^2) 𝐊(
r_-^2/r_+^2) .
λ_-^2=-r_-^2/r_+^2.
See Appendix <ref> for a careful derivation of these expressions. We plot these results in Figure <ref>. The various panels of Figure <ref> show some conventional and other less conventional behaviours. First, note that the expression for L̂_QQ in eq.(<ref>) very well approximates the exact expression in eq.(<ref>). As is usual, the concavity of the curve E_QQ(L_QQ) is 'downwards', indicating that the Nambu-Goto string configuration of eq.(<ref>) is stable, as confirmed by the Z(r_0) in eq.(<ref>). Note also that for large separations between the quark pair L_QQ, the energy grows linearly (signalling confinement). What is less conventional is that there is a minimal separation, given by the Little String Theory scale. This indicates that the far UV of the QFT dynamics is not field theoretical (but has the dynamics of the LST). The plot of the strings profiles confirms this. Indeed, strings that barely explore the bulk (with r_0/r_+ large) show a minimal fixed separation between the quark pair. On the other hand, the strings that explore deeper into the bulk display a bigger quark separation and carry higher energy.
Let us now focus on a second interesting observable, the 't Hooft loop. In the next section we propose a string-like object (for the gauge theory observer) with magnetic charge. This characteristically is represented by a Dp brane that wraps a (p-1) cycle in the internal space. Once the Born-Infeld action for this Dp brane is written and integrals over the internal space are performed, we arrive at an action for the 'effective string'. This action is studied with the same formalism as that used for Wilson loops, described in Section <ref>.
§.§ 't Hooft loops
The 't Hooft loop can be calculated by proposing an object with magnetic charge that effectively appears like a string for the (1+1) dimensional QFT. The ends of this magnetic string appear as a pair of monopoles of oposite charge. To study this object, we adapt the formulas
summarised in eqs.(<ref>)-(<ref>) for the effective magnetic string.
We propose to calculate the 't Hooft loop by studying the effective magnetic string obtained when extending a D5 brane along the directions [t,x,φ,θ_A,ϕ_A,ψ_A], with r(x). There is an analog magnetic string for the second gauge group, for the D5 in the configuration [t,x,φ,θ_B,ϕ_B,ψ_B], with r(x). We do not discuss the latter object, as its result is analog to the one obtained below. Note that this object becomes string-like when we consider the size of the S^1_φ to be small enough.
The induced metric for the above D5 is,
ds^2 = r{ -dt^2+( 1+4/r^2f_s( r)
r^' 2) dx^2+( f_s( r) +2Q_B^2ζ( r) ^2) dφ ^2.
. +2/e_A^2[ ω̂_1^2+ω̂
_2^2+( ω̂_3-e_AQ_Aζ (r)dφ) ^2
] } .
Then we calculate,
e^-Φ√(- g_MN)=( 2/e_A^2) ^3/2r^2sinθ_A√(( 1+4r^' 2/r^2f( r) )
( f( r) +2Q_B^2ζ( r) ^2) ) .
The action for this D5 is,
S_D̂_5[ r] =T_D_5L_φ( 4π) ^2( 2/e_A^2) ^3/2∫ dtdx√((
f_s( r) +2Q_B^2ζ( r) ^2)
r^4+4r^2( 1+2Q_B^2ζ( r) ^2/f_s(
r) ) r^' 2) .
Comparing with (<ref>)-(<ref>) we identify
F^2 =( f_s(r) +2 Q_B^2ζ(r)^2) r^4 , G^2=4/
r^2f_s( r) F^2 , T_eff =T_D_5L_φL_t( 4π) ^2(
2/e_A^2) ^3/2 .
Where T_eff is the effective tension of the magnetic string.
Following eq.(<ref>), the effective potential
V_eff( r) =r^3/2C√(f_s( r) (
f_s( r) +2Q_B^2ζ( r) ^2-C^2/r^4
) ).
The constant C= F(r_0)= √(( f_s(r_0) +2 Q_B^2ζ^2(r_0)) )r_0^2. In the asymptotic region the effective potential in eq.(<ref>) scales as V_eff( r→∞) ∼ r^3, satisfying the condition in eq.(<ref>).
The intuitive criteria discussed in the previous section applied for this case indicate a screening behaviour. In fact F(r_+)=0 and V_eff∼(r-r_+)^1/2, signalling a vanishing tension of the magnetic QCD-string in the IR, and a finite maximal separation, after which the monopoles are screened (more about this is discussed below).
We perform a similar analysis using the approximate expression for the separation between the monopole-anti-monopole pair L̂_MM. Replacing in eq.(<ref>) the functions in eq.(<ref>), we find an involved expression that asymptotes as,
L̂_MM(r_0→∞)∼π√(2/e_A^2+ e_B^2), L̂_MM(r_0→ r_+ )∼ 8π√(r_+(r_0-r_+)/(e_A^2+e_B^2)(r_+^2-r_-^2)).
These asymptotic behaviours indicate that at high energies in the field theory, the pair is separated by a maximum distance characteristic of the UV completion (Little String Theory). In this sense, the magnetic string behaves oppositely to the electric one used to compute the Wilson loop (which shows a minimum separation). On the other hand, at low energies the separation decreases to zero (again, oppositely to the electric string case). We calculate Z(r_0) in eq.(<ref>) for this configuration and find that is positive in all the range. This indicates the instability of the U-shaped embeddings. A more general embedding that the one proposed here should drive the dynamics. Instead of finding this more complicated embedding, below we introduce a UV-cutoff. New short configurations appear close to a cutoff, that dominate the dynamics and produce a phase transition (to a deconfining behaviour).
More formally, we write expressions for the separation between the monopole-anti-monopole pair as those in eqs.(<ref>)-(<ref>). A careful analysis of these integrals is performed in Appendix <ref>. Let us quote the exact expression for the separation for the monopole-anti-monopole pair. We leave the study of the expression for the energy between the pair of monopoles for Appendix <ref>. In terms of the elliptic integral of the first kind
F(y| x)= ∫_0^y dθ/√(1-x sin^2θ),
and the definition for 𝐊 (x) in eq.(<ref>), working in the BPS limit and defining η =e_A/e_B, we find
L_MM^BPS( r_0) = 2/e_B√(2(
r_0^2/r_+^2-1) ( ( η ^2+2)
r_0^2/r_+^2+η ^2) /( η ^2+1) (
r_0^2/r_+^2( η ^2+2) -1) )
×[ 𝐅( . arcsin√(2(
-1+( 2+η ^2) r_0^2/r_+^2) /(
2r_0^2/r_+^2+η ^2( 1+r_0^2/r_+^2) ) )
|( 1+r_0^2/r_+^2) (
2r_0^2/r_+^2+η ^2( 1+r_0^2/r_+^2) ) /
-4+4( 2+η ^2) r_0^2/r_+^2) .
. +i𝐊( 1-(
1+r_0^2/r_+^2) ( 2r_0^2/r_+^2+η ^2(
1+r_0^2/r_+^2) ) /-4+4( 2+η ^2)
r_0^2/r_+^2) ] .
In Figure <ref> we compare this exact expression with the approximating function whose asymptotics we write in eq.(<ref>). As stated above, opposite to the Wilson loop case, we encounter a maximal separation for the monopole pair. This maximal separation is associated with the Little String Theory scale. We also plot the string profiles as they enter the bulk. We observe a different behaviour to that found in the case of the Wilson loop–compare with the lower right panel of Figure <ref>.
For magnetic strings that barely explore the bulk, the separation between the monopole pair is large (equal to the LST scale). As we decrease the separation between the monopole pair, the magnetic string dives into r_0→ r_+. These unconventional behaviours, together with the instability of the string embedding–note that E_MM(L_MM) has upwards concavity (and Z(r_0) is positive), indicate the presence of a second 'disconnected' configuration for which the pair of monopoles separate without energy expense, which is indicative of screening. The transition to a disconnected configuration is dynamically favoured.
To avoid the 't Hooft loop to be driven by the UV (LST) dynamics and to realise explicitly these short configurations, we introduce a hard cutoff in the radial direction and recalculate things. Doing so, the behaviour changes qualitatively.
In fact, after introducing the UV cut-off, the separation between the pair of monopoles does not show a maximum value, instead we find a 'double valued' behaviour as displayed in the left panel of Figure <ref>.
This leads to a phase transition in the curve E_MM(L_MM). Note that this curve now has the correct 'downwards' concavity, indicating that the configuration is stable. This phase transition is the physical manifestation of the magnetic string suddenly changing into two disconnected magnetic strings that move without energy expense. This is deconfinement for the pair monopole anti-monopole. We are finding confinement for the quark-anti-quark pair and screening for a pair of monopoles. These behaviours are consistent with the (electric) confining behaviour of the dual QFT.
The introduction of the UV cutoff might seem unsatisfactory. Here we use it as a device to show that the correct five brane embedding must be more elaborated than the one we proposed above. It is also used to avoid the LST overtaking the dynamics. The effect of the cutoff is clear considering the integral needed to calculate L(r_0). This integral vanishes for r_0→ r_MAX. This produces a double-valued L(r_0) and a consequent phase transition.
For the Entanglement Entropy a very similar behaviour occurs. We study this next.
§.§ Entanglement Entropy
The Entanglement Entropy (EE) between two regions for field theories with a string dual can be calculated as shown in <cit.>. The method is to find a minimal area eight-surface (Σ_8, a codimension-two surface to which we refer below as RT surface) such that the boundary of the surface coincides with the two entangled regions. We focus on the case in which one of the regions is a strip of size L_EE and the other region is the complement.
The EE between these regions is given in <cit.>, <cit.>, minimising the quantity
S_EE=1/4 G_N∫_Σ_8 d^8σ√(e^-4Φ[g_Σ_8]).
There are various eight-surfaces that minimise S_EE in eq.(<ref>). Due to this, in some cases there is a phase transition between different extremal surfaces.
It was suggested in <cit.> that a criterium for confinement is the presence of a phase transition in the EE. This proposal was critically analysed in <cit.>. It was found that for the case of field theories that confine, but have a non-local high energy behaviour, the phase transition in the EE is absent. The point is subtle, as introducing a UV cutoff or UV-completing the QFT to avoid the non-locality, recovers the phase transition.
It is in this way that the EE can serve as an order parameter for confinement, but also as a tool to diagnose non-locality in the UV-behaviour of the QFT (when used together with a confining Wilson loop).
In <cit.> it was found that introducing a UV-cutoff implies the existence of new configurations realising the phase transition (and resolving a stability issue with the original eight-surface). Below, we perform an analysis of these features in our background of eq.(<ref>).
We follow the approach of <cit.>, in particular the treatment for non-AdS backgrounds developed in <cit.>, <cit.>.
We calculate the Entanglement Entropy on a strip by computing the area of an eight-surface [x,φ, θ_A,ϕ_A,ψ_A,θ_B,ϕ_B,ψ_B] with r=r(x) in the background of eq.(<ref>).
The induced metric on the RT eight-surface, its determinant and the Entanglement Entropy are,
ds^2_st = r{ dx^2(1+4 r'^2/r^2 f_s(r)) +f_s( r) dφ ^2+2/e_A^2[ ω̂_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ
(r)dφ) ^2] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3^2-e_BQ_Bζ( r)
dφ) ^2] } ,
√(e^-4Φ[g_8]) = ( 8/e_A^3e_B^3)√(r^4 f_s(r) + 4 r^2 r'^2)sinθ_Asinθ_B.
S_EE = 1/4 G_N∫ d^8x √(e^-4Φ[g_8])=( 2 (4π)^4 L_φ/e_A^3e_B^3 G_N)∫_-L/2^L/2 dx √(r^4 f_s(r) + 4 r^2 r'^2).
From eqs.(<ref>)-(<ref>), this implies
F(r)= r^2√(f_s(r)), G(r)= 2 r.
To minimise the S_EE above, we follow the usual conserved Hamiltonian treatment. The Entanglement Entropy needs to be regularised by the area of two eight-surfaces that hang straight from infinity. Then, computing the regulated area for a surface that turns around at r_0, we find for the length of the interval and the Entanglement Entropy,
L= 4 r_0^2 √(f_s(r_0))∫_r_0^∞dr/√(r^2 f_s(r) ( r^4 f_s(r) - r_0^4 f_s(r_0) )),
S_EE= N/G_N[∫_r_0^∞√(r^6 f_s(r)/r^4 f_s(r) -r_0^4 f_s(r_0) ) dr -∫_r_+^∞ r dr ].
As in eq.(<ref>), we can write a simple expression that approximates L_EE in eq.(<ref>)—see <cit.>,
L̂_EE= π G(r_0)/F'(r_0)=2πH(r)√(β(r))/H'(r)|_r_0, with H(r)= N^2 r^4 f_s(r), β(r)=4/r^2 f_s(r).
Using eq.(<ref>) we find
L̂_EE= (π√(8/e_A^2+e_B^2))
√((r_0^2-r_+^2)(r_0^2-r_-^2))/(2r_0^2-r_+^2-r_-^2) .
This function is monotonous, going from a vanishing value at r_0=r_+ to a constant value at r_0→∞. This behaviour prevents the possibility of phase transitions, which require that for a given L_EE there are two possible values of r_0.
In fact, the conditions for the presence of a phase transition (see section 2.4 of the work <cit.>) are not satisfied, in particular equations (2.26)-(2.29) of <cit.> imply j=2 preventing a phase transition. The absence of a phase transition in a confining model was interpreted in <cit.> as an effect of the non-locality of the completion of the QFT, in this case, by a LST.
Since Z(r_0) defined in eq.(<ref>) gives
Z(r_0)= √(8)π (r_+^2-r_-^2)^2/r_0 f_s(r_0) (r_+^2+r_-^2-2r_0^2)^2>0,
the proposed embedding is unstable. Upon the introduction of a cutoff, new surfaces appear as found in <cit.>, <cit.>. These cure the instability problem of the embedding and give place to the phase transition, in agreement with confinement.
The treatment in the papers <cit.>, <cit.> applies to our background, even when the IR dynamics is different, the UV dynamics is similarly driven by a Little String Theory.
Before discussing the presence (or absence) of phase transitions we write the analytic expressions for the values of the separation between the two entangled regions L_EE(r_0) and the Entanglement Entropy S_EE(r_0). These expressions are explicitly derived in Appendix <ref>. Using the definitions in eqs.(<ref>),(<ref>) and recalling that λ_-^2=-r_-^2/r_+^2, we find
L_EE( r_0) = 2√(8/e_A^2+e_B^2)√(
r_0^2/r_+^2-1/r_0^2/r_+^2+λ _-^2)[ i
𝐊( ( r_0^2/r_+^2-1) ^2/(
r_0^2/r_+^2+λ _-^2) ^2) .
. +𝐅( . arcsin√(
r_0^2/r_+^2-r_-^2/r_+^2/1-r_-^2/r_+^2)|( 1-r_-^2/r_+^2) (
-r_-^2/r_+^2+2r_0^2/r_+^2-1) /(
r_0^2/r_+^2-r_-^2/r_+^2) ^2) ] ,
and
S_EE^BPS( r_0) =𝒩/G_Nr_+^2[
r_+^2/2r_0^2( -r_0^4/r_+^4𝐄(
r_+^4/r_0^4) -𝐊( r_+^4/r_0^4)
+r_0^4/r_+^4𝐊( r_+^4/r_0^4) ) +
1/2] .
To analyse these expressions, it is useful to show some plots. First, we check that the approximate L̂_EE(r_0) in eq.(<ref>) approximates well the analytic expression in eq.(<ref>), see the left panel of Figure <ref>.
We also plot S_EE(r_0), see the right panel of Figure <ref>. The plot of S_EE in terms of L_EE in the left panel of Figure <ref>, shows an upwards concavity indicating that the configuration is unstable. This follows the prediction of <cit.> that indicates that new configurations should appear as we introduce a UV-cutoff in the geometry. The profiles of the effective strings shown in the right panel of Figure <ref>, display a behaviour similar to the one we encountered in the study of 't Hooft loops (and opposite to that of the Wilson loop), again suggesting the need for a phase transition.
In analogy with the case of the 't Hooft loop, if we introduce a UV-cutoff, the separation between the two entangled regions becomes multiple-valued, as shown in Figure <ref> (right panel). This is at the root of the phase transition. The plot of S_EE (L_EE) shows the correct concavity and the presence of a transition to the disconnected configuration is clearly displayed. See Figure <ref>
Following the findings of <cit.>, we state that if a field theoretical UV completion to our system (that is completed by a LST) were found, the phase transitions for the 't Hooft loops and the EE would become apparent. In this sense the UV-cutoff captures the correct dynamics.
§ SIGMA MODEL
In this section we review some results <cit.>, <cit.>, concerning the string σ-model on the background in eq.(<ref>)[We gratefully acknowledge conversations with Lewis Cole and Gastón Giribet on the topics discussed here.]. To properly study the string action we move to the S-dual frame, and work in terms of NS5 branes. After suitable coordinate changes detailed in Appendix <ref>, the background reads,
ds^2_st = -dt^2+ dx^2+dφ^2+ dρ^2
+ N_B ds^2(S^3_A)
+ N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = -√(1/N_A+1/N_B) ρ.
For a careful derivation we refer the reader to Appendices <ref> and <ref>. The background is a product space of the form
ℝ^2,1×ℝ_ρ× S^3_N_A× S^3_N_B,
where ℝ_ρ denotes the direction (with the linear dilaton), and the subscript on the S^3 denotes the square of their radius. The metric on the spheres together with the H_3 flux on each of them allows us to write the σ-model on them as a WZW model on SU(2). This is due to the fact the the S^3 is a group manifold. Naively, the contribution of the ρ coordinate to the string action is (for clarity we reinstate the α'-factor)
S_ρ = 1/4πα'∫ d^2σ√(-h)( h^ab∂_aρ∂_bρ - α' √(1/N_A + 1/N_B)R^(2)ρ),
where R^(2) is the world-sheet Ricci scalar. However when ρ→ -∞ the theory becomes non-perturbative, since g_s∼ e^-ρ. In order to avoid the strong coupling region, it is necessary to add the tachyon operator e^2bρ to the action, so that
S_ρ = 1/4πα'∫ d^2σ√(-h)( h^ab∂_aρ∂_bρ - α' Q R^(2)ρ + Λ e^2bρ).
Here, b is related to the background charge Q = √(1/N_A + 1/N_B) as
Q = b+1/b,
such that it does not have a strong coupling region. Thus the contribution of the ρ direction to the σ-model corresponds to a Liouville field. The complete σ-model on this geometry is
U(1)^3×Liouville× SU(2)_N_AWZW× SU(2)_N_BWZW
where the subscripts denote the WZW level. One can check that this is indeed a good σ-model by computing the central charge. Here we use
c(U(1))=1, c(Liouville) = 1+ 6Q^2, c(SU(2)_k) = 3(k-2)/k.
In total we have
c_total=3 + 1 + 6 ( 1/N_A + 1/N_B)
+ 3(N_A-2)/N_A+ 3(N_B-2)/N_B = 10.
To this we should add the ten free fermions, that contribute to the central charge c_ferm=5. The central charge of the SUSY system c_SUSY=15, is then cancelled by the b-c and β-γ ghosts.
We now study the interesting case Q_A=Q_B=0, but with m>0. The configuration of interest is obtained by S-dualising
eq.(<ref>),
ds^2_st = -dt^2+ dx^2+ f_s(r)dφ^2+ 4 dr^2/r^2 f_s(r)
+ N_B ds^2(S^3_A) + N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = - log(r).
Recall that here φ∼φ + π/Q^2, see eq.(<ref>), with Q^2=1/N_A+1/N_B, as above. Before proceeding to the σ-model analysis, it is convenient to perform some changes of variables. First, we want to rewrite the background (<ref>) in such a way that it reduces to (<ref>) when m=0. For this use
r = e^Q ρ, m̃ = m/4Q^2,
which leads to
ds^2_st = -dt^2+ dx^2+ 4Q^2(1-m̃ e^-2Qρ)dφ^2
+ dρ^2/1-m̃ e^-2Qρ
+ N_B ds^2(S^3_A) + N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = - Q ρ .
We change coordinates as,
tanh^2(λ) = 1- m̃ e^-2Qρ, φ = ϕ/2Q^2,
which puts the geometry in the usual cigar form (note that ϕ has period 2π)
ds^2_st = -dt^2+ dx^2+ 1/Q^2( tanh^2(λ)dϕ^2 + dλ^2)
+ N_B ds^2(S^3_A) + N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = - log( cosh(λ) ) - 1/2log(m̃) .
As explained in <cit.>, for more details see Section 2 of <cit.>, the above backgrounds leads to an exact σ-model
U(1)^2×SL(2,ℝ)_k/U(1)× SU(2)_N_B WZW× SU(2)_N_A WZW,
where k^-1 = Q^2=1/N_A+1/N_B. To check that this is also a good string σ-model we use
c( SL(2,ℝ)_k) = 3(k+2)/k, c( G/H ) = c(G)-c(H),
so that
c( SL(2,ℝ)_k/U(1)) = 2 + 6/k=2+6/N_A+6/N_B ,
from where is easy to see that c_total=10. We leave for future research the study of the σ-model in the background with m, Q_A,Q_B arbitrary.
§ CONCLUSIONS AND FUTURE RESEARCH
The I-brane QFT, defined as the (1+1) field theory on the intersection of two stacks of D5 branes, was studied in <cit.>. The field theory has the remarkable behaviour that as the coupling is increased, the system gains one more dimension and enhances its SUSY (with a peculiar SUSY algebra in (2+1) dimensions <cit.>). The background dual to this strongly coupled QFT was written in <cit.>, see our eq.(<ref>). This presents a singular behaviour for large values of the radial coordinate, where the dilaton diverges and string coupling effects cannot be neglected. This is solved by performing an S-duality and working with the NS branes system. The background of eq.(<ref>) is also singular for small values of the radial coordinate, r→ 0, as indicated by eq.(<ref>). This ill-defined IR behaviour is amended by our background in eq.(<ref>). Our simple and explicit solution describes the holographic dual to a (2+1) QFT that gets compactified to (1+1) dimensions, preserving four supercharges and ending the flow with a confining and gapped behaviour.
We holographically studied different aspects of this peculiar QFT. Maldacena-Wilson loops, 't Hooft loops, Entanglement entropy were discussed in dedicated sections, with emphasis on the effects of the UV-completion in terms of LST. R-symmetry and its breaking, a suitably defined gauge coupling and a quantity measuring the number of degrees of freedom as a function of the radial coordinate (the energy) are presented and discussed.
By a double Wick rotation, a black hole solution is found. Also, some of the NS string σ-model aspects are briefly mentioned.
It would be interesting to dedicate future efforts to
* The careful study of the black hole solution in Section <ref>. In particular if its entropy can be computed in terms of a (2+1) field theory compactified on a torus.
* The study of the string σ-model for the full solution in eq.(<ref>), in the NS5 branes frame.
* To achieve a cleaner understanding of the R-symmetry breaking in terms of anomalies in two dimensional QFT. To relate this to the Chern Simons coefficients discussed in eq.(<ref>). Note that while we find an anomalous breaking U(1)_A× U(1)_B→ℤ_N_B×ℤ_N_A, we do not find a further spontaneous breaking to ℤ_2×ℤ_2, as it normally occurs in holographic models to four dimensional N=1 dynamics. Understanding the reason of the absence of the spontaneous breaking would be of interest.
* Geometrically, it would be interesting to generalise the metric and fluxes in eq.(<ref>) adding warp factors in front of the ω_i's and more general fibrations. Finding a more general classification is of interest. It may be possible to relate this to the material in <cit.>.
* A fair amount of papers have been written studying the background in eq.(<ref>). See for example <cit.>-<cit.>. It would be interesting to understand the effects of the resolution provided in eq.(<ref>) on some of these observables.
We hope to report on some of these problems in the near future.
§ ACKNOWLEDGEMENTS:
The contents and presentation of this work much benefitted from extensive discussion with various colleagues. We are very happy to thank: Andres Anabalon, Adi Armoni, Fabrizio Canfora, Lewis Cole, Gaston Giribet, Nicolas Grandi, Nabil Iqbal, Prem Kumar, Juan Maldacena, Anibal Neira, Leo Pando Zayas, Julio Oliva, Niels Obers, Dibakar Roychowdhury, Kostas Skenderis, Christoph Uhlemann who shared their knowledge with us.
We are supported by STFC grant ST/T000813/1. The work of M.O. is partially funded by Beca ANID de
Doctorado 21222264. The work of R.S. is supported by STFC grant ST/W507878/1. The authors have applied to a Creative Commons Attribution (CC BY) licence.
§ DETAILS OF THE SUPERGRAVITY BACKGROUNDS
In this appendix we set some of the conventions used in this paper and study the SUSY preserved by the background in eq.(<ref>).
§.§ Type IIB Supergravity
We start this appendix by explicitly writing the Type IIB Supergravity action
and its SUSY variations. The field content of Type IIB is split into two sector. In the NS-NS sector we have: the metric g_μν, 2-form potential B_2 with field strength H_3, and the Dilaton Φ. In the R-R sector we have a set of Abelian p-form gauge fields: C_0, C_2, C_4. By defining
F_1 = dC_0, F_3 = dC_2 - C_0∧ H_3,
F_5 = dC_4 - C_2∧ H_3.
The bosonic part of the Type IIB action in String frame is
S_IIB = 1/2κ^2∫ d^10x√(-g)[e^-2Φ( R + 4∂_μΦ∂^μΦ - 1/2 |H_3|^2) - 1/2|F_1|^2 - 1/2|F_3|^2
- 1/4|F_5|^2]
= - 1/4κ^2∫ C_4∧ H_3∧ F_3
where |F_p|^2 = F_μ_1...μ_pF^μ_1...μ_p/p! and analogously for H_3. On the solutions of this theory we need to impose [To be precise, there is no covariant action for the effective theory of the Type IIB Superstring, but the presented here is close enough. The issue is that is not possible to implement the self-duality condition for F_5 at the level of the action.] self-duality of the F_5=⋆ F_5. The equations of motion of this theory are
∇^2Φ - ∂_μΦ∂^μΦ + 1/4R - 1/8|H_3|^2=0,
d( e^-2Φ⋆ H_3) = - F_5∧ F_3 - F_1∧ F_7,
d F_5- H_3∧ F_3 = 0,
d F_7 - H_3∧ F_5 = 0,
d F_9 - H_3∧ F_7 = 0,
R_μν + 2∂_μ∂_νΦ
- 1/2|H_3|^2_ μν
- e^2Φ/2( 1/2|F_1|^2_ μν
+ 1/2|F_3|^2_ μν
+1/2|F_5|^2_ μν -1/2g_μν( |F_1|^2+ |F_3|^2)) = 0
where |F_p|^2_μν = F_μν_1...ν_p-1F_ν^μν_1...ν_p-1/(p-1)!, similarly for H_3, and
F_7 = - ⋆ F_3, F_9 = ⋆ F_1.
The equations of motion are complemented by the Bianchi Identities
dF_1 = 0, dF_3 - H_3∧ F_1 = 0.
Due to the self-duality of F_5, its equation of motion and its the Bianchi identity are the same.
Solutions of the purely bosonic part of Type IIB has all the fermionic partners, the dilatino λ and gravitino Ψ_μ, set to zero. If we are interested in finding SUSY solutions, we need to be consistent with the fact that we turn off the fermions by asking for the SUSY variations of these fields to vanish. In string frame, the SUSY variations of the fermionic fields are <cit.>,
δλ = 1/2( Γ^μ∂_μΦ
+ 1/2· 3!H_μνλΓ^μνλσ^3 -e^Φ(F_μΓ^μ(iσ_2) + 1/2· 3!F_μνλΓ^μνλσ^1))ϵ,
δΨ_μ = ∂_μϵ
+ 1/4ω^μab_μΓ_abϵ + 1/4· 2!H_μνλΓ^νλσ^3ϵ
= +e^Φ/8(
F_νΓ^ν(iσ_2)
+ 1/3!F_νλρΓ^νλρσ^1
+ 1/2·5!F_νλρστΓ^νλρστ(iσ_2))Γ_μϵ.
Here ω^μab_μ is the spin connection of the 10D background, where the a,b indexes are flat space ones, and σ^1, σ^2 and σ^3 are Pauli matrices. Also
Γ^μ_1...μ_p = Γ^[μ_1...Γ^μ_p]
Here ϵ is a 64 component spinor,
ϵ = [ ϵ_1; ϵ_2 ]
where both 32-component parts are left-handed.
§.§ Checking SUSY for the Fibered Background
Here we aim to compute how many supercharges are preserved by the the backgrounds presented in this paper. For the un-fibered background in eq.(<ref>) we refer the reader to <cit.>, <cit.>, where it is shown that this solution preserves 16 Supercharges in an interesting way: the anti-commutator of two supercharges includes the R-Symmetry generators. Now we present the analysis for the fibered background in eq.(<ref>). We perform all the analysis in the S-dual system, in terms of NS5 branes, where we only have H_3 flux.
First, note that the dilatino variation is a matrix equation of the form Mϵ=0. In order to have non-trivial solutions to this equation, we require M to be non-invertible, for which we need to impose det(M)=0. It is also possible to obtain a matrix equation from the gravitino variation. Noting that we can write the gravitino variation as a covariant derivative, for which we define the connection
W_μ = 1/4ω^μab_μΓ_ab + 1/4· 2!H_μνλΓ^νλσ^3 +e^Φ/8( F_μΓ^μ(iσ_2)
+ 1/3!F_μνλΓ^μνλσ^1
+ 1/2·5!F_μνλρσΓ^μνλρσ(iσ_2))Γ_μ,
then we can write the gravitino variation as
δψ_μdx^μ = ( ∂_μϵ
+ W_μϵ)dx^μ≡𝒟ϵ.
We can get rid of the partial derivative of the spinor by acting with 𝒟 a second time
𝒟∧𝒟ϵ = ( dW + W∧ W )ϵ = 1/2Θ_μνdx^μ∧ dx^νϵ.
Each of the components of Θ_μν defines a matrix equation, giving a total of 45 independent equations. We need to make sure that det(Θ_μν)=0 for each of the components. The equations
M ϵ = 0 , Θ_μνϵ =0,
constrain the number of independent components of the spinor. After this procedure we use the gravitino variation to solve the dependence of the spinor on the spacetime coordinates.
Specialising to our background, the determinant of the Dilatino variation for the background in eq.(<ref>) reads
det(M) ∼( 4(e_BQ_A-e_AQ_B)^2+m^2)^8(4(e_BQ_A+e_AQ_B)^2+m^2)^8.
In order to have non-trivial solutions we need to impose the following BPS conditions on the parameters of the background
e_AQ_B = ± e_BQ_A, m = 0.
With this conditions it is possible to check that det(Θ_μν)=0 is also satisfied. Solving these matrix equations shows that the spinor has 8 independent components. Then, solving for the gravitino variation shows that these components are not independents, and in fact, the total number of independent components its reduced to 4. The solution for the spinor is
ϵ_1 = 0⃗
and
ϵ_2 = [ c_1 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; 0; 0; c_1 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; 0; i c_2 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; i c_2 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; 0; 0; c_1 e^-1/4 i φ(e_A^2+e_B^2)√((e_A^2+e_B^2) (e_A r^2-2 Q_A))/r (e_A-i e_B); c_1 (e_A+i e_B) e^-1/4 i φ(e_A^2+e_B^2)(2 Q_A-e_A r^2)/r √((e_A^2+e_B^2) (e_A r^2-2 Q_A)); 0; i c_2 (e_A+i e_B) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; 0; 0; c_2 (e_B-i e_A) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; 0; c_3 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; -c_3 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; 0; i c_4 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; 0; 0; -i c_4 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; -c_3 e^-1/4 i φ(e_A^2+e_B^2)√((e_A^2+e_B^2) (e_A r^2-2 Q_A))/r (e_A+i e_B); 0; 0; -c_3 e^-1/4 i φ(e_A^2+e_B^2)√((e_A^2+e_B^2) (e_A r^2-2 Q_A))/r (e_A+i e_B); 0; -i c_4 (e_A-i e_B) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; -i c_4 (e_A-i e_B) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; 0 ]
We have found a spinor with four arbitrary constants (c_1,c_2,c_3,c_4). Being the spinor complex, we count four preserved supercharges.
§ HOW ARE THE BACKGROUNDS OBTAINED
In this appendix we describe the procedure followed to obtain the backgrounds in eqs.(<ref>) and (<ref>). These solutions are originally obtained in gauged supergravity, together with a lift procedure. Below, we review these steps.
§.§ 4D N=4 SU(2)xSU(2) Gauged Supergravity
The action of the bosonic part of the Freedman-Schwarz (FS) gauged Supergravity is
S_FS = ∫ d^4x√(-g_(4))(-R^(4)/4 + 1/2(∂ϕ)^2
+ 1/2e^4ϕ(∂a)^2 - V(ϕ)
.
=.
-e^-2ϕ/4Tr( F_(A)mnF_(A)^mn
+ F_(B)mnF_(B)^mn)
- a/2 Tr( F̃_(A)mnF_(A)^mn
+ F̃_(B)mnF_(B)^mn) ).
Here g_(4) and R^(4) are the determinant of the 4D metric and the 4D Ricci scalar, ϕ is the 4D dilaton, a is a pseudo-scalar called axion and F_(A)mn and F_(B)mn are the field strengths of two SU(2) gauge fields A_m and B_m,
F^i_(A)mn = ∂_mA^i_n - ∂_nA^i_m + e_Aϵ_ijkA^j_mA^k_n
F^i_(B)mn = ∂_mB^i_n - ∂_nB^i_m + e_Bϵ_ijkB^j_mB^k_n
where e_A and e_B are the gauge couplings of A_m and B_m respectively. Here the index i=1,2,3 transform in the adjoint of each of the copies of SU(2). Also, the duals of the field strengths are
F̃_(A)mn = 1/2√(-g)ϵ_mnλρ F_(A)mn, F̃_(B)mn = 1/2√(-g)ϵ_mnλρ F_(B)mn,
§.§.§ A BPS Solution
This theory admits a series of BPS solution preserving some amount of supersymmetry. Our main focus is the 1/4 BPS soliton presented in <cit.>. As we will review, this solution is particularly interesting because it manages to resolve a singularity by introducing a thermal cycle that preserves some SUSY (in the usual case, the non-extremal factor completely breaks SUSY). The field configuration of the Soliton is[In <cit.> the solution was presented in the mostly-minus signature. We write the solution in the mostly-plus one.]
ds^2_4D = -ρ dt^2 + dρ^2/g(ρ) + g(ρ)dφ^2 +ρ dx^2,
ϕ(ρ) = -1/2log(ρ),
A^1 =0, A^2=0, A^3= Q_Aζ(ρ)dφ,
B^1 =0, B^2=0, B^3= Q_Bζ(ρ)dφ,
where
g(ρ) = e^2_A+e^2_B/2ρ - m -2Q^2_A+Q^2_B/ρ
which has a zeros
ρ_± = m±√(4(e^2_A+e^2_B)(Q^2_A+Q^2_B)+m^2)/(e^2_A+e^2_B).
Also
ζ(ρ) = 1/ρ - 1/ρ_+,
where the last term ensures that both of the gauge fields vanish at ρ=ρ_+.
In order for the cycle φ to close smoothly at ρ=ρ_+, the period of φ needs to be
β_φ = 4π/g'(ρ_+)
= 8πρ^2_+/(e^2_A+e^2_B)ρ^2_++
4(Q^2_A+Q^2_B).
This solution was shown to preserve 4 supercharges <cit.>, when the parameters (e_A,e_B, Q_A, Q_B,m) satisfy
e_AQ_B = ± e_BQ_A, m=0.
In what follows it is convenient to perform the change of coordinates ρ = r^2. The background configuration now reads
ds^2_4D = -r^2 dt^2 + 4dr^2/f_s(r) + r^2f_s(r)dφ^2 +r^2 dx^2,
ϕ(r) = -log(r),
A^1 =0, A^2=0, A^3= Q_Aζ(r)dφ,
B^1 =0 B^2=0, B^3= Q_Bζ(r)dφ,
with
f_s(r) = e^2_A+e^2_B/2 - m/r^2 -2Q^2_A+Q^2_B/r^4, ζ(r) = 1/r^2-1/r^2_+.
§.§ Lift to 10D Supergravity
It was shown in <cit.> that the FS Supergravity has
a Kaluza-Klein interpretation as a compactification of 𝒩=1 Supergravity in 10D on the group manifold S^3× S^3. The action of the 10D theory is
S_10D = -1/4∫ d^10x√(-g)( R
-2∂_μΦ̃∂^μΦ̃
-e^2Φ̃/3H̃_μνλH̃^μνλ),
where R is the 10D Ricci scalar, Φ̃ the 10D Dilaton and H̃_μνλ is a 3-form field strength H̃_3=dB̃_2. We split the indexes as
x^μ ={ x^m=t,r,φ,x ;
z_A^i = ψ_A,θ_A,ϕ_A ;
z_B^i = ψ_B,θ_B,ϕ_B} .
The of the lift to 10D is
ds^2_10D=e^3ϕ /2ds^2_4D+2e^-ϕ /2(
Θ_A^iΘ_A^i
+Θ_B^iΘ_B^i)
where the 1-forms
Θ^i_A = A^i+1/e_Aω_A^i,
Θ^i_B = B^i+1/e_Bω_B^i,
with ω_A^i and ω_B^i are the Maurer-Cartan forms of the two different SU(2), is so that Θ^i_A and Θ^i_B realise a fibration of the two 3-spheres.
The 10D Dilaton Φ̃ is written in terms of the four dimensional one ϕ
Φ̃=-ϕ/2,
while the 3-form field strength is written in term of the non-Abelian gauge fields and the SU(2) Maurer-Cartan forms as
H̃_3= -∑^3_i=1F_A^i∧Θ_A^i
-∑^3_i=1F_B^i∧Θ_B^i
+ e_AΘ_A^1∧Θ_A^2∧Θ_A^3
+e _BΘ_B^1∧Θ_B^2∧Θ_B^3.
§.§.§ A Note on Conventions
We are interested in lifting the theory to Type II Supergravity, when the field content is purely of the NS-NS sector. The action in eq.(<ref>) can be mapped to Type II, after the field redefinitions
Φ̃→Φ =- 2Φ̃, H̃_3→ H_3= 2 H̃_3.
The action corresponds to the Type II in Einstein frame
S_Type II, E= -1/4∫ d^10x√(-g)(R
-1/2∂_μΦ∂^μΦ
- e^-Φ/12 H_μνρ H^μνρ).
We move to String frame by g^(S)_μν = e^1/2Φg^(E)_μν, then the action reads
S_Type II, S=-1/4∫ d^10x√(-g) e^-2Φ( R
+ 4∂_μΦ∂^μΦ
-1/2· 3!H_μνρ H^μνρ).
In this frame, the lift of the 4D FS Supergravity reads
ds^2_st = e^2ϕds^2_4D+2(
Θ_A^iΘ_A^i
+Θ_B^iΘ_B^i),
H_3 = 2(-∑^3_i=1F_A^i∧Θ_A^i
-∑^3_i=1F_B^i∧Θ_B^i
+ e_AΘ_A^1∧Θ_A^2∧Θ_A^3
+e _BΘ_B^1∧Θ_B^2∧Θ_B^3.)
,
Φ = ϕ(x^m).
It is convenient to write the lift in the S-dual frame, where instead of H_3 flux, we have a F_3 flux, the Dilaton is Φ'=-Φ and the metric now is g_μν' = e^-Φg_μν, explicitly this is
ds^2_st = e^ϕds^2_4D+2e^-ϕ(
Θ_A^iΘ_A^i
+Θ_B^iΘ_B^i) ,
F_3 = 2(-∑^3_i=1F_A^i∧Θ_A^i
-∑^3_i=1F_B^i∧Θ_B^i
+ e_AΘ_A^1∧Θ_A^2∧Θ_A^3
+e _BΘ_B^1∧Θ_B^2∧Θ_B^3.)
,
Φ = -ϕ(x^m).
§.§.§ Lift of the BPS Solution
Following the explicit construction of the lift, we now read the lift of the 4D solution that preserves 4 Supercharges. In the S-dual frame of eqs. (<ref>), we have
ds^2_st = r{ -dt^2+dx^2+f_s( r) dφ ^2+4/r^2f_s( r)
dr^2+2/e_A^2[ ω̂_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ
(r)dφ) ^2] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3-e_BQ_Bζ( r)
dφ) ^2] } ,
F_3 = dC_2= 2 ζ'(r)dr∧ dφ∧(
Q_A/e_Aω̂_3+Q_B/e_Bω̃
_3) +2/e_A^2ω̂_1∧ω̂
_2∧( e_AQ_Aζ (r)dφ -ω̂_3)
+2/e_B^2ω̃_1∧ω̃_2∧( e_BQ_Bζ (r)dφ -ω̃_3) ,
C_2 = ψ_A( 2Q_A/e_Aζ ^'(
r) dr∧ dφ -2/e_A^2sinθ_Adθ_A
∧ dϕ_A) +2/e_Acosθ_AQ_Aζ( r) dφ∧ dϕ_A
+ψ_B( 2Q_B/e_Bζ ^'( r)
dr∧ dφ -2/e_B^2sinθ_Bdθ_B
∧ dϕ_B) +2/e_Bcosθ_BQ_Bζ( r) dφ∧ dϕ_B .
C_6 = -2e_Ar^2/e^3_Bdt∧ dx∧ dφ∧(S^3_B)
+2e_Br^2/e^3_Adt∧ dx∧ dφ∧(S^3_A) ,
= -8Q_B/e^3_Ae^2_Bcos(θ_B)dt∧ dx ∧(S^3_A)∧ dϕ_B
+8Q_A/e^3_Be^2_Acos(θ_A)dt∧ dx ∧(S^3_B)∧ dϕ_A,
Φ = log r .
This is the background in eq.(<ref>).
In the case for which Q_A=Q_B=m=0, the background fields read (note that we S-dualise moving to the NS5 brane frame),
ds^2_st = -dt^2+dx^2+ e^2_A+e^2_B/2dφ^2 + 8/e^2_A+e^2_Bdr^2/r^2
+ 8/e^2_Ads^2(S^3_A)
+ 8/e^2_Bds^2(S^3_B),
H_3 = -8/e^2_A(S^3_A)-8/e^2_B(S^3_B),
Φ = -log(r).
This is the background in eq.(<ref>).
We can rescale φ to absorb the prefactor. Also, it is convenient to set as in eq.(<ref>)
N_A = 8/e^2_B, N_B = 8/e^2_A,
and perform the change of coordinates
r = e^√(1/N_A+1/N_B) ρ,
after changing H_3→ -H_3, the background reads
ds^2_10D = - dt^2 + dx^2 + dφ^2 + dρ^2
+ N_B ds^2(S^3_A)
+ N_Ads^2(S^3_B),
H_3 = N_AS^3_A + N_BS^3_B,
Φ = -√(1/N_A+1/N_B) ρ.
This is the background written in Section <ref> to study the string σ-model on this field configuration.
§ ON THE UNFIBERED GEOMETRY
Here, we review a different derivation of the background (<ref>). The S-dual of this background (<ref>) was first introduced in <cit.>, <cit.>, here we review the derivation of the pure NS-NS frame for simplicity. Let us consider two stacks of NS5-branes, the first extended in (t,x,y_1,y_2,y_3,y_4) and while the second one spans (t,x,w_1,w_2,w_3,w_4). These stacks intersect in the (t,x) directions, thus in the weak coupling regime, the effective theory on the intersection is 1+1 dimensional and preserves 8 Supercharges.
We now move to the strong coupling regime. For this, we write the space ℝ^4_y = (y_1,y_2,y_3,y_4) in spherical coordinates (r_A, S^3_A), and similarly for ℝ^4_w we use (r_B, S^3_B). In terms of the harmonic functions
H_A(r_A) = 1 + N_B/r^2_A,
H_B(r_B) = 1 + N_A/r^2_B,
the backreacted fields are given by
ds^2_st = dx^2_1,1
+ H_A(r_A)( dr^2_A + r^2_Ads^2(S^3_A))
+ H_B(r_B)( dr^2_B + r^2_Bds^2(S^3_B)),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = 1/2log(H_A(r_A)H_B(r_B)).
By taking the near-horizon geometry we are led to
ds^2_st = dx^2_1,1
+ N_Bdr^2_A/r^2_A + N_Adr^2_B/r^2_B
+ N_Bds^2(S^3_A) + N_Ads^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = -log(r_A) -log(r_B) + 1/2log(N_AN_B).
Under the change of coordinates r_A = e^ρ_A/√(N_B) and r_B = e^ρ_B/√(N_A) we obtain
ds^2_st = dx^2_1,1
+ dρ^2_A + dρ^2_B
+ N_Bds^2(S^3_A) + N_Ads^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = -√(1/N_B)ρ_A -√(1/N_A)ρ_B + 1/2log(N_AN_B).
Finally, taking the linear combination
√(1/N_A+1/N_B)ρ_A = √(1/N_B)ρ - √(1/N_A)φ,
√(1/N_A+1/N_B)ρ_B = √(1/N_A)ρ + √(1/N_B)φ,
we reach the background fields in (<ref>)
ds^2_st = -dt^2+ dx^2+dφ^2+ dρ^2
+ N_B ds^2(S^3_A)
+ N_A ds^2(S^3_B),
H_3 = N_B (S^3_A) + N_A (S^3_B),
Φ = -√(1/N_A+1/N_B) ρ,
where we cancelled the constant term of the Dilaton by a suitable shift of its zero mode. Note that the Dilaton in (<ref>) has functional dependence in two of the coordinates of the background, while the one in (<ref>) only depends only on one coordinate. This allows us to interpret the extra flat direction of (<ref>) as being part of the Field Theory ones. In this way, we see that in the strong coupling regime, the theory on the intersection acquires an extra dimension, becoming (2+1) dimensional. There is also a SUSY enhancement from 8 to 16 supercharges. We refer the reader to <cit.> to cover this matter.
§ CHARGES OF THE BLACK HOLE BACKGROUND
We compute the charges of the configuration considering the Noether-Wald method
<cit.>. The bulk action principle in string frame of IIB in the metric-dilaton-F_3 sector is
S_IIB,bulk=1/2κ ^2∫ d^10x√(-g)( R-
1/2( ∂Φ) ^2-1/12e^ΦF_μνρF^μνρ) =∫ d^10x√(-g)ℒ ,
A general variation of the action gives
δ S_IIB,bulk=∫ d^10x√(-g)[ δ g^μν
ℰ_μν^( g) +δΦℰ^(
Φ) +δ C_νρℰ_( F_3) ^νρ+∇ _μΘ ^μ( 𝐟,δ𝐟
) ] .
where 𝐟 denotes the fields collectively and
Θ ^μ( 𝐟,δ𝐟) = 1/
2κ ^2( g^δηδΓ _ ηδ^μ-g^δμδΓ _ λδ^λ-δΦ∂ ^μΦ -e^Φδ C_νρF^μνρ)
,
ℰ_μν^( g) = 1/2κ ^2[
R_μν-1/2g_μνR.
-1/2( ∂ _μΦ∂ _νΦ -1/2
g_μν∂ _ρΦ∂ ^ρΦ)
. -1/2e^Φ( 1/2F_μδρF_ν^ δρ-1/12g_μνF_δρσF^δρσ) ] ,
= 1/2κ ^2( G_μν-1/2T_μν^(
Φ) -1/2T_μν^( F_3) ) ,
ℰ^( Φ) = 1/2κ ^2( ∇
^ρ∇ _ρΦ -1/12e^ΦF_μνρF^μνρ) ,
ℰ_( F_3) ^νρ = 1/2κ ^2
∇ _μ( e^ΦF^μνρ) .
The Noether current is defined by
J^μ=Θ ^μ( 𝐟,ℒ_ξ𝐟)
-ξ ^μℒ ,
where ℒ is the Lagrangian scalar under diffeomorphisms in (<ref>) and ℒ_ξ is the Lie derivative along the
vector ξ. The Noether current is conserved on-shell, thus it can be
written locally as J^μ=∇ _νq^μν. The Noether
current (<ref>) for our system gives
J^μ =-1/κ ^2∇ _ν( ∇ ^μξ ^ν ]+1/2e^Φ2C_λρξ ^λF^μνρ) +2ξ ^λℰ_( g) λ^μ-2ξ
^λC_λρℰ_( F_3) ^μρ ,
on-shell it defines the Noether pre-potential:
q^μν( ξ) =-1/κ ^2( ∇
^μξ ^ν ]+1/2e^Φ2C_λρξ
^λF^μνρ) .
The Hodge dual of the Noether pre-potential gives the 8-form
Q[ ξ] =1/21/8!√(-g)ϵ
_μνρ _1…ρ _8q^μνdx^ρ _1∧…∧ dx^ρ _8 ,
that in differential forms is
Q[ ξ] =-1/κ ^2( ⋆ dξ
+e^Φξ¬ C_2∧⋆ F_3) .
¬ stands for the contraction operator.
The boundary term that allow us to have a well posed action principle and
finite mass is
S_full=S_IIB,bulk+∫_∂ Md^9x√(-h)1/κ ^2(
𝒦-e^-1/4Φ) ,
where the first term in the integral is the Gibbons-Hawking-York term and
the last term corresponds to a counter term, that depends only on intrinsic
quantities, that allow us to renormalize the mass term. The extrinsic
curvature is defined in terms of the normal unit outwards vector n^μ
to the boundary of the spacetime by
𝒦_μν=h_ μ^ρh_ ν^σ∇ _ρn_σ ,
and the induced metric is h_μν=g_μν-n_μn_ν for
our case.
Following <cit.> the energy, angular momentum and entropy are defined by
ℰ[ t] = ∫_∞( Q[
t] -ξ¬B) ,
𝒥[ ψ] = -∫_∞Q[
ψ] ,
S[ ξ] = 1/T∫_ℋ
Q[ ξ] .
The boundary terms are in the 9-form
B=-1/κ ^2( 𝒦-e^-1/4Φ) ⋆ n .
t is the time-like killing vector at infinity properly
normalized, ψ is the rotation generator and ξ is the horizon generator
ξ=t+Ωψ .
ξ is null at the Horizon which defines the angular
velocity Ω and satisfies the geodesic equation at the horizon
ξ ^μ∇ _μξ ^ν=κ _sξ ^ν ,
defining the surface gravity κ _s that is related to the
temperature as T=κ _s/2π. In this case we are in general
relativity, therefore the entropy give one-quarter of the horizon area.
Let us consider the black hole configuration in Einstein frame (<ref>), with
ζ(r)=1/r^2 , Φ→Φ-2log(e_A^2+e_B^2/2) , F_3 →e_A^2+e_B^2/2F_3 .
The in-going Eddington-Finkelstein coordinates are well-defined
at the horizon, which are defined by
dt = dv-2dr/rf_bh( r) ,
dψ _A = dψ _A^'-2e_AQ_Aζ( r) /
rf_bh( r) dr ,
dψ _B = dψ _B^'-2e_BQ_Bζ( r) /
rf_bh( r) dr .
Then, the metric becomes
ds_E^2 = √(r){ dy^2+dx^2-f_bh( r) dv^2+
4/rdrdv.
+2/e_A^2[ dθ _A^2+sin ^2θ _Adϕ
_A^2+( dψ _A^'+cosθ _Adϕ
_A-e_AQ_Aζ( r) dv) ^2]
. +2/e_B^2[ dθ _B^2+sin ^2θ
_Bdϕ _B^2+( dψ _B^'+cosθ _Bdϕ
_B-e_BQ_Bζ( r) dv) ^2] } ,
In these coordinates we consider the vector
ξ=t+Ω _Aψ_A+Ω
_Bψ_B
where
t=1/2∂/∂ v ,
ψ_A=∂/∂ψ _A , ψ
_B=∂/∂ψ _B .
The vector ξ is null at the horizon located at r_+
when
Ω _A=e_AQ_A/r_+^2 , Q_B=e_BQ_B/
r_+^2 .
Due to the fact that the spacetime that we are considering is not asymptotically Minkowski times S^3× S^3, instead is conformal to Minkowski times S^3× S^3, it is not clear how we should normalize the vector t time-like at infinity. This ambiguity propagates to the energy and the temperature. Therefore we expect to obtain the temperature in 4D up to a factor.
The energy (<ref>), angular momentum (<ref>), temperature
defined through (<ref>) and the entropy give
E = ℰ[t]=2m/e_A^3e_B^3r_+^2κ ^2( 16π
^2) ^2L_xL_y ,
J_A = 𝒥[ψ_A]=8Q_A/e_A^4e_B^3κ ^2( 16π
^2) ^2L_xL_y ,
J_B = 𝒥[ψ_B]=8Q_B/e_B^4e_A^3κ ^2( 16π
^2) ^2L_xL_y ,
T = e_A^2+e_B^2/16π-4(
Q_A^2+Q_B^2) /16π r_+^4 ,
S =2r_+^2/e_A^3e_B^3G_10( 16π)
^2L_xL_y .
They satisfy the first law of thermodynamics
dE=TdS+Ω _AdJ_A+Ω _BdJ_B .
§ R-SYMMETRY BREAKING
In this appendix we give a detailed derivation of the symmetry breaking pattern of three U(1) directions present in our background. In order to do this, we gauge this symmetries by introducing a gauge field A and a scalar ϵ. The presence of a mass term in the effective action of the gauge field, i.e. an explicit symmetry breaking of the gauge symmetry, signals the breaking of this U(1) symmetry on the dual field theory. In the QFT, the breaking of the global symmetry can be either spontaneous or anomalous.
§.§ U(1) R-Symmetry of psi A and psi B
Let us start by recalling the R-R C_2 potential
C_2 = ψ_A( 2Q_A/e_Aζ ^'(
r) dr∧ dφ -2/e_A^2sinθ_Adθ_A
∧ dϕ_A) +2/e_Acosθ_AQ_Aζ( r) dφ∧ dϕ_A
+ψ_B( 2Q_B/e_Bζ ^'( r)
dr∧ dφ -2/e_B^2sinθ_Bdθ_B
∧ dϕ_B) +2/e_Bcosθ_BQ_Bζ( r) dφ∧ dϕ_B .
Since this potential is not invariant under ψ_A,B→ψ_A,B+4π, we expect this symmetry to be broken in the dual field theory. We gauge these isometries by doing the following replacements in the R-R potential and the metric
dψ_A,B→ dψ_A,B + A_A,B, ψ_A,B→ψ_A,B + ϵ_A,B.
Where A_A,B is a U(1) gauge field and ϵ_A,B is a scalar charged under the gauged U(1) symmetry, which makes the combination D_A,Bϵ = ∂_A,Bϵ_A,B - A_A,B is gauge invariant (here ∂_A,B = ∂/∂ψ_A,B). These fields only depend on the coordinates of the field theory directions, i.e. (t,x). After these replacements the metric and the R-R 3-form read
ds^2 = ds^2_(0) +4r/e^2_A( ω̂_3-e_AQ_Aξ(r)dφ) A_A μdx^μ + 2r/e^2_AA_A μ A_A ν dx^μdx^ν
+4r/e^2_B( ω̃_3-e_BQ_Bξ(r)dφ) A_B μdx^μ + 2r/e^2_BA_B μ A_B ν dx^μdx^ν
F_3 = F^(0)_3 - 2 dϵ_A∧( -Q_A/e_Aξ'(r)dr∧ dφ + 1/e^2_A(S^2_A) )
- 2 dϵ_B∧( -Q_B/e_Bξ'(r)dr∧ dφ + 1/e^2_B(S^2_B) )
where ds^2_(0) and F^(0)_3 denotes the metric and the 3-form of the configuration before gauging the U(1) symmetries and (S^2_A,B) = sin(θ_A,B)dθ_A,B∧ dϕ_A,B. Now we want to obtain an effective lagrangian for A_A,B and ϵ_A,B. In order to do this, we consider how the Ricci scalar and the kinetic term of F_3 change under the gauging of the symmetry. Explicitly, the Ricci scalar transforms as
R = R^(0) - 1/42r/e^2_A F^2_A - 1/42r/e^2_B F^2_B,
where F^2_A,B = F_A,B μνF^μν_A,B and F_A,B μν is the field strengh of A_A,B μ, while the kinetic term of the R-R potential reads
1/12 F_μνλF^μνλ
= 1/12 F^(0)_μνλF_(0)^μνλ
+ 1/2r^2( Q^2_Ar^2/e^2_Aξ'(r)^2+1)( A_A-dϵ_A)^2
+ 1/2r^2( Q^2_Br^2/e^2_Bξ'(r)^2+1)( A_B-dϵ_B)^2 + Q_AQ_B/e_Ae_B( A_A-dϵ_A)·( A_B-dϵ_B).
Finally, replacing this expression in the Type IIB action (in string frame), leads to the following effective lagrangian
ℒ =
- 1/42/e^2_Ar F^2_A - 1/42/e^2_BrF^2_B
- 1/2r^2( Q^2_Ar^2/e^2_Aξ'(r)^2+1)( A_A-dϵ_A)^2
= - 1/2r^2( Q^2_Br^2/e^2_Bξ'(r)^2+1)( A_B-dϵ_B)^2
- Q_AQ_B/e_Ae_B( A_A-dϵ_A)·( A_B-dϵ_B)
Due to the coupling between A_A,B and ϵ_A,B the gauge field obtains a mass. This is the same as the Stueckelberg mechanism. Defining W_A,B = A_A,B - dϵ_A,B, we obtain an action for the massive gauge field.
ℒ =
- 1/42/e^2_Ar F^2_A - 1/42/e^2_BrF^2_B
- 1/2r^2( Q^2_Ar^2/e^2_Aξ'(r)^2+1)
W_A μW^μ_A
= - 1/2r^2( Q^2_Br^2/e^2_Bξ'(r)^2+1)W_B μW^μ_B
- Q_AQ_B/e_Ae_BW_A μW^μ_B
§.§ U(1)) R-Symmetry of varphi
Now we repeat the same procedure as above for the φ direction. The only difference is at the starting point. The potential (<ref>) does not depend on φ. We need to perform a gauge transformation to give it φ dependance, after which
C_2 = 2Q_A/e_Aφ[ ξ'(r)( dψ_A + cos(θ_A)dϕ_A)∧ dr + ξ(r) (S^2_A)] - 2/e^2_Aψ_A(S^2_A)
= + 2Q_B/e_Bφ[ ξ'(r)( dψ_B + cos(θ_B)dϕ_b)∧ dr + ξ(r) (S^2_b)] - 2/e^2_Aψ_B(S^2_B).
As before, we gauge the symmetry along φ by shifting the metric and the R-R potential as follows
dφ→ dφ + A_φ, φ→φ + ϵ_φ.
Repeating the procedure of the previous section lead to the following shifts for Ricci scalar
R = R^(0) - 1/4( r f_s(r) + 2(Q^2_A+Q^2_B)rξ(r)^2)F^2_φ
where F^2_φ = F_φ μνF^μν_φ, with F_φ μν the field strenght of A_φ, and the kinetic term of the R-R potential
1/12 F_μνλF^μνλ = 1/12 F^(0)_μνλF_(0)^μνλ
= +1/4( 2/r^2ξ(r)^2( e^2_AQ^2_A + e^2_BQ^2_B
+ (Q^2_A + Q^2_B)^2r^2ξ'(r)^2) + (Q^2_A + Q^2_B)f_s(r)ξ'(r)^2) ( A_φ - dϵ_φ)^2
which leads to the effective lagrangian
ℒ = - 1/4( f_s(r) + 2(Q^2_A+Q^2_B)ξ(r)^2)F^2_φ
= -1/4( 2/r^2ξ(r)^2( e^2_AQ^2_A + e^2_BQ^2_B
+ (Q^2_A + Q^2_B)^2r^2ξ'(r)^2) + (Q^2_A + Q^2_B)f_s(r)ξ'(r)^2) ( A_φ - dϵ_φ)^2
As before, we see from the action that after a gauge transformation the gauge field obtains a mass via Stueckelberg mechanism. Explicitly by defining W_φ = A_φ - dϵ_φ we obtain
ℒ = - 1/4( f_s(r) + 2(Q^2_A+Q^2_B)ξ(r)^2)F^2_φ
= -1/4( 2/r^2ξ(r)^2( e^2_AQ^2_A + e^2_BQ^2_B
+ (Q^2_A + Q^2_B)^2r^2ξ'(r)^2) + (Q^2_A + Q^2_B)f_s(r)ξ'(r)^2) W_φ μW^μ_φ
§ MALDACENA-WILSON, 'T HOOFT LOOPS AND EE. DETAILED CALCULATIONS
In this appendix, we study the integrals needed to compute the Wilson loops, 't Hooft loops and Entanglement Entropy. We express the analytic results in terms of r_0.
Let us define the following quantities that allow us to write the integrals
in a simpler way
λ _0 = r_0/r_+ , ξ =r/r_+ , η =e_A/e_B ,
λ _-^2 = -r_-^2/r_+^2≡ 1-m/
r_+^2( e_A^2+e_B^2) ,
where λ _-∈[ 0,1], λ _0>1 and η >0.
All the problems we will address here can be reduced to a one-dimensional
problem for the function r=r( x) which minimises the functional
in eq.(<ref>) once we impose that the parameter x(σ)=σ. Then, the equation for the function r(σ)=r(x) reduces to
dr/dx=± V_eff( r) ,
for a suitable effective potential which is case-dependent. In most
cases the function r( x) can be interpreted as a string (or a
section of a higher dimensional surface) with end points at r→∞. The profile of the string subject to the initial condition x( r_0) =0, can be obtained by performing the integral
x( r) =±∫_r_0^rdr/V_eff( r) .
From here we compute the end points separation as
L( r_0) ≡lim_r→∞2x( r) .
The definition in eq.(<ref>) coincides with the
quark-anti-quark separation, monopole-anti-monopole separation and the
interval length for the Maldacena-Wilson loop, t' Hooft loop and
entanglement entropy, respectively.
The results of the integrals that we compute analytically are given
in terms of elliptic integrals. The elliptic integral of first
kind 𝐅( ϕ |m) and the complete elliptic integral
of first kind 𝐊( m) are defined as
𝐅( ϕ |m) = ∫_0^ϕdθ1/√(
1-msin ^2θ) ,
𝐊( m) = 𝐅( . π/2
| m) ,
respectively for -π/2<ϕ <π/2. The elliptic
integral of second kind 𝐄( ϕ |m) and the complete
elliptic integral 𝐄( m) are defined respectively as
𝐄( ϕ |m) = ∫_0^ϕ√(1-msin ^2θ)dθ ,
𝐄( m) = 𝐄( . π/2
| m) ,
where -π/2<ϕ <π/2.
§.§ Wilson loop
The effective potential in terms of the variable ξ defined in (<ref>) reads
V_eff( ξ) =√(e_A^2+e_B^2/8)r_+/
λ _0ξ√(( ξ ^2-λ _0^2) ( ξ
^2+λ _-^2) ( ξ ^2-1) ) .
The string profile, considering the change of variables in eq.(<ref>)
x( ξ) = ±√(8/e_A^2+e_B^2)∫_λ
_0^ξλ _0ξ dξ/√(( ξ ^2-λ
_0^2) ( ξ ^2+λ _-^2) ( ξ
^2-1) ) ,
= ±√(8/e_A^2+e_B^2)λ _0/√(λ
_0^2-1)
×[ -𝐅( . arcsin√(λ
_0^2-1/ξ ^2-1)|1+λ _-^2/1-λ
_0^2) +𝐊( 1+λ _-^2/1-λ
_0^2) ] .
The definition of the quark-antiquark separation given in eq.(<ref>) can
be expressed in terms of the limit (<ref>) of
the string profile. Replacing in eq.(<ref>) we find
L_QQ( λ _0) =2√(8/e_A^2+e_B^2)
λ _0/√(λ _0^2-1)𝐊( 1+λ
_-^2/1-λ _0^2) .
This is our result in eq.(<ref>).
In order to get an analytic expression for the energy in eq.(<ref>) we compute the integrals
E_QQ=F( r_0) L_QQ( r_0) +I_2+I_3
where
I_2 = √(32/e_A^2+e_B^2)∫_r_0^+∞dz z
√(z^2-r_0^2/( z^2-r_-^2) (
z^2-r_+^2) ) ,
I_3 = -√(32/e_A^2+e_B^2)∫_r_+^∞dz
z^2/√(( z^2-r_-^2) ( z^2-r_+^2) ).
Considering the change of variable ξ =z/r_+ and the definitions (<ref>) the integrals become
I_2 = 4√(2)r_+/√(e_A^2+e_B^2)∫_λ
_0^+∞dξ ξ√(ξ ^2-λ _0^2/( ξ
^2+λ _-^2) ( ξ ^2-1) ) ,
I_3 = -4√(2)r_+/√(e_A^2+e_B^2)∫_1^∞dξξ ^2/√(( ξ ^2+λ _-^2) (
ξ ^2-1) ) .
We perform the indefinite integral of I_2 giving
∫^ξdξ ξ√(ξ ^2-λ _0^2/( ξ
^2+λ _-^2) ( ξ ^2-1) ) = √(
( ξ ^2-λ _0^2) ( λ _-^2+ξ
^2) )/√(ξ ^2-1)
+√(( λ _0^2-1) )𝐄𝐄( . arcsin√(λ _0^2-1/ξ ^2-1)|1+λ
_-^2/1-λ _0^2) .
Taking the limits
lim_ξ→∞∫^ξdξ ξ√(ξ
^2-λ _0^2/( ξ ^2+λ _-^2) ( ξ
^2-1) ) = lim_ξ→∞ξ +𝒪(
1/ξ) ,
lim_ξ→λ _0∫^ξdξ ξ√(ξ
^2-λ _0^2/( ξ ^2+λ _-^2) ( ξ
^2-1) ) = √(( λ _0^2-1) )𝐄𝐄
( 1+λ _-^2/1-λ _0^2) .
Hence,
I_2=4√(2)r_+/√(e_A^2+e_B^2)[ lim_ξ→∞ξ -√(( λ _0^2-1) )𝐄𝐄
( 1+λ _-^2/1-λ _0^2) ].
The indefinite integral of I_3 gives
∫^ξdξξ ^2/√(( ξ ^2+λ
_-^2) ( ξ ^2-1) )=iλ _m[ 𝐄
( arcsinξ| -1/λ _-^2. ) -
𝐅( arcsinξ| -1/λ _-^2.
) ] .
Computing the limits we get
lim_ξ→∞∫^ξdξξ ^2/√((
ξ ^2+λ _-^2) ( ξ ^2-1) ) = lim_ξ→∞ξ +ℐ_λ _-+O( ξ ^-1)
,
lim_ξ→ 1∫^ξdξξ ^2/√(( ξ
^2+λ _-^2) ( ξ ^2-1) ) = iλ _-
[ 𝐄( -λ _-^-2) -𝐊( -λ
_-^-2) ] ,
where
ℐ_λ _- = [ iλ _-𝐄( -λ
_-^-2) -𝐄( -λ _-^2) -λ _-
𝐊( 1+λ _-^-2) .
. -2iλ _-𝐊( -λ _-^-2) +𝐊
( -λ _-^2) +λ _-^2𝐊( -λ
_-^2) ].
Therefore, the integral becomes
I_3 = -4√(2)r_+/√(e_A^2+e_B^2)[lim_ξ→∞ξ -𝐄( -λ _-^2) -λ
_-𝐊( 1+λ _-^-2)
. -iλ _-𝐊( -λ _-^-2) +
𝐊( -λ _-^2) +λ _-^2𝐊(
-λ _-^2) ] .
Replacing into the energy in eq. (<ref>), we find the result in eq.(<ref>)
E_QQ( λ _0) = 2r_+√(8/e_A^2+e_B^2
)[ λ _0^2/√(λ _0^2-1)𝐊(
1+λ _-^2/1-λ _0^2) .
+𝐄( -λ _-^2) +λ _-𝐊(
1+λ _-^-2) +iλ _-𝐊( -λ
_-^-2)
. -√(( λ _0^2-1) )𝐄(
1+λ _-^2/1-λ _0^2) -( 1+λ
_-^2) 𝐊( -λ _-^2) ]
§.§ t' Hooft loop
The effective potential V_eff is given in eq.
(<ref>).
Replacing explicitly the functions and using the definition in eq. (<ref>) leads to
V_eff = r_+e_B/2√(η ^2+1)1/ξ√((
ξ ^2-λ _0^2) ( ξ ^2-1) ( ξ
^2+λ _-^2) )
×√(( 4Q_B^2r_+^-4e_B^-2( ξ
^2+λ _0^2-2) +( η ^2+1) ( ξ
^2+λ _0^2-1+λ _-^2) ) /2( λ
_0^2-1) [ 4Q_B^2r_+^-4e_B^-2( λ
_0^2-1) +( η ^2+1) ( λ _0^2+λ
_-^2) ] ) .
We compute analytically the integrals in the BPS bound in
which λ _-=1 and Q_B=±e_B/e_AQ_A implying Q_A=e_A/2r_+^2. In this limit the effective potential
simplifies to
V_eff^BPS = r_+1/2√(e_A^2+e_B^2/2(
λ _0^2-1) ( ( e_A^2+2e_B^2) λ
_0^2+e_A^2) )
×√(1/ξ ^2( ξ ^2-λ _0^2)
( ξ ^4-1) ( e_A^2( ξ ^2+λ
_0^2) +2e_B^2( ξ ^2+λ _0^2-1) ) .
)
The indefinite integral (<ref>) gives
∫^ξdξ r_+/V_eff^BPS = 2/e_B√(
2( λ _0^2-1) ( ( η ^2+2) λ
_0^2+η ^2) /η ^2+1)
×∫ξ dξ/√(( ξ ^2-λ _0^2)
( ξ ^4-1) ( η ^2( ξ ^2+λ
_0^2) +2( ξ ^2+λ _0^2-1) ) ) ,
= 1/e_B√(2( λ _0^2-1) (
( η ^2+2) λ _0^2+η ^2) /( η
^2+1) ( λ _0^2( η ^2+2) -1) )
×𝐅( . arcsin√(2( -1+(
2+η ^2) λ _0^2) ( ξ ^2-1) /(
2λ _0^2+η ^2( 1+λ _0^2) ) (
ξ ^2-λ _0^2) )|( 1+λ
_0^2) ( 2λ _0^2+η ^2( 1+λ
_0^2) ) /-4+4( 2+η ^2) λ _0^2
).
Taking the limit ξ→ r_0 we find
lim_ξ→λ _0r_+∫^ξdξ/V_eff
= 1/e_B√(2( λ _0^2-1) (
( η ^2+2) λ _0^2+η ^2) /( η
^2+1) ( λ _0^2( η ^2+2) -1) )
( -i) 𝐊( 1-( 1+λ _0^2)
( 2λ _0^2+η ^2( 1+λ _0^2) ) /
-4+4( 2+η ^2) λ _0^2) .
Therefore, the profile of the string is
± x( ξ) = 1/e_B√(2( λ
_0^2-1) ( ( η ^2+2) λ _0^2+η
^2) /( η ^2+1) ( λ _0^2( η
^2+2) -1) )
×[ 𝐅( . arcsin√(2(
-1+( 2+η ^2) λ _0^2) ( ξ
^2-1) /( 2λ _0^2+η ^2( 1+λ
_0^2) ) ( ξ ^2-λ _0^2) )
|( 1+λ _0^2) ( 2λ
_0^2+η ^2( 1+λ _0^2) ) /-4+4(
2+η ^2) λ _0^2) .
. +i𝐊( 1-( 1+λ _0^2)
( 2λ _0^2+η ^2( 1+λ _0^2) ) /
-4+4( 2+η ^2) λ _0^2) ] .
The monopole-anti-monopole separation can be deduced easily from the above
expression by taking the limit (<ref>). We obtain,
L_MM^BPS( λ _0) = 2/e_B√(
2( λ _0^2-1) ( ( η ^2+2) λ
_0^2+η ^2) /( η ^2+1) ( λ
_0^2( η ^2+2) -1) )
×[ 𝐅( . arcsin√(2(
-1+( 2+η ^2) λ _0^2) /( 2λ
_0^2+η ^2( 1+λ _0^2) ) )|( 1+λ _0^2) ( 2λ _0^2+η
^2( 1+λ _0^2) ) /-4+4( 2+η ^2)
λ _0^2) .
. +i𝐊( 1-( 1+λ _0^2)
( 2λ _0^2+η ^2( 1+λ _0^2) ) /
-4+4( 2+η ^2) λ _0^2) ] .
We compare it with the approximate function
(<ref>)
for the separation, replacing the functions explicitly we find
L̂_MM( r_0) =π√(2)/√(
e_A^2+e_B^2)( e_A^2+2e_B^2)
r_0^2/r_+^2+e_A^2/( e_A^2+2e_B^2)
r_0^2/r_+^2-e_B^2√(r_0^2/r_+^2-1/
r_0^2/r_+^2+1) .
For the energy of the t' Hooft loop we have a similar expression to the one obtained when computing the
energy of the Wilson loop,
E_MM( r_0) =F( r_0) L_MM( r_0)
+I_2+I_3 ,
where
I_2 = 4r_+^2/√(η ^2+1)∫_λ _0^+∞dξ√(ξ ^2( ξ ^2-λ _0^2) (
η ^2( ξ ^2+λ _0^2) +2( ξ ^2+λ
_0^2-1) ) /ξ ^4-1) ,
I_3 = -2∫_r_+^+∞dzG( z) =-4r_+^2/
√(( 1+η ^2) )∫_1^∞dξ ξ√(
2ξ ^2+η ^2( ξ ^2+1) /( 1+ξ ^2) ) .
These integrals are quite involved and present technical
difficulties to be performed in terms of known functions. Therefore we
compute them numerically up to a large value of the upper limit ξ̃
_max. Since ξ̃_max is finite the integrals are
convergents and we can write them together in terms of a single integral which depends
on ξ̃_max
E_MM( r_0) . =. F( r_0)
L_MM( r_0) -[ 4r_+^2/√(1+η ^2)
∫_1^λ _0 ξ√(2ξ ^2+η ^2( ξ
^2+1) /( 1+ξ ^2) ).
-lim_ξ̃_max→∞∫_λ _0^ξ̃_maxdξ( √(ξ ^2( ξ ^2-λ
_0^2) ( η ^2( ξ ^2+λ _0^2)
+2( ξ ^2+λ _0^2-1) ) /ξ ^4-1).
. . -ξ√(2ξ
^2+η ^2( ξ ^2+1) /( 1+ξ ^2) ))
] .
We verify that the integral converges to a limiting value for ξ̃
_max large enough and much bigger than λ _0=r_0/r_+.
The limiting value of the function L_MM( r_0) when r_0→∞ is non-zero and is given in terms of a
characteristic length of the Little String Theory (LST). This asymptotic
behavior matches with the asymptotic behavior of the background
(<ref>)
, to see this fact explicitly we compute the separation length of the 't
Hooft loop of the background
(<ref>)
by taking the limit of the end-points separations
(<ref>)
. The relevant functions for the t' Hooft loop of the background
(<ref>)
are the metric function
f_s( r) =1/2( e_A^2+e_B^2) ,
and the effective potential
V_eff( r,r_0) =√(2( e_A^2+e_B^2)
)/4r_0^2√(r^2( r^4-r_0^4) ) .
Then, the profile of the string in the bulk is
± x( r) =√(2/e_A^2+e_B^2)arctan(
√(r^4-r_0^4)/r_0^2) .
The end-point separation is given by the limit
(<ref>)
and leads to the following constant
L_MM( r_0) =π√(2/e_A^2+e_B^2) .
Thus, all the strings in the background
(<ref>)
that explores the bulk has the same end points separation. This value
coincides with the limiting value of the separation length of the background
(<ref>)
and with a LST characteristic length. Therefore the UV behavior of the dual
theory is driving by the LST. To capture the field theory behavior we
introduce a cut-off to rule out the non-local effects of the LST. In that
case the system present a phase transition between the unstable
configurations to the short strings configuration.
The energy of the t' Hooft loop of the background
(<ref>)
is given by
(<ref>)
with
F( r) =√(e_A^2+e_B^2/2) r^2, G(
r) =2√(e_A^2+2e_B^2/e_A^2+e_B^2)r .
We find that the energy is zero. This implies that energy of the t' Hooft
loop of the disconnected solution is the same to the connected one.
§.§ Entanglement entropy
The profile of the 8-dimensional surface is governed by the function r=r( x) with equations coming from the minimisation of eq.
(<ref>) which gives an equation like (<ref>) for the effective potential
V_eff=1/2r_0^2√(f_s( r_0) )√(
r^2f_s( r) ( r^4f_s( r)
-r_0^4f_s( r_0) ) ) .
The integration of (<ref>) for this potential subject to the initial
condition x( r_0) =0 gives
± x( ξ) = ∫_r_0^r2r_0^2√(f_s(
r_0) )dr/√(r^2f_s( r) ( r^4f_s(
r) -r_0^4f_s( r_0) ) )
= √(8/e_A^2+e_B^2)√(( λ
_0^2-1) ( λ _0^2+λ _-^2) )
×∫_λ _0^ξξ dξ/√(( ξ
^2-λ _0^2) ( ξ ^2-1) ( ξ
^2+λ _-^2) ( ξ ^2+λ _0^2+λ
_-^2-1) ).
Performing the indefinite integral
ℐ_L( ξ) ≡ ∫^ξξ dξ/√(
( ξ ^2-λ _0^2) ( ξ ^2+λ
_-^2) ( ξ ^2-1) ( ξ ^2+λ
_0^2+λ _-^2-1) )
= 1/λ _0^2+λ _-^2𝐅( .
arcsin√(( λ _0^2+λ _-^2) (
ξ ^2-1) /( λ _-^2+1) ( ξ ^2-λ
_0^2) )|( 1+λ _-^2) (
λ _-^2+2λ _0^2-1) /( λ
_0^2+λ _-^2) ^2) .
The limit ξ→λ _0 gives
lim_ξ→λ _0ℐ_L( ξ) =-
i/λ _0^2+λ _-^2𝐊( ( λ
_0^2-1) ^2/( λ _0^2+λ _-^2) ^2
) .
Thus, the profile of the surface in the bulk is
± x( ξ) = √(8/e_A^2+e_B^2)√(
λ _0^2-1/λ _0^2+λ _-^2)[ i𝐊
( ( λ _0^2-1) ^2/( λ
_0^2+λ _-^2) ^2) .
. +𝐅( . arcsin√(( λ
_0^2+λ _-^2) ( ξ ^2-1) /( λ
_-^2+1) ( ξ ^2-λ _0^2) )|( 1+λ _-^2) ( λ _-^2+2λ
_0^2-1) /( λ _0^2+λ _-^2) ^2
) ].
The length of the interval is given by the limit in eq.(<ref>). In term of the variables in eq.(<ref>) gives
L_EE( λ _0) = 2√(8/e_A^2+e_B^2)
√(λ _0^2-1/λ _0^2+λ _-^2)[ i
𝐊( ( λ _0^2-1) ^2/( λ
_0^2+λ _-^2) ^2) .
. +𝐅( . arcsin√(( λ
_0^2+λ _-^2) /( λ _-^2+1) )
|( 1+λ _-^2) ( λ
_-^2+2λ _0^2-1) /( λ _0^2+λ
_-^2) ^2) ] .
This is our expression in eq.(<ref>).
We cannot see a phase transition in this background. However, if we put a
cutoff at ξ _cutoff=r_cutoff/r_+ the
coordinate λ _0<ξ <ξ _cutoff, the double-valued character of L_EE shows, as in Figure <ref>.
The renormalized EE in eq.(<ref>) written in the variables of eq.(<ref>) reads
S_EE( λ _0) =𝒩/G_Nr_+^2[
∫_λ _0^∞dξ√(ξ ^2( ξ
^2+λ _-^2) ( ξ ^2-1) /( ξ
^2-λ _0^2) ( ξ ^2+λ _0^2+λ
_-^2-1) )-∫_1^∞ξ dξ] .
This integral can be done analytically. In the BPS limit it becomes
particularly simple,
S_EE^BPS( λ _0) =𝒩/G_Nr_+^2
[ ∫_λ _0^∞dξ√(ξ ^2( ξ
^4-1) /( ξ ^2-λ _0^2) ( ξ
^2+λ _0^2) )-∫_1^∞ξ dξ] .
The indefine integral reads
ℐ_S^BPS( ξ) = ∫^ξdξ√(ξ
^2( ξ ^4-1) /( ξ ^2-λ _0^2)
( ξ ^2+λ _0^2) )
= 1/2𝐄( . arcsinξ ^2/λ
_0^2|λ _0^4) .
Expanding for large ξ and ξ→λ _0 we find
lim_ξ→∞ℐ_S( ξ)
= lim_ξ→∞1/2ξ ^2+1/2λ
_0^2[ -λ _0^4𝐄( λ _0^-4)
+λ _0^2𝐄( λ _0^4) -𝐊(
λ _0^-4) +λ _0^4𝐊( λ
_0^-4) ]
+𝒪( ξ ^-2) ,
lim_ξ→λ _0ℐ_S( ξ) =
1/2𝐄( λ _0^4) .
Replacing in these expressions in the entanglement entropy of the BPS
configuration (<ref>) we obtain the expression in eq.(<ref>),
S_EE^BPS( λ _0) =𝒩/G_Nr_+^2
[ 1/2λ _0^2( -λ _0^4𝐄(
λ _0^-4) -𝐊( λ _0^-4) +λ
_0^4𝐊( λ _0^-4) ) +1/2]
.
The limiting of the interval length
(<ref>)
value when r_0→∞ is non-zero and coincides with the
characteristic length of the Little String Theory. The background
(<ref>)
and the fibred one coincides in the UV leading to a regime in which the LST
dominates the behavior of dual theory. To verify this point we compute
profiles of the strings in the bulk
± x( r) =√(2/e_A^2+e_B^2)arctan(
√(r^4-r_0^4)/r_0^2) ,
which end points separation at r→∞ gives interval length
L_EE( r_0) =π√(2/e_A^2+e_B^2) .
One again, in order to capture the field theoretical behaviour of the dual
theory we add a cut-off to the observables which allow us to recover
expected behavior of a confining field theory.
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|
http://arxiv.org/abs/2307.04978v1 | 20230711023526 | Diffusion idea exploration for art generation | [
"Nikhil Verma"
] | cs.CV | [
"cs.CV"
] |
top=2.5cm,bottom=2.5cm,outer=2.5cm,inner=2.5cm
1
./references/IEEEtran
|
http://arxiv.org/abs/2307.04105v1 | 20230709055525 | Towards Assumption-free Bias Mitigation | [
"Chia-Yuan Chang",
"Yu-Neng Chuang",
"Kwei-Herng Lai",
"Xiaotian Han",
"Xia Hu",
"Na Zou"
] | cs.LG | [
"cs.LG",
"cs.CY"
] |
Texas A&M University
[email protected]
Rice University
[email protected]
Rice University
[email protected]
Texas A&M University
[email protected]
Rice University
[email protected]
Texas A&M University
[email protected]
Despite the impressive prediction ability, machine learning models show discrimination towards certain demographics and suffer from unfair prediction behaviors. To alleviate the discrimination, extensive studies focus on eliminating the unequal distribution of sensitive attributes via multiple approaches. However, due to privacy concerns, sensitive attributes are often either unavailable or missing in real-world scenarios. Therefore, several existing works alleviate the bias without sensitive attributes. Those studies face challenges, either in inaccurate predictions of sensitive attributes or the need to mitigate unequal distribution of manually defined non-sensitive attributes related to bias. The latter requires strong assumptions about the correlation between sensitive and non-sensitive attributes. As data distribution and task goals vary, the strong assumption on non-sensitive attributes may not be valid and require domain expertise.
In this work, we propose an assumption-free framework to detect the related attributes automatically by modeling feature interaction for bias mitigation. The proposed framework aims to mitigate the unfair impact of identified biased feature interactions.
Experimental results on four real-world datasets demonstrate that our proposed framework can significantly alleviate unfair prediction behaviors by considering biased feature interactions.
Our source code is available at: https://anonymous.4open.science/r/fairint-5567
<ccs2012>
<concept>
<concept_id>10002951.10003227.10003351.10003269</concept_id>
<concept_desc>Information systems Collaborative filtering</concept_desc>
<concept_significance>300</concept_significance>
</concept>
<concept>
<concept_id>10010147.10010257.10010293.10010319</concept_id>
<concept_desc>Computing methodologies Learning latent representations</concept_desc>
<concept_significance>300</concept_significance>
</concept>
</ccs2012>
Towards Assumption-free Bias Mitigation
Na Zou
Accepted 08-Jul-2023. Received 18-Jun-2023; in original form 23-May-2023
============================================================================
§ INTRODUCTION
Machine learning models have shown superiority in various high-stake decision-makings <cit.>, and have been deployed in many real-world applications, such as credit scoring <cit.>, loan approval <cit.>, criminal justice <cit.>, education opportunity <cit.>.
However, machine learning models show discrimination towards certain demographics and suffer from biased prediction behavior, which may negatively impact the minority groups in those application fields.
For example, COMPAS, a recidivism prediction system, shows discrimination towards African-American offenders with a higher possibility of becoming a recidivist two years after leaving prison <cit.>. Recent works focus on bias mitigation techniques to alleviate discrimination in machine learning models.
Existing works to tackle the fairness issues are generally based on two groups of assumptions, i.e., bias assumptions and correlation assumptions.
For the works based on bias assumptions, they mitigate bias with known distributions of sensitive attributes by fairness regularization <cit.>, contrastive learning <cit.>, adversarial learning <cit.>, disentanglement representations <cit.>, and representation neutralization <cit.>.
However, due to privacy concerns, sensitive attributes are often missing <cit.>. Therefore, existing works adopt clustering methods <cit.> and an auxiliary module <cit.> to simulate the sensitive attributes. However, they often suffer from the inaccuracy of the predicted sensitive attributes when adopting clustering algorithms <cit.>.
Thus, the work based on correlation assumptions, FairRF <cit.>, addresses the unfair issues with a strong assumption that the unfair model prediction actually comes from the relationship between sensitive attributes and a set of predefined related non-sensitive attributes.
In this paper, we argue that correlation assumptions between sensitive and non-sensitive attributes may not be valid as data distribution and task goals vary. For example, FairRF <cit.> predefines (inherently assumes) that the related features of gender in the Adult dataset are age, relationship, and marital status. To show this assumption is invalid, we conducted an experiment to explore the relationship between gender and all other features. This assumption is not consistent with the linear relationships between gender and all other features, as shown in Figure <ref>. Additionally, domain expertise and knowledge are required to predefine the related features.
Therefore, we raise the following question: Can we achieve fairness without assuming a predefined relation between sensitive and non-sensitive attributes?
To tackle the limitations of 1) the correlation assumption that unfair model prediction comes from the handcrafting predefined related attributes and 2) the further fairness problems caused by feature interactions, we aim to develop an assumption-free framework to automatically detect and integrate feature interactions for bias mitigation.
It is nontrivial to achieve our goal due to the following challenges.
First, in the real-world scenario, implicit bias of feature interactions are difficult to be detected, especially when sensitive attributes are missing.
Specifically, it is hard to find the high-order statistical interactions that may lead to biased predictions of deep neural networks due to the complex model structures.
Thus, when neither the sensitive attributes are available nor make strong correlation assumptions on related features, it becomes very challenging to identify the biased feature interactions.
For example, identifying the biased feature interactions among all the combinations of features without the sensitive attributes may lead to numerous candidate features interactions, which make models extremely hard to learn the distribution of actual biased feature interactions.
Second, it is challenging to mitigate bias in feature interactions due to the uneven distribution among feature interactions.
For example, without considering the potential uneven distribution of the feature interactions, trained prediction models may fail to detect and mitigate the bias in feature interactions.
To address the aforementioned challenges, we propose FairInt, an assumption-free framework to automatically identify and further mitigate the bias in feature interactions.
Specifically, we develop a sensitive attribute reconstructor for tackling a situation where sensitive attributes are unavailable during the inference stage.
By designing a sensitive-oriented attention score, we develop a biased interaction detection layer to automatically identify the biased feature interactions and then embed the biased interaction information into the latent representation.
It is different from traditional deep neural networks that model feature interactions among all possible feature combinations and cannot identify specific biased feature interactions.
To equalize the probability distribution of sensitive attributes, we design two bias regularizations for debiasing the latent representation that contains biased interaction information.
These two regularizations debias the feature interactions by minimizing the divergence of latent space and the model predictions between different sensitive attribute groups.
We evaluate our framework on four real-world datasets across three different application domains, which include finance, education, and healthcare.
Compared with baseline models, the experimental results demonstrate that the FairInt can successfully further mitigate the biased prediction behaviors while providing similar performances of downstream tasks by considering biased feature interactions.
Moreover, by observing the modeled feature interaction, the FairInt shows the ability to provide better explainability via the designed sensitive-oriented attention score. We highlight our contributions as follows:
* We argue that the related attributes with high correlations to sensitive attributes that can be identified by prior knowledge is problematic. Because the correlations between sensitive and non-sensitive attributes will be changed with different models.
* We propose an assumption-free framework to automatically identify and further mitigate the biased feature interactions. Our framework does not need to handcraft related attributes for mitigating the unfair model prediction that comes from the interactions between sensitive and non-sensitive attributes. Instead, the proposed framework automatically identifies related attributes without prior knowledge during the inference stage.
* Experimental results on several real-world datasets demonstrate the effectiveness of the proposed FairInt framework.
Additionally, our framework provides better explainability via observing the attention weights between sensitive and non-sensitive attributes.
§ PRELIMINARIES
In this section, we introduce the existing bias mitigation strategies for deep neural networks and feature interaction modeling methods that inspire our proposed framework.
§.§ Bias Mitigation
To tackle the prejudicial decisions problem in deep learning models, there is increased attention to bias mitigation methods in recent studies <cit.>.
Extensive approaches apply regularization-based methods to the objective function of the proposed models, which require pre-hoc assumptions to develop.
Existing alleviating techniques are generally based on two groups of assumptions.
Bias Assumptions.
Because machine learning models show discrimination towards certain demographics, people assume that machine learning models have biased behaviors against certain groups.
With a known distribution of a sensitive attribute set, there are several advancements proposed to mitigate bias, such as:
1) Fairness regularization: the objective function of the bias mitigated models generally adds the fairness-related constraint terms <cit.>, which may penalize the prejudiced behaviors of the prediction models. Another existing work <cit.> compares the distributions of model predictions of different sensitive attributes and then minimizes KL-divergence between each sensitive attribute.
2) Adversarial learning: adversarial learning alleviates the biased effects from the known sensitive attributions by simultaneously building an Adversary with the Predictor of machine learning models.
One previous work <cit.> aims to leverage bias alleviation by proposing an adversarial learning strategy with the given distribution of sensitive attributes.
The model includes a Predictor, which accomplishes the downstream task predictions, and an Adversary, which predicts the target sensitive attributes.
The framework adopts adversarial training by minimizing Predictor and maximizing adversary, which aims to debias the unfair situations brought from Predictor.
3) Latent representation neutralization: one latent representation neutralization work <cit.> is to implicitly mitigate bias by adjusting the distribution of latent representations during the model training.
Correlation Assumptions.
In the real-world scenario, it is hard to get the true distribution of sensitive attributes due to privacy concerns, we thus assume that the unfair model predictions are caused by certain related attributes that have high correlations to sensitive attributes.
Specifically, when we face the fairness issue for model prediction, it is challenging to leverage the model bias if we lack sensitive feature information.
Thus, there are some works that focus on eliminating prediction bias under the constraint of unknown sensitive attributes' distribution.
ARL <cit.> utilizes adversarial learning based on Rawlsian Max-Min fairness objectives. However, this approach could be too strict in enhancing fairness across groups, and it is hard to maintain the performance of downstream tasks.
FairRF <cit.> addresses the biased issues by leveraging the relatedness between a set of related non-sensitive attributes and sensitive attributes.
This work assumes that the bias of model prediction actually comes from the high correlation between non-sensitive attributes and sensitive features.
In this manner, a fair model can be achieved by the proposed objective function of alleviating the relatedness between non-sensitive attributes and sensitive attributes. Formally, the objective function of FairRF can be illustrated as follows:
Let f_i ∈ F_n be a set of predefined related non-sensitive attributes, where F_n is a set of non-sensitive features, FairRF applies correlation regularization R_related on each f_i to make trained model fair toward sensitive attribute s by calculating the following function:
min_θℛ_related = ∑_i=1^Kλ_i ·ℛ(f_i, ŷ),
where λ is the weight for regularizing correlation coefficient between x^i and ŷ.
However, this correlation assumption between sensitive and non-sensitive attributes may be sub-optimal, because it requires strong assumptions on feature dependencies.
In other words, data-specific and distribution similarity are necessary.
For example, when we define the related features of sensitive features Gender are three non-sensitive features, which are Age, Relation, and Marital-Status, an accompanying assumption comes up that the three non-sensitive features have top-3 highest correlation with the given sensitive feature Gender.
Nevertheless, it is possible that the true highest correlation-related features of the sensitive feature in the dataset are not obvious for human beings, so we cannot define it correctly.
For instance, maybe the highest related features of the sensitive feature gender are color of eyes and sleeping quality in a certain dataset, and it is hard for humans to associate the two features as related features of gender.
In our work, instead of adopting assumptions on bias feature distribution with its related features, we propose an assumption-free framework for automatically detecting the related features for bias mitigation.
§.§ Learning Feature Interactions
One major advantage of neural networks is their ability to model complex interactions between features by automatic feature learning.
In the territory of click-through rate prediction, CTR prediction, feature interaction modeling has been playing a key role in improving downstream task performances by modeling different orders of feature combinations.
Instead of multiple layers of non-linear neural network approaches which suffer from inefficient and lack of good explanation of feature interactions <cit.>, there are popular approaches that are able to explicitly model different orders of feature combinations and meanwhile offer good model interpretability.
One of the previous works models feature interactions by calculating the inner products between a feature embedding and a trainable matrix, afterward calculating the Hadamard product of another feature embedding <cit.>.
AutoInt <cit.> models feature interactions by adopting the key-value attention mechanism and using the conducted attention weights between all feature pairs to weighted-sum the all input feature embedding.
AutoInt utilizes the inner product operator ψ(·, ·) to define the similarity between two feature embeddings e_j and e_c, and leverages it to compute the attention weights under a specific attention head h by the following equation:
a^(h)_j, c = exp(ψ^(h)(e_j, e_c))/∑_n=1^Nexp(ψ^(h)(e_j, e_n)),
where N represents the number of input features.
The classic self-attention-based approach considers all feature pairs for feature interaction learning, therefore it is difficult to significantly identify the bias between feature pairs containing target sensitive attributes.
In our work, we only consider the feature pairs which treat target sensitive attributes as a Query of attention components to identify the feature interactions between sensitive and non-sensitive attributes for further alleviating the biased interactions.
Our framework can automatically detect the related features for bias mitigation.
§.§ Problem Definition
We first define the notations used in this work.
Let X be the input data set and Y be the ground truth label set of the model output, where X = { x_1, …, x_p} is the p-kind attribute set and Y∈{0, 1} is the binary label set.
Among the input attribute set X = S∪C, where sensitive attributes set S (e.g. gender, race, marital status) and non-sensitive attributes set C.
We observe that the biased feature interactions are the influential factor in yielding fairness of predictive results.
Formally, we define the sensitive feature interaction set as ℐ_s = {ℐ(s, c_1), … , ℐ(s, c_p-1) | ∀ c_j ∈C}, where ℐ(·, ·) denotes an feature interaction between any two features, and s ∈S is a sensitive attribute.
For example, an interaction between a sensitive attribute gender and non-sensitive attribute job can be denoted as ℐ(gender, job).
Based on modeling the feature interactions throughout the prediction models, the biased interactions from ℐ_s eventually lead to bias on prediction tasks.
Based on the definitions and the intuitions above, we consider the interaction bias from prediction model f(X, θ) ≡ p(g(X)), where θ is the model parameters and p(·) is a single-layer prediction head of d-dimensional feature embedding encoder g(·): X→ℝ^d.
In our work, let ℐ_s be the sensitive feature interaction set learned from prediction model f(·), we aim to identify the biased interaction that appears in ℐ_s such that the detected biased interactions are alleviated during the prediction model training.
§ METHODOLOGY
In this section, we introduce an assumption-free fair mitigation framework, FairInt, to alleviate the biased feature interactions.
Figure <ref> illustrates our FairInt framework with two components: Assumption-free Bias Detection, which includes Sensitive Attributes Reconstructor (SAR) and Bias Interaction Detection (BID) layer, and Interaction-wise Bias Mitigation, which includes the regularizations Fairness Constraint (FC) and Interaction Fairness Constraint (IFC).
Our goal is to encourage the classifier to disentangle the biased interaction between sensitive and non-sensitive attributes and instead focus more on learning task-relevant information.
Assumption-free Bias Detection aims at identifying bias within feature interactions without predefined related features, and Interaction-wise Bias Mitigation focuses on alleviating the identified feature interaction bias.
In the following sections, we give a comprehensive description of our FairInt framework.
We first illustrate the details of the proposed bias detection component (Sec. <ref>).
Then, we introduce our two bias mitigation components (Sec. <ref>).
Finally, we demonstrate how to learn the fair predictor through our FairInt framework (Sec. <ref>).
§.§ Assumption-free Bias Detection
sensitive attributes s ∈S are generally unavailable in real-world scenario during the inference stage. Many existing works mitigate the interaction bias under the assumption of the known distribution of sensitive attributes. However, in real-world scenarios, the unavailability of sensitive attributes exists due to various reasons, such as legal issues, which make most of the existing advancements unworkable. To tackle the problems, we develop two corresponding components: Sensitive Attributes Reconstructor (SAR) for sensitive attributes bias assumption-free, and Bias Interaction Detection (BID) for feature interaction assumption-free. Our assumption-free framework aims to disentangle the hand-crafted assumptions of the feature dependency between sensitive and specific non-sensitive attributes during the debiasing process.
Sensitive Attributes Reconstructor (SAR).
Since sensitive attributes s ∈S are generally unavailable in real-world scenario during the inference stage, we design Sensitive Attributes Reconstructor (SAR) to simulate the sensitive attributes for alleviating the implicit interaction bias obtaining in non-sensitive attributes.
Specifically, we aim to generate a pseudo-sensitive attribute ŝ by imitating the distribution of sensitive attributes s ∈S throughout our proposed reconstructor, which brings out the biased interaction between the sensitive attributes and all other non-sensitive features.
Let the input attribute set be x ∈X without the sensitive attributes s ∈S. The objective of Sensitive Attributes Reconstructor (SAR) is to construct a reconstructor f to generate a pseudo-sensitive attribute ŝ for identifying the implicitly biased interactions toward non-sensitive features. The generating process of a pseudo-sensitive attribute can be formally illustrated as follows:
ŝ = SAR(e_x/s; Θ_r),
where Θ_r is the trainable parameters of reconstructor r, and e_x/s denotes the latent representation set of input features x without sensitive attribute s.
Specifically, we leverage the embeddings of all non-sensitive attributions to generate a pseudo-sensitive attribute vector. This makes the reconstructor extract the correlated information between sensitive and non-sensitive features. During training stage, the reconstructor loss ℒ_SAR can be shown as follows:
ℒ_SAR≡min_Θ_r∑_i=1^N (ŝ_i - s_i)^2,
where N is the number of training instance.
The effectiveness of SAR was evaluated by predicting unavailable sensitive attributes using non-sensitive features from Adult and Law School datasets. SAR achieved 87% accuracy for predicting Sex in Adult and 94% for predicting Race in Law School.
The results show that SAR can achieve impressive performance by capturing the correlations between non-sensitive attributes and unobserved sensitive attributes.
Besides predicting the pseudo-sensitive attributes ŝ, SAR advantages our FairInt to better capture the interactions between unobserved sensitive and non-sensitive attributes.
Bias Interaction Detection (BID) Layer.
Optimizing Eq. <ref> in SAR generates a pseudo-sensitive attribute ŝ as a sensitive sensitive attribute, which allows our proposed FairInt to quantitatively analyze the interaction between pseudo-sensitive attributes and non-sensitive attributes. Thus, we propose Bias Interaction Detection (BID) to identify the highly potential biased interactions with the generated pseudo-sensitive attribute.
We first let all the input features be the p-kind attribute set X = {x_1, …, x_p} which contains categorical and numerical features.
Because categorical features are too sparse to learn, we map all the input features into low-dimensional spaces with the unique feature embeddings e_i. The formula can be illustrated as e_i = M_i x_i,
where M_i is an embedding lookup matrix corresponding to feature x_i with dimension d.
Feature interactions are typically modeled by either the inner product similarity or attention scoring mechanism between two feature embeddings <cit.>. For instance, AutoInt <cit.> utilizes the multi-head self-attention mechanism to model high-order feature interactions for improving the downstream task's performance.
AutoInt learns feature interaction within a hierarchical representation structure, which has proven to be effective in several machine learning territories <cit.>.
Especially, self-attention-based mechanism has been utilized in several machine learning areas for capturing the importance within features of input instances <cit.>.
In our work, we exploit self-attention machanism <cit.> to model feature interactions. The main goal of our framework is to mitigate the biased feature interaction for the model predictions but without predefined assumptions.
Therefore, based on the ability of self-attention mechanism to identify important feature interactions, we design Bias Interaction Detection (BID) to point out the key biased interactions of pseudo-sensitive attributes.
Unlike original self-attention mechanism that calculates the attention weights between all the feature two by two, we focus on modeling the feature interactions only between the sensitive pseudo-sensitive attribute ŝ and other non-sensitive features by computing their attention weights.
Specifically, we model the interactions between a pseudo-sensitive attribute ŝ and one non-sensitive features c ∈C with attention head h as a_ŝ, c, which can be calculated as follows:
a_ŝ, c = exp(ψ^h(ê_̂ŝ, e_c))/∑_c ∈Cexp(ψ^h(ê_̂ŝ, e_c)),
where ê_̂ŝ and e_c are the low-dimensional embedding of ŝ and c, and ψ^h(ê_̂ŝ, e_c) denotes as the scoring operator to evaluate the similarity between ê_̂ŝ and e_c.
In this paper, we adopt dot product as an example for ψ^h(ê_̂ŝ, e_c), which can be illustrated as follows:
ψ^h(ê_̂ŝ, e_c) = ⟨ W^h_Queryê_̂ŝ, W^h_Key e_c ⟩,
where ⟨· , ·⟩ is inner product operator, and W^h_Query and W^h_Key are embedding matrices for ê_̂ŝ and e_c. The biased interaction scores can now be defined as a_ŝ, c in this manner.
After obtaining the biased interaction scores between the sensitive and non-sensitive features, we generate the biased interaction embeddings ê^H_s to represent the biased interactions for bias mitigation. We formally define the biased interaction embeddings as following formula:
ê^H_s = _h=1^|H| ∑_c=1^C a_ŝ, c (W^h_value· e_c),
where W^h_value is a trainable embedding matrix, and ‖ denotes the concatenation operator for all biased interaction embeddings of each attention layer h ∈H.
§.§ Interaction-wise Bias Mitigation
After receiving the detected bias interaction embeddings ê^H_s, we focus on alleviating the bias from feature interactions.
Our goal is to equalize the conditional probability distribution of bias interaction embeddings given different sensitive attributes s ∈S. However, the sensitive attribute information in ê^H_s can be easily perturbed due to the imbalance amounts of sensitive and non-sensitive attributes. This may affect the bias mitigation performance since the alleviation process requires an explicate sensitive attribute as a pivot to mitigate. Hence, we adopt a residual neural network (ResNet) <cit.> to enrich the information of pseudo-sensitive attributes, which we can formally reveal as follows:
e_ŝ = ReLU(ê^H_s + W_Res·ê_̂ŝ),
where W_Res is the residual model weight and ê_̂ŝ is the embedding of pseudo-sensitive attributes.
In this work, we design two fairness constraints: Interaction Fairness Constraint and Fairness Constraint for biased interaction mitigation.
Interaction Fairness Constraint (IFC) Loss.
In order to mitigate the detected bias interactions from different sensitive attribute groups, we design the Interaction Fairness Constraint (IFC) loss to minimize the KL-divergence between the sensitive attribute groups. IFC can then ensure the equivalent information gained from each feature interaction.
Formally, IFC can be formulated as follows:
ℒ_IFC = ∑_i ∈S∑_j ∈S/iKL(e_[ŝ≈ i], e_[ŝ≈ j]),
where KL(·) denotes the KL-Divergence, and e_[ŝ≈ i] is the subset of e_ŝ that is more similar to sensitive attributes i ∈S. To the convenience of our work, we set the hierarchical boundary with expected value of uniform distributed S to distinguish which group ŝ belongs to in S.
IFC loss mitigates the bias information of the latent representation by calculating the KL-divergence scores as biased scores between each group in pseudo-sensitive attributes S.
Therefore, by adding ℒ_IFC as a regularization term to our framework, the bias feature interaction of latent representation e_ŝ can be alleviated.
Fairness Constraint (FC) Loss.
Although our proposed IFC mitigates most of the biased interaction information from the embedding aspect, the remaining biased interaction may be amplified by prediction models and generate unfair task predictions.
To alleviate the unfairness of model predictions on downstream tasks, we adopt the Fairness Constraint (FC) loss toward pseudo-sensitive attributes ŝ. In this work, we focus on classification tasks.
Our proposed FC aims to mitigate biased prediction behaviors ŷ by computing the absolute differences of the cross entropy between every two of each pseudo-sensitive attribute (ŝ_i, ŝ_j) ∈S.
Formally, FC can be formulated as follows:
ℒ_FC = ∑_i ∈S∑_j ∈S/i |CE_[ŝ≈ i] - CE_[ŝ≈ j]|,
where CE_[ŝ≈ i] is cross entropy which belongs to a certain sensitive attribute i ∈S.
Based on the meaning of cross entropy, it reflects the correctness of classification prediction ŷ. The idea of FC loss is to ensure the discrepancy of the correctness of ŷ by giving every two of each pseudo-sensitive attribute.
Thus, ℒ_FC can effectively alleviate the prejudiced model predictions among the pseudo-sensitive attribute set.
§.§ Fair Classifier with FairInt
Here we discuss how to incorporate the IFC loss ℒ_IFC and the FC loss ℒ_FC with a classifier to alleviate the biased interaction.
We adopt the reconstructor loss ℒ_SAR to our framework for training the SAR to generate the pseudo-sensitive features.
As the IFC loss can be a stand-alone optimization objective function, it is capable of mitigating bias feature interaction in latent representations for any kinds of classification model.
In our work, we evaluate the effectiveness of our framework on a one-layer multi-layer perceptron as the classification model, which can be replaced by any deeper or more powerful models.
To train a classification model with our proposed FairInt framework, we optimize the cross entropy loss ℒ_0.
We then incorporate ℒ_0 with the Interaction Fairness Constraint (IFC) loss ℒ_IFC, Fairness Constraint (FC) loss ℒ_FC, and reconstructor loss ℒ_SAR as the final objective function to fair classifier training.
Our proposed IFC loss and FC loss help the classification models mitigate the bias feature interactions from the views of latent representations and alleviate the prejudiced model predictions with given different kinds of sensitive attributes during training.
Specifically, we optimize the proposed FairInt by illustrating the following joint loss function:
ℒ_FairInt = ℒ_0 + λ_IFCℒ_IFC + λ_FCℒ_FC + ℒ_SAR,
where ℒ_FairInt denotes as the loss function to the proposed FairInt and λ_IFC and λ_FC are the weighting hyper-parameters to balance the biased interaction mitigating and feature interactions modeling.
By optimizing ℒ_FairInt, we can alleviate the bias model predictions by mitigating the detected bias feature interactions without defining any related and potentially biased features interactions in advance.
In inference stage, the trained FairInt framework can provide fair predictions without sensitive attributes.
§ EXPERIMENT
In this section, we empirically evaluate the proposed FairInt framework. We mainly focus on the following research questions:
* Compared with the existing baseline methods, can our assumption-free FairInt framework mitigate the unfair model prediction on the downstream tasks (Sec. <ref>)?
* Can our proposed Bias Interaction Detection layer identify the bias feature interaction and encode it in the latent representation (Sec. <ref>)?
* How do the proposed Interaction Fairness Constraint loss and Fairness Constraint loss in Eq. <ref> and Eq. <ref> impact the fairness of the classification model (Sec. <ref>)?
* How do the hyper-parameters impact the fairness performance of the proposed FairInt (Sec. <ref>)?
* How does our assumption-free FairInt framework automatically detect the related features and further mitigate the bias feature interaction (Sec. <ref>)?
§.§ Datasets
We consider four real-world tabular datasets <cit.> that are commonly used for studying fairness-aware classification, which include three application domains as shown in Table <ref>.
§.§ Baselines and Fairness Metrics
Besides the ARL <cit.> and FairRF <cit.> mentioned in Sec <ref>, we also leverage two fairness constraint regularization methods to train vanilla MLP models as baselines for comparing with our framework.
We adopt the Fair Constraint (FC) loss as a regularization to a vanilla MLP model as a baseline, where the FC loss is to mitigate biased behaviors toward model predictions, and it can be calculated as Eq. <ref>.
For another baseline, we apply a regularization-based mitigation method to a vanilla MLP model, Prejudice Remover <cit.>, which considers the mutual information for equalizing the distributions between two variables to alleviate biases.
For each dataset, both the two baselines are leveraged to the same vanilla MLP model which we will describe in the Sec. <ref>.
We also compare the proposed FairInt to two other baselines including vanilla MLP classification models and the CTR prediction model AutoInt, which modeling feature interaction by adopting the key-value attention mechanism to improve the performance on CTR prediction tasks.
We use two group fairness metrics to evaluate the fairness of prediction models: Demographic Parity (Δ DP) <cit.> and Equalized Odds (Δ EO) <cit.>.
Δ DP measures the difference in the probability of a positive outcome between different sensitive groups and it is better to be closer to 0, which can be calculated as follows:
Δ DP = p(ŷ = 1|s = s_i) - p(ŷ = 1|s = s_j),
where s_i and s_j represent different sensitive groups.
Equalized Odds require the probability of positive outcomes to be independent of the sensitive group s, conditioned on the ground truth label y.
Specifically, Δ EO calculates the summation of the True Positive Rate difference and False Positive Rate difference as follows:
Δ EO = |P(ŷ = 1|s = s_i, y = 1) - P(ŷ = 1|s = s_j, y = 1)|
+ |P(ŷ = 1|s = s_i, y = 0) - P(ŷ = 1|s = s_j, y = 0)|,
where Δ EO is better to be closer to 0.
§.§ Implementation Details
In FairInt, we set the embedding dimension d=4 and the number of attention heads in BID layer as 1 among all four datasets.
For the Adult dataset, we use a two-layer MLP with 64 and 32 units of each hidden layer as the vanilla MLP model.
For the Law School dataset, we use a two-layer MLP with 64 and 32 units of each hidden layer as the vanilla MLP model.
For the Bank Marketing dataset, we use a two-layer MLP with 40 and 20 units of each hidden layer as the vanilla MLP model.
For the Diabetes dataset, we use a four-layer MLP with 512, 256, 64, and 64 units of each hidden layer as the vanilla MLP model.
As for the AutoInt, we set the embedding dimension of each feature to 4, the number of interaction layers to 2, and the number of attention heads in each interaction layer to 2 for all four datasets.
To prevent overfitting, we search dropout rate from the set {0.1, 0.3, 0.5, 0.7, 1.0} and search l_2 norm regularization from the set {5e-1, 1e-1, 5e-2, 1e-2, 5e-3, 1e-3, 5e-4, 1e-4, 5e-5, 1e-5} for all the models.
To address the situation of the unavailablity of sensitive attributes in the inference stage, we utilize a four-layer MLP as the SAR on both vanilla MLP predictor and AutoInt models.
§.§ (Q1) Performance Comparison on Real-world Datasets
In this section, we provide the results of classification prediction by using a binary classifier performance measure AUC <cit.> for evaluating imbalanced data.
The fairness testings are evaluated with the two aforementioned fairness metrics: Δ DP and Δ EO.
Table <ref> summarizes the performance of the FairInt and the baselines on the four real-world datasets, where FC and PR refer to two vanilla MLP models which are debiased by two regularization-based bias alleviation methods Fair Constraint proposed in Sec. <ref> and Prejudice Remover <cit.>.
We observe that our FairInt significantly and consistently mitigates the bias predictions with the lower Δ DP and Δ EO across all four datasets.
The best fairness results are highlighted in bold.
Given the limitations demonstrated by FairRF <cit.> and ARL <cit.> in balancing the trade-off between AUC and fairness performance as assessed in the Law School and Bank Marketing, a comparison of their DP and EO performance with other methods is not performed.
Compared with the best bias alleviated baselines between FC and PR, our FairInt improves Δ DP by 5.37%, 36.36%, 8.08%, and 19.61% on Adult, Law School, Bank Marketing, and Diabetes, respectively.
As for Δ EO, our FairInt improves Δ EO by 4.89%, 37.70%, 17.82% and 40.35% on the four datasets, respectively.
We also make the following observations of the experimental results.
First, AutoInt can slightly improve the AUC performance with the attention-based feature interaction modeling mechanism, but it also augments the biased prediction behaviors.
As we can see from Table <ref>, AutoInt can improve the AUC of vanilla MLP models by 0.44%, 0.15%, 0.36% and 0.74% on the four datasets, respectively.
However, it at the same time increases Δ DP and Δ EO on all four datasets.
Compared with the vanilla MLP models, AutoInt increases Δ DP on three out of four datasets and increases Δ EO on all four datasets.
The reason is that the modeled feature interactions not only improve the downstream task performances but also contain the biased feature interactions that will augment the biased behaviors of predictions.
Second, our FairInt can maintain the competitive classification performance compared with the other debiased baselines.
As we can see from Table <ref>, the fairness performances of our proposed FairInt are improved significantly while the classification performances are slightly decreased.
We compare our FairInt with the Vanilla MLP model, AutoInt, and the debiased baselines PR in Figure <ref>, which illustrates their fairness-AUC curves for the four datasets.
The hyper-parameter λ_IFC and λ_FC in Eq. <ref> controls the trade-off between AUC and fairness for FairInt.
For the debiased vanilla MLP with PR, the hyper-parameter in front of the regularization term also controls the trade-off.
From Figure <ref> we also can observe that our proposed FairInt can achieve the best Δ DP and Δ EO in all four datasets while remaining competitive AUC compared to PR.
§.§ (Q2) Analysis of Bias Interaction Detection Layer
We analyze the ability of the Bias Interaction Detection (BID) layer that can identify the biased feature interactions.
In Table <ref>, the Vanilla FairInt refers to the FairInt framework without the two interaction-wise bias mitigation regularization IFC and FC, and it keeps the Bias Interaction Detection (BID) layer which is designed to identify biased feature interactions.
Compared with the vanilla MLP models, the Vanilla FairInt significantly augments biased behaviors of model predictions.
For all four datasets, the Vanilla FairInt increases Δ DP by 2.99%, 16.06%, 22.22% and 38.67%, and it increases Δ EO by 33.10%, 48.13%, 37.24% and 55.29%, respectively.
The reason Vanilla FairInt can remarkably augment the biased predictions is that BID focuses on detecting the biased feature interactions and embedding them into the latent representation.
A similar scenario can be observed in AutoInt because it models all the interactions between all the feature pairs that include biased feature interactions.
Unlike the AutoInt, our proposed FairInt focuses on learning to model the biased feature interactions only among the feature pairs which contain a sensitive attribute.
By doing so, the latent representations in FairInt embed the biased feature interaction information without other noising knowledge.
§.§ (Q3) Analysis of Fairness Constraint Components
After the latent representations in FairInt embed the bias feature interaction information, we leverage the two fairness constraints to mitigate the embedded bias feature interactions.
To better understand the effects of the two fairness constraints, Interaction Fairness Constraint and Fairness Constraint, in the proposed FairInt, we conduct the ablation studies to analyze and verify their contributions to the FairInt framework.
In Table <ref>, the Vanilla FairInt refers to the FairInt framework without the two interaction-wise bias mitigation regularization IFC and FC, + FC refers to the Vanilla FairInt with Fairness Constraint, and + IFC refers to the Vanilla FairInt with Interaction Fairness Constraint.
Although the debiasing effects of the + FC are not as significant as the FairInt, it can achieve the same level of Δ DP and Δ EO as the vanilla MLP models debiased by FC in all the four datasets.
Compared with the + FC, the + IFC more focuses on improving Δ EO than Δ DP.
The reason is that the implicit mitigation regularization IFC focuses on optimizing the latent representation rather than directly mitigating bias behaviors against the model predictions.
Therefore, when the FairInt adopts the IFC with the FC, it can markedly improve the fairness with the lower Δ DP and Δ EO than only leverage one of the two bias regularization components.
§.§ (Q4) Analysis of Sensitive Hyper-parameter
In this section, we study the impact of the hyper-parameter λ_IFC and λ_FC in the Eq. <ref> to answer the research question Q4.
We conduct the sensitivity analysis for both the two hyper-parameters on the Adult and Bank Marketing datasets.
To analyze the influence of λ_FC, we fix the best λ_IFC to see the trend of AUC, Δ DP, and Δ EO when changing λ_FC on the two datasets, respectively.
As shown in Figure <ref>, in the proposed FairInt, λ_FC is not sensitive to downstream task performances AUC.
As for the two fairness metrics Δ DP and Δ EO, they will be improved when the λ_FC increases, and the improvement will gradually converge to a certain level.
And to analyze λ_IFC, we fix the best λ_FC to observe the trend of AUC, Δ DP and Δ EO when changing λ_IFC on the Adult and Bank Marketing datasets, respectively.
According to the observations from Figure <ref>, in the FairInt, λ_IFC is not sensitive to downstream task performance AUC.
At the same time, the best λ_IFC can typically achieve the best Δ DP when reaching the best Δ EO on both Adult and Bank Marketing datasets.
§.§ (Q5) Key Observations on Interaction
One of the benefits of modeling feature interaction is that it provides better interpretability by observing the pattern of modeled feature interactions.
Therefore, in this section, we provide the key observations on the feature interactions, which refer to the attention weights a_s,k calculated by Eq. <ref> in our proposed FairInt.
Here, we show the feature interactions between the sensitive and non-sensitive attributes on the Adult dataset, and we treat the FairInt w/o Both as a biased model, the FairInt w/o FC as a slightly fair model, the FairInt w/o IFC as a fair model, and FairInt as a fairer model.
The feature interactions of FairInt w/o Both, FairInt w/o FC, FairInt w/o IFC and FairInt are shown in the Figure <ref>.
In the four figures, the yellow points represent the mean values of each attention weight between the sensitive attribute gender and a non-sensitive attribute.
By comparing the feature interactions between biased and fair models, we conclude with the two factors of the feature interactions, which are variance and mean value.
Fair models have a lower variance of each feature interaction between sensitive and non-sensitive attributes, and a mean value of one feature interaction represents the correlation between the sensitive and the non-sensitive attribute.
For example, comparing the attention weights of FairInt, the fairest one among the four models, with FairInt (w/o Both), the most unfair one among the four models, the feature interactions between gender and all other non-sensitive attributes have lower variances in the more fair model.
Also, the mean value of the feature interaction between gender and relationship is lower in the fairest model, which implies the fairer model treats relationship as a less relevant attribute against gender.
§ CONCLUSION AND FUTURE WORKS
In this paper, we proposed FairInt, an assumption-free framework that automatically identifies and mitigates the biased feature interactions. Our framework doesn't need prior knowledge to identify the related attributes in advance for mitigating the unfair model predictions. FairInt is composed of Sensitive Attribute Reconstructor, Bias Interaction Detection, and Interaction-wise Bias Mitigation, which aims to predict pseudo-sensitive attributes, model the information of identified bias feature interactions, and mitigate biased interaction with FC and IFC, respectively. Experiments on four real-world datasets demonstrate that FairInt can alleviate the unfair model predictions while maintaining the competitive classification performance.
As for the future direction, we will explore the novel fairness constraint by limiting the variance of feature interaction, which implies the fairness extent of the proposed FairInt.
acm
|
http://arxiv.org/abs/2307.05067v1 | 20230711071309 | Exploiting Asymmetry in Logic Puzzles: Using ZDDs for Symbolic Model Checking Dynamic Epistemic Logic | [
"Daniel Miedema",
"Malvin Gattinger"
] | cs.LO | [
"cs.LO",
"cs.AI",
"I.2.4; I.2.11"
] |
Retrieval-augmented GPT-3.5-based Text-to-SQL Framework with Sample-aware Prompting and Dynamic Revision Chain
Chunxi Guo, Zhiliang Tian (), Jintao Tang, Shasha Li, Zhihua Wen,
Kaixuan Wang and Ting Wang ()
August 12, 2023
==============================================================================================================
Binary decision diagrams (BDDs) are widely used to mitigate the state-explosion problem in model checking.
A variation of BDDs are Zero-suppressed Decision Diagrams (ZDDs) which omit variables that must be false, instead of omitting variables that do not matter.
We use ZDDs to symbolically encode Kripke models used in Dynamic Epistemic Logic, a framework to reason about knowledge and information dynamics in multi-agent systems.
We compare the memory usage of different ZDD variants for three well-known examples from the literature: the Muddy Children, the Sum and Product puzzle and the Dining Cryptographers.
Our implementation is based on the existing model checker SMCDEL and the CUDD library.
Our results show that replacing BDDs with the right variant of ZDDs can significantly reduce memory usage.
This suggests that ZDDs are a useful tool for model checking multi-agent systems.
§ INTRODUCTION
There are several formal frameworks for reasoning about knowledge in multi-agent systems, and many are implemented in the form of epistemic model checkers.
Here we are concerned with the data structures used in automated epistemic reasoning.
This is a non-issue in theoretical work, where Kripke models are an elegant mathematical tools.
But they are not very efficient: models where agents know little tend to be the largest.
More efficient representations are often based on Binary Decision Diagrams (BDDs), which use the idea that a representation of a function not depending on p can simply ignore that variable p.
This fits nicely to the models encountered in epistemic scenarios, such as the famous example of the Muddy Children: If child 2 does not observe whether it is muddy, i.e. whether p_2 is true or false, then we can save memory by omitting p_2 in the encoding of the knowledge of child 2.
However, which variables matter may change, and in many examples the claim that “many variables do not matter” only holds in the initial model.
This motivates us to look at Zero-suppressed Decision Diagrams (ZDDs) which use an asymmetric reduction rule to omit variables that must be false, instead of the symmetric reduction rule targeting variables that do not matter.
Our informal research question is thus: Is it more memory efficient to have a default assumption that “anything we do not mention does not matter” or, for example “anything we do not mention must be false”?
Obviously, the answer will depend on many aspects.
Here we make the question precise for the case of Dynamic Epistemic Logic, and consider three well-known examples from the literature.
The article is structured as follows.
We discuss related work in the rest of this section, then we provide the relevant background in Sections <ref> and <ref>.
Section <ref> describes our experiment design and the formal models used.
We present our results in Section <ref> and conclude in Section <ref>.
Related work
Model checking aims to verify properties of formally specified systems.
Standard model checking methods search through a whole state transition graph and thus suffer from the state explosion problem: the number of states grows exponentially with the number of components or agents.
To tackle this problem symbolic methods were developed <cit.>.
These reduce the amount of resources needed, by reasoning about sets instead of individual states.
Starting with SMV from <cit.>, most approaches use Binary Decision Diagrams (BDDs) <cit.> to encode Boolean functions.
Zero-suppressed Decision Diagrams (ZDDs) are an adaption of BDDs, introduced by Minato <cit.>.
ZDDs naturally fit combinatorial problems and many comparisons between BDDs and ZDDs have been done.
For both an elegant introduction into the topic of BDDs and many more references we refer to <cit.>.
Symbolic model checking using ZDDs has not been studied much, partly due to underdeveloped construction methods <cit.>.
Most existing symbolic model checkers use temporal logics such as LTL or CTL.
Yet problems come in many forms and for examples typically described using epistemic operators (e.g. in multi-agent systems), Dynamic Epistemic Logic (DEL) is an established framework <cit.>.
Also DEL model checking can be done symbolically <cit.>, by encoding Kripke models as so-called knowledge structures. This lead to its implementation, SMCDEL, which is extended in this work.
Another encoding, sometimes also called “symbolic models”, is based on mental programs <cit.>.
In concrete applications such as “Hintikka's World” these also get encoded as BDDs <cit.>.
To our knowledge no previous work used ZDDs or other BDD variants for DEL model checking, with the exception of <cit.> where Algebraic Decision Diagrams (ADDs) are used for probabilistic DEL.
Here our main research questions is:
Can ZDDs be more compact than BDDs when encoding the Kripke models for classical logic puzzles?
We answer this question by adding ZDD functionality to SMCDEL and then comparing the sizes for three well-known examples from the literature.
§ THEORY: DECISION DIAGRAMS
Symbolic model checkers, including SMCDEL, rely on efficient representations of Boolean functions.
The most widely used data structure for this are Binary Decision Diagrams (BDDs).
In this section we recall their definition and explain the difference between standard BDDs and ZDDs.
How Boolean functions are then used for model checking DEL will be explained in the next section.
Before we get to decision diagrams we define Boolean formulas and functions.
The Boolean formulas over a set of variables P (also called vocabulary) are given by
φ ::= ⊤| p |¬φ|φφ
where p ∈ P.
We define
:= ¬⊤,
φψ := ¬ (¬φ¬ψ)
and
φ→ψ := ¬ (φ¬ψ).
We write for the usual Boolean semantics using assignments of type P →{0, 1}.
When P is given we identify an assignment (also called state) with the set of variables it maps to 1.
A Boolean function is any f 𝒫(P) →{0,1}.
For any φ we define the Boolean function
f_φ(s) := {if s φ then 1 else 0 }.
For example, if our vocabulary is P = {p,q,r} and s(p) = 0, s(q)=1 and s(r)=0 then we identify s with {q} and we have s (¬ p q) r.
In the following we will also just write φ for f_φ.
Notably, two different formulas can correspond to the same Boolean function, but not vice versa.
For any φ, ψ, and p,
let φ(p/ψ) be the result of replacing every occurrence of p in φ by ψ.
For any A={p_1,…,p_n},
let φ(A/ψ) := ψ (p_1/ψ) (p_2/ψ) … (p_n/ψ).
We use ∀ p φ to denote φ(p/⊤) ∧φ(p/).
For any A={p_1,…,p_n}, let ∀ A φ := ∀ p_1 ∀ p_2 …∀ p_n φ.
Decision Diagrams
A decision diagram is a rooted directed acyclic graph, used to encode a Boolean function.
Any terminal node (i.e. leaf) is labelled with 0 or 1, corresponding to the result of the function.
Any internal node n is labelled with a variable and has two outgoing edges to successors denoted by (n) and (n) — each representing a possible value for the variable.
A path from the root to a leaf in a decision diagram corresponds to an evaluation of the encoded function.
A decision diagram is called ordered if the variables are encountered in the same order on all its paths.
The first (left-most) decision diagram in Figure <ref> is a full decision tree for q ¬ r.
To evaluate it at state {p,q} we start at the root and then go along the solid -edge because p is true, then again along a -edge as q is true and then along the dashed -edge as r is false.
We get 1 as a result, reflecting the fact that {p,q} q ¬ r.
Similarly we can use the second and third diagram.
Binary Decision Diagrams (BDDs) were introduced by <cit.> and are particularly compact decision diagrams, obtained using two reduction rules.
The first rule identifies isomorphic subgraphs, i.e. we merge nodes that have the same label and the same children.
In Figure <ref> we get from the first to the second diagram.
The second rule eliminates redundant nodes.
A node is considered redundant if both its - and -edge go to the same child.
In Figure <ref> this gets us from the second to the third diagram.
Zero-suppressed Decision Diagrams (ZDDs) were introduced by <cit.> and use a different second rule than BDDs.
While in BDDs a node n is eliminated when (n) = (n), in ZDDs a node is eliminated when (n) = 0.
In Figure <ref> this rule gets us from the second to the fourth diagram called ZDD_T0(f).
The idea is to not ignore the variables that “do not matter” (as p in q ¬ r), but to remove the nodes of variables that must be false (as r in q ¬ r).
To evaluate ZDD_T0(f) at state {p,q} we again start at the root and twice follow a solid edge because p and q are true, but then we notice that the solid edge goes from q to 1, without asking for the remaining variable r.
When evaluating a ZDD_T0 such a transition demands that the variable we “jump over” must be false — hence the name “zero-suppressed”.
Indeed r is false in our state, so we do reach 1.
If r would have been true, the result would have been 0.
Generalizing Elimination Rules
The elimination rule “remove nodes that have a -edge leading to 0” can be modified in two obvious ways: instead of - we could consider -edges, and instead of 0 we could consider 1.
This leads us to three additional elimination rules.
We denote five different node elimination rules as follows.
A node n with pairs of children ((n), (n)) is eliminated if it matches the left side of the rule,
and any edges leading to n are diverted to the successor s on the right side of the rule.
[ EQ: (s, s) ⇒ s T0: (0, s) ⇒ s E0: (s, 0) ⇒ s; T1: (1, s) ⇒ s E1: (s, 1) ⇒ s; ]
Here EQ is the rule for BDDs, while T0 (for “Then 0”) is the traditional ZDD rule.
The remaining three are variations.
For example, E0 says that any node with an -edge to 0 is removed, and any edge that led to the removed node should be diverted to where the -edge of the removed node led.
In Figure <ref> the E0 rule gets us from the second to the sixth diagram ZDD_E0(f).
Note that we used the rule twice: After deleting an r node the q node has an -branch to 0, so it is also eliminated.
All diagrams encode the same function f, but when evaluating them we must interpret “jumps” differently.
A crucial feature of BDDs and ZDDs is that they are canonical representations: given a fixed variable order there is a unique BDD and a unique ZDD for each variant.
It also becomes clear that for different Boolean functions a different kind of diagram can be more or less compact.
For any Boolean function f, recall that f denotes its complement.
Let f denote the result of complementing all atomic propositions inside f.
(For example, (q ¬ r) = ¬ q r.)
For any decision diagram d,
let (d) be the result of changing the labels of all leaves from 0 to 1 and vice versa;
and let (d) be the result of changing the labels of all edges from to and vice versa.
There is a correspondence between and , and between and .
Moreover, we can use these operations to relate the four different variants of ZDDs as follows.
For any Boolean function f we have:
[ DD_T1(f) = DD_T0 ( f); DD_E0(f) = DD_T0 ( f); DD_E1(f) = DD_T0 ( f) ]
We illustrate Fact <ref> using our running example f := q ¬ r with vocabulary {p,q,r}.
Figure <ref> shows the T0 decision diagrams mentioned in Fact <ref>.
We see that for example DD_T1(f) shown in Figure <ref> is the same graph as DD_T0( f) with only the labels of the leaf nodes exchanged.
Similarly, DD_E1(f) in Figure <ref> is the same graph as DD_T0( f) with flipped edges and leaves.
Fact <ref> is crucial for our implementation, because the CUDD library we use does not support T1, E0 and E1 explicitly.
Hence instead we always work with T0 diagrams of the negated or flipped functions.
§ THEORY: SYMBOLIC MODEL CHECKING DEL
Kripke Models
We recap the standard syntax and semantics of Public Announcement Logic (PAL), the most basic version of Dynamic Epistemic Logic (DEL).
Fix a vocabulary V and a finite set of agents I.
The DEL language ℒ(V) is given by
φ ::= p |¬φ|φφ| K_i φ| [φ]φ
where p ∈ V, i ∈ I.
As usual, K_iφ is read as “agent i knows that φ”.
The formula [ψ]φ says that after a public announcement of ψ, φ holds.
The standard semantics for ℒ(V) on Kripke models are as follows.
A Kripke model for a set of agents I={1,…,n} is a tuple
M=(W,π , K_1 ,…, K_n ), where
W is a set of worlds,
π associates with each world a state π(w),
and
K_1,…, K_n are equivalence relations on W.
A pointed Kripke model is a pair ( M, w) consisting of a model and a world w ∈ W.
Semantics for ℒ(V) on pointed Kripke models are given inductively as follows.
* ( M,w) p iff π ^M (w)(p) = ⊤.
* ( M,w)φ iff not ( M,w)φ
* ( M,w)φψ iff ( M,w)φ and ( M,w)ψ
* ( M,w)K_i φ iff for all w'∈ W,
if w K_i^M w', then ( M,w')φ.
* ( M,w) [ψ] φ iff
( M,w)ψ implies ( M^ψ,w)φ
where M^ψ is a new model based on the set W^ M^ψ := { w ∈ W^ M| ( M,w)ψ}
and appropriate restrictions of K_i and π to W^ M^ψ.
More expressive versions of DEL also include common knowledge and complex epistemic or ontic actions such as private communication, interception, spying and factual change.
Moreover, DEL can work both with S5 models and with arbitrary Kripke models.
All of this is compatible with the symbolic semantics we recall in the next section, but for our purposes in this article the restricted language above is sufficient, and we only consider S5 models.
Knowledge Structures
While the semantics described above is standard, it has the disadvantage that models are represented explicitly, i.e. the number of worlds also determines the amount of memory needed to represent a model.
To combat this well-known state-explosion problem we can replace Kripke models with symbolic knowledge structures.
Their main advantage is that knowledge and results of announcements can be computed via purely Boolean operations, as shown in <cit.>.
Suppose we have n agents.
A knowledge structure is a tuple
ℱ = (V,θ,O_1,…,O_n) where
V is a finite set of atomic variables,
θ is a Boolean formula over V and
for each agent i, O_i⊆ V.
The set V is the vocabulary and the formula θ is the state law of ℱ.
The O_i are called observational variables.
An assignment over V that satisfies θ is a state of ℱ.
A scene is a pair (ℱ,s) where s is a state of ℱ.
Consider the knowledge structure ℱ := ( V={p, q}, θ = p → q, O_1={p}, O_2={q} ).
The states of F are the three assignments ∅, {q} and {p,q}.
Moreover, ℱ has two agents who each observe one of the propositions: agent 1 knows whether p is true and agent 2 knows whether q is true.
We now give semantics for ℒ(V) on knowledge structures.
Semantics for L(V) on scenes are defined as follows.
* ( F,s) p iff s p.
* (ℱ,s)φ iff not (ℱ,s)φ
* (ℱ,s)φψ iff (ℱ,s)φ and (ℱ,s)ψ
* (ℱ,s) K_i φ iff
for all t of ℱ, if s∩ O_i=t∩ O_i, then (ℱ,t)φ.
* (ℱ,s) [ψ] φ iff (ℱ,s)ψ implies (ℱ^ψ, s) φ
where
ℱ^ψ:=(V,θψ_ℱ, O_1, …, O_n).
where ·_ℱ is defined in parallel in the following definition.
For any knowledge structure ℱ = (V, θ, O_1, … , O_n) and any formula φ we define its local Boolean translation φ_ℱ as follows.
[ p _ℱ := p K_i ψ_ℱ := ∀(V ∖ O_i)(θ→ψ_ℱ); ψ_ℱ := ψ_ℱ [ ψ ] ξ_ℱ := ψ_ℱ→ξ_ℱ^ψ; ψ_1 ψ_2 _ℱ := ψ_1_ℱψ_2 _ℱ; ]
where the case for K_i ψ quantifies over the variables not observed by agent i,
using Boolean quantification as defined in Definition <ref> above.
A main result from <cit.> based on <cit.> is that for any finite Kripke model there is an equivalent knowledge structure and vice versa.
This means we can see knowledge structures as just another, hopefully more memory-efficient, data structure to store a Kripke model.
An additional twist is that we usually store the state law θ not as a formula but only the corresponding Boolean function — which can be represented using a decision diagram as discussed in Section <ref>.
§ METHODS: LOGIC PUZZLES AS BENCHMARKS
Our leading question is whether ZDDs provide a more compact encoding than BDDs for models encountered in epistemic model checking.
To answer it we will work with three logic puzzles from the literature.
All examples start with an initial model which we encode as a knowledge structure with the state law as a decision diagram.
Then we make updates in the form of public announcements, changing the state law.
We record the size of the decision diagrams for each update step.
As a basis for our implementation and experiments we use SMCDEL, the symbolic model checker for DEL from <cit.>.
SMCDEL normally uses the BDD library CacBDD <cit.> which does not support ZDDs.
Hence we also use the library CUDD <cit.> which does support ZDDs.
However, also CUDD does not support the generalized elimination rules from Definition <ref>.
Therefore we use Fact <ref> to simulate the T1, E0 and E1 variants.
Our new code — now merged into SMCDEL — provides easy ways to create and update knowledge structures where the state law is represented using any of the four ZDD variants.
An additional detail is that CUDD always uses so-called complement edges to optimize BDDs, but not for ZDDs.
To compare the sizes of ZDDs to BDDs without complement edges we still use CacBDD.
Altogether in our data set we thus record the sizes of six decision diagrams for each state law: the EQ rule with and without complement edges (called BDD and BDDc) and the four ZDD variants from Definition <ref>.
We stress that by size of a diagram we mean the node count and not memory in bytes, because the former is independent of what libraries are used, whereas the latter depends on additional optimisations.
It now remains to choose examples.
We picked three well-known logic puzzles from the literature with different kinds of state laws, such that we also expect the advantage of ZDDs to vary between them.
Muddy Children
The Muddy Children are probably the best-known example in epistemic reasoning, hence we skip the explanation here and refer to the literature starting with <cit.>.
A formalisation of the puzzle can be found in <cit.> and the symbolic encoding in <cit.>.
Dining Cryptographers
This problem and the protocol to solve it was first presented by <cit.>:
“Three cryptographers gather around a table for dinner. The waiter informs them that the meal has been paid for by someone, who could be one of the cryptographers or the National Security Agency (NSA). The cryptographers respect each other's right to make an anonymous payment, but want to find out whether the NSA paid.”
The solution uses random coin flips under the table, each observed by two neighbouring cryptographers but not visible to the third one.
A formalisation and solution using Kripke models can be found in <cit.>.
To encode the problem in a knowledge structure we let p_0 mean that the NSA paid, p_i for i ∈{1,2,3} that i paid.
Moreover, p_k for k ∈{4,5,6} represents a coin.
The initial scenario is then
(V = {p_0,…,p_6}, θ = ⊗_1 {p_0,p_1,p_2,p_3}, O_1 = {p_1,p_4,p_5}, O_2 = {p_2,p_4,p_6}, O_3 = {p_3,p_5,p_6} )
where the state law θ says that exactly one cryptographer or the NSA must have paid.
In the solution then each cryptographer announces the XOR (⊗) of all bits they observe, with the exception that the payer should invert their publicly announced bit.
Formally, we get a sequence of three public announcements
[?!(⊗p_1,p_4,p_5)] [?!(⊗p_2,p_4,p_6)] [?!(⊗p_3,p_5,p_6)]
where [?!ψ] φ := [!ψ] φ [ !ψ] φ abbreviates announcing whether.
The protocol can be generalised to any odd number n instead of three participants.
Sum and Product
The following puzzle was originally introduced in 1969 by H. Freudenthal.
The translation is from <cit.> where the puzzle is also formalised in DEL:
A says to S and P:
I have chosen two integers x,y such that 1 < x < y and x+y ≤ 100.
In a moment, I will inform S only of s = x + y, and P only of p = xy.
These announcements remain private.
You are required to determine the pair (x, y).
He acts as said.
The following conversation now takes place:
P says: “I do not know it.” —
S says: “I knew you didn’t.” —
P says: “I now know it.” —
S says: “I now also know it.” —
Determine the pair (x, y).
Solving the puzzle using explicit model checking is discussed in <cit.>.
To represent the four variables and their values in propositional logic we need a binary encoding, using ⌈log_2 N ⌉ propositions for each variable that take values up to N.
For example, to represent x ≤ 100 we use p_1,…,p_7 and encode the statement x = 5 as p_1 p_2 p_3 p_4 p_5 p_6 p_7, corresponding to the bit-string 0000101 for 5.
The initial state law for Sum and Product is a big disjunction over all possible pairs of x and y with the given restrictions, and the observational variables ensure that agents S and P know the values of s and p respectively.
For a detailed definition of the knowledge structure, see <cit.>.
The announcements in the dialogue are formalised as follows, combining the first two into one:
First S says K_S ⋁_i + j ≤ 100 K_P ( x=i y=j ), then P says
⋁_i + j ≤ 100 K_P ( x=i y=j ) and finally S says
⋁_i + j ≤ 100 K_S ( x=i y=j ).
Solutions to the puzzle are states where these three formulas can be truthfully announced after each other.
A common variation on the problem is to change the upper bound for x+y.
We use this to turn obtain a scalable benchmark, starting with 65 to ensure there exists at least one answer.
It is well known that ZDDs perform better on sparse sets <cit.>.
In our case, sparsity is the number of states in the model divided by the total number of possible states for the given vocabulary.
Our three examples vary a lot in their sparsity:
Muddy Children's sparsity is 0.5 on average (going from 0.875 to 0.125, for 3 agents),
Dining Cryptographers is fairly sparse from start to finish (0.25 to 0.0625, for 3 agents),
and Sum and Product is extremely sparse (e.g. starting with < 1.369·10^-7 for x+y ≤ 100).
§ RESULTS
For each example we present a selection of results we deem most interesting, showing differences between BDD and ZDD sizes.
The full data set for two examples can be found in the appendix where we also include instructions how all of the results can be reproduced.
Muddy children
We vary the number of children n from 5 to 40, in steps of 5.
We can also vary the number of muddy children m ≤ n, but mostly report results here where m=n.
Given any number of children, we record the size of the decision diagrams of the state law after the kth announcement, where k ranges from 0 (no announcements made yet) to m-1 (after which all children know their own state).
As an example, let us fix n=m=20.
Figure <ref> shows the size of the decision diagrams after each announcement.
The lines all follow a similar curve, with the largest relative differences in the initial and final states.
Initially the most compact variant is T1 whereas at the end E0 is the most compact.
This matches the asymmetry in the Muddy Children story: at the start the state law is p_1 … p_n, hence all edges lead to 1 and T1 removes all nodes.
In contrast, at the end the state law is p_1 … p_n which means that all edges lead to 0 and thus E0 eliminates all nodes.
Hence at different stages different variants are more compact.
But we want a representation that is compact throughout the whole process.
We thus consider the average size over all announcements, varying n from 5 to 40.
Figure <ref> shows the relative size differences, with standard BDDs as 100%.
The T0/E1 and the BDDc/E0/T1 lines overlap.
We see that T1 and E0 are more compact for small models, but not better than BDDs with complement edges and this advantage shrinks with a larger number of agents.
We also computed sizes for m < n, i.e. not all children being muddy.
In this case the sizes for each update step stay the same but there are fewer update steps because the last truthful announcement is in round m-1.
As expected this is in favour of the T1 variant.
Dining cryptographers
For 13 agents we show the sizes after each announcement in Figure <ref>.
It becomes clear that there is little difference between the variants, which can be explained by the sparsity of the model throughout the whole story.
Still, the T0/E0 variants slightly outperform the BDD(c) and the T1/E1 variants.
This makes sense as most variables saying that agent i paid will be false.
For lower numbers of agents the difference is larger, as visible in Figure <ref> where we vary the number of agents from 3 to 13.
Note that T1 and E1 overlap here, and T0 provides the best advantage.
Sum and Product
In this last example we can vary the upper bound of x+y from 50 to 350, but not the number of agents and announcements.
Figure <ref> shows the sizes averaged over all four stages.
We note that the BDD(c), T1 and E1 lines all overlap (with insignificant differences), and that T0 and E0 perform the best here.
In contrast to the first two examples, this advantage does not disappear for larger instances of the puzzle, as can be seen in Figure <ref> where we show the relative differences.
Interestingly, we see that T0 and E0 meet up and diverge again wherever the bound for x+y is a power of 2 (i.e. 64, 128 or 256) which we mark by vertical dashed lines.
This is due to the bit-wise encoding where just above powers of two an additional bit is needed, but it must be false for almost all values.
§ CONCLUSION
In all experiments we find a ZDD elimination rule that can reduce the number of nodes compared to BDDs, with the exception that in the Muddy Children example complement edges provide the same advantage.
This leads us to conclude that ZDDs are a promising tool for DEL model checking.
Specifically, if domain knowledge about the particular model allows one to predict which ZDD variant will be more compact, ZDDs can outcompete BDDs.
The BDD elimination rule treats true and false atomic propositions symmetrically, whereas ZDD rules are asymmetric.
This means their success depends on asymmetry in the model.
When translating an example from natural language to a formal models we usually try to avoid redundant variables, which already reduces the number of BDD-eliminable nodes.
This is likely the reason why using ZDDs provides an advantage or, for examples with a sparsity around 0.5 like the Muddy Children, at least the same performance as BDDs with complement edges.
Specifically for logic puzzles, usually all variables are needed, and models become asymmetric and sparse as information is revealed and possible answers are ruled out.
Our results confirm that sparsity and the kind of asymmetry prevalent in the model can predict which ZDD variant is most beneficial.
In this article we only considered S5.
SMCDEL also provides modules for K and in further experiments we compared the sizes of ZDDs and BDDs of the state law of belief structures.
As an example we used the famous Sally-Anne false belief task.
The results were similar to those here and can be found in <cit.>.
Future work
An obvious limitation is that we only compared memory and not computation time.
The size of a decision diagram correlates with the computation time needed to build it.
But the step-wise construction techniques in SMCDEL are slower for ZDDs than for BDDs.
For example, to compute the Sum and Product result we rather convert each state law BDD to ZDDs instead of computing ZDDs directly.
Before a meaningful comparison of computation time can be done, the construction methods for ZDDs need to be further optimized.
We found some indicators which elimination rule is most compact in which case, but a more general approach to formalise domain knowledge and use it to make a correct prediction would be a powerful tool.
Acknowledgements
This work is based on the master thesis <cit.> by the first author, written at the University of Groningen and co-supervised by Rineke Verbrugge and the second author.
We thank the TARK reviewers for their careful reading and helpful comments on this article.
eptcs
§ APPENDIX
The ZDD encoding of knowledge structures has been integrated into SMCDEL itself.
All our results can be reproduced using the Haskell Tool Stack from <https://haskellstack.org> as follows.
The last three commands will create files containing the results.
On a system with a 4.8 GHz CPU the last three commands above take approximately 10 seconds, one minute and three hours.
We include the results for Dining Crytographers and Sum and Product here, but omit the (several pages long) results for the Muddy Children.
Results for Dining Cryptographers
.
experimentResults/dining.dat
Results for Sum and Product
.
experimentResults/sap.dat
|
http://arxiv.org/abs/2307.07640v1 | 20230714215719 | $ \mathrm{SE} (3) $ Synchronization by Eigenvectors of Dual Quaternion Matrices | [
"Ido Hadi",
"Tamir Bendory",
"Nir Sharon"
] | math.SP | [
"math.SP"
] |
[
[
August 12, 2023
===================
In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or “rounded,” onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups.
In this paper, we develop a spectral approach to synchronization over the non-compact group (3), the group of rigid motions of ^3.
We based our method on embedding (3) into the algebra of dual quaternions, which has deep algebraic connections with the group (3).
These connections suggest a natural rounding procedure considerably more straightforward than the current state-of-the-art for spectral (3) synchronization, which uses a matrix embedding of (3).
We show by numerical experiments that our approach yields comparable results to the current state-of-the-art in (3) synchronization via the spectral method.
Thus, our approach reaps the benefits of the dual quaternion embedding of (3), while yielding estimators of similar quality.
§ INTRODUCTION
Synchronization problems arise as part of data processing pipelines in several contexts, including single-particle reconstruction in cryogenic electron microscopy <cit.>, structure from motion problems <cit.> and simultaneous localization and mapping (SLAM) problems <cit.>.
A synchronization problem is an estimation problems over a group G in which group elements g_1, …, g_n∈ G are estimated from measurements of their ratios g_i g_j^-1.
These measurements are inherently ambiguous, since the set of ratios of g_1 g, …, g_n g ∈ G for any g ∈ G are the same as the ratios of the original group elements.
Thus, in synchronization problems, the goal is to estimate the group elements up to right-multiplication by an arbitrary element of G.
The spectral method is a widely used method for solving synchronization problems. It seeks to approximate a solution by finding eigenvectors of a block matrix formed by embedding the group into a matrix algebra. The embedding enables the richer algebraic structure of matrix algebras, and in particular their spectral decomposition, to be leveraged to solve the synchronization problem.
Importantly, the eigenvectors of the observation matrix are not themselves a solution of the synchronization problem. They have to be projected back onto the group, a projection that is typically non-linear. This procedure is often referred to as “rounding” in the synchronization literature. In typical rounding procedures, a matrix formed by these eigenvectors is chopped into blocks and these blocks are subsequently mapped onto elements of the group.
Consider a synchronization problem over a relative simple compact group in the absence of measurement noise.
As we explain in greater detail in <Ref>, even in this ideal setting,
typically there is a gap between the synchronization problem and the eigenproblem used to solve it.
Oversimplifying to an extent, we can say that every solution to synchronization problem is a solution of the eigenproblem used by the spectral method, but not every solution of the eigenproblem is a solution of the synchronization problem.
This issue is exacerbated for synchronization over non-compact group, the focal point of this paper.
The solutions of the eigenproblem form a compact set, the set of unit eigenvectors which span the relevant eigenspaces of the measurement matrix.
Yet, unlike the case of synchronization over compact groups, the possible solutions of a synchronization problems over non-compact groups form a non-compact set. Therefore, the gap between the two problems can be said to be much larger.
In practice, this gap between the eigenproblem and the synchronization problem must be bridged by the rounding procedure.
It ensures that the output of the synchronization method is comprised of elements of the group.
However, existing rounding procedures are made to do so in an ad hoc manner. Specifically, they do not stem from a well characterized relationship between the synchronization problem and the eigenproblem or the relationship between their respective algebraic contexts, the group and the algebra in which it is embedded.
They are merely constructed to ensure that the result of a synchronization problem is an element of the group.
In this paper, we address the issues we raised above in the non-compact case of (3) synchronization, the group of rigid motions, rotations and translations, of ^3.
In <cit.>, an analog of the spectral method was developed for (3) synchronization, which utilized an embedding of (3) into a matrix algebra.
We replace this matrix algebra embedding with an embedding into the algebra of dual quaternions.
Unlike the matrix algebra embedding used by <cit.>, the algebra of dual quaternions is much closer to the structure of (3), in a sense we explain in <Ref>.
Two recent advances in dual quaternion linear algebra were a spectral theorem for matrices of dual quaternions <cit.> and a power iteration capable of approximating its spectra <cit.>.
These two results allow us to develop an elegant spectral synchronization method. The algebraic context eliminates the gap between the synchronization problem and the eigenproblem, at least in the absence of measurement noise.
In addition, the rounding step itself stems directly from the algebraic relationship between elements of the algebra itself and the set on which (3) is represented. In addition, it is computationally simpler than the rounding procedure of <cit.>.
The structure of this paper is as follows. We begin in <Ref> with a survey of the necessary background material. Among other things, we define the algebra of dual quaternions, matrix algebras with dual quaternion entries and their properties.
In <Ref>, we delve deeper into synchronization problems and the spectral method. We define the problem rigorously, introduce the spectral method and conclude by laying out our approach, relying on dual quaternion embedding of (3).
Finally, in <Ref>, we demonstrate empirically that our approach yields comparable results to the current state-of-the-art spectral method for (3) synchronization developed by <cit.>. Thus, we offer a better theoretical foundation tying (3) synchronization problems with eigenproblems, while still maintaining the quality of estimation of the current state-of-the-art spectral method.
§ BACKGROUND: DUAL QUATERNIONS AND MATRICES OF DUAL QUATERNIONS
We provide a succinct background information on dual quaternions and dual quaternion matrices.
In <Ref> and <Ref> we construct the dual quaternions and describe some of their properties.
In <Ref> we survey the well known representation of (3) on the algebra of quaternions and in <Ref> we survey the lesser known representation of (3) on the algebra of dual quaternions.
In <Ref> we define the free modules of dual quaternions and algebras of matrices with dual quaternion elements. This section culminates in the spectral theorem for matrices of dual quaternions, which was recently proved by <cit.>.
Finally, in <Ref> we survey the power iteration for matrices of dual quaternions, which was recently developed by <cit.>.
§.§ Dual Numbers
The algebra of dual numbers is the set = { a + b a, b ∈} with addition and multiplication defined by
(a_1 + b_1) + (a_2 + b_2)
= (a_1 + a_2) + (b_1 + b_2)
( a_1 + b_1) ( a_2 + b_2)
= a_1 a_2 + ( a_1 b_2 + b_1 a_2) .
If a + b ∈, it is convenient to refer to a as the real coordinate and b as the dual coordinate.
It is easy to see from their definition that the dual numbers are defined analogously to the complex numbers.
Whereas the complex numbers are the algebra over the reals generated by 1 and i such that i^2 = - 1, the algebra of dual numbers is generated by 1 and such that ^2 = 0.
This similarity accounts for the similar properties of the algebras of dual numbers and complex numbers.
Namely, the underlying vector space of both is ^2, addition and multiplication operations of both are associative and commutative and both algebras have a unit, 1.
Despite the aforementioned similarities, these two algebras are fundamentally very different. The algebra of complex numbers is a field, while the algebra of dual numbers is not.
It has zero divisors, i.e., non-zero elements whose product is zero.
Indeed, recall that an element x of an algebra is said to be nilpotent if there is a positive integer n such that x^n = 0.
The only nilpotent complex number is 0.
The following proposition characterizes the nilpotent dual numbers, and it is obvious it has many non-zero nilpotents.
An x = a + b ∈ is nilpotent if and only if a = 0.
Also, x^2 = 0 for every nilpotent.
If a 0, it is easy to show that (<ref>) implies that x^n = a^n + (…) 0.
If a = 0, then x^2 = b^2^2 = 0.
As the next proposition shows, nilpotent dual numbers are the only dual numbers which are not invertible.
If x = a + b ∈,
x is invertible if and only it is not nilpotent.
In that case, x^-1 = a^-1 - b a^-2.
Let y = c + d. From (<ref>) it follows that x y = 1 if
ac = 1, ad + bc = 0.
The first equation has a solution if and only if a 0. By <Ref>, this holds if and only if x is not nilpotent.
The formula for x^-1 is easily obtained by solving this system of equations when a 0.
A final point of analogy between the dual and complex numbers, is that both admit a square root.
Following <cit.>, given x = a + b and y = c + d, we write x > y if a > c, or a = c and b > d.
We say a dual number x = a + b ∈ is non-negative if x ≥ 0 and positive if x > 0.
In <cit.>, the following definition of a square root was given:
√(x)
= √(a) + b/2 √(a) x ,
0 x = 0.
In all other cases, the square root is undefined.
Using this definition, it is possible to show that √(x y ) = √(x)√(y).
This multiplicative identity implies that for any x = a + b ∈ with a 0, we have
√(x^2) = |a| + (a) b, where (a) = 1 for a>0, zero when a = 0 and -1 when a<0.
Such considerations motivated <cit.> to define the following absolute value on dual numbers:
|x|
= |a| + (a) b a 0
|b| a = 0.
It has the following properties <cit.>, which are generalized forms of the usual absolute value defined on the real line:
For any x, y ∈:
* |x| = 0 if and only if x = 0.
* |x| = x if x ≥ 0 and |x| > x otherwise.
* |x| = √(x^2) if x is invertible.
* |x y| = |x| |y| for any x, y ∈.
* | x + y| ≤|x| + |y|.
§.§ Dual Quaternions
The algebra of dual quaternions is easiest to define using the algebra of quaternions.
The algebra of quaternions is the set = { a + b + c + d a, b, c, d ∈}.
We typically refer to b, c and d as the , or coordinates, respectively.
In the literature, a is sometimes referred to as the real part of the quaternion, but in order to eliminate confusion with the real and dual parts of a dual number, we simply refer to it as the first coordinate of the quaternion.
Addition in is defined coordinate-wise and multiplication is defined by the following relations among its generators:
^2
= ^2
= ^2
=
= -1.
As a vector space, ≅^4. Also, ⊂ by constraining the , and coordinates to zero.
The center of algebra, the set of elements for which multiplication commutes with all other elements of the algebra, is .
There is an involution on the quaternions, defined analogously to the complex conjugation, q^* = a - b - c - d for q = a + b + c + d.
This involution is referred to as quaternion conjugation and we say that q^* is the conjugate of q.
Similarly to the complex norm, there is a norm on the quaternions defined as
|q| √(q^* q) = √(a^2 + b^2 + c^2 + d^2),
which can be shown to imply ≅^4 as a normed vector space.
We note here that |q|^2 = q^* q = q q^*, and so the inverse of every non-zero q ∈ is q^*/q^* q.
Overall, the algebra of quaternions is an involutive normed division algebra.
The algebra of dual quaternions is = { a + b a, b ∈}.
Essentially, it means a dual quaternion is a dual number with quaternion coordinates instead of real coordinates.
From this, we immediately see that ⊂ and also that ⊂, the former by constraining a, b ∈ and the latter by setting b = 0.
Addition and multiplication are still defined by (<ref>) and (<ref>) with the quaternion binary operations replacing the real ones.
The trivial involution on , the identity, and quaternion conjugation, induce an involution on , x^* = a^* + b^* for x = a + b ∈.
Because is the center of , we can use the usual notation for division whenever d = a + b ∈ is invertible and x ∈, and so we write
x/d x d^-1
= d^-1 x.
As a vector space, ≅^8.
We use the same names for the coordinates we used for the dual numbers. Given a + b ∈, we call a the real coordinate and b the dual coordinate.
The algebra of dual numbers is a unital algebra, since both and are.
It is non-commutative, since is.
It is not a division algebra, since is not.
This combination of features set it apart from both the algebra of quaternions and the algebra of dual numbers.
Despite this, many of their features generalize well to the dual quaternions.
As we did for the dual numbers, we begin by noting that its nilpotents are characterized in exactly the same way as the nilpotents of the dual numbers.
Indeed, one need only replace with in <Ref> to obtain the proper characterization.
Note here that zero is the only nilpotent in the algebra of quaternions.
Therefore, one can say the first part of <Ref> holds for quaternions, namely, that a quaternion is invertible if and only if it is not nilpotent.
We show now that the dual quaternions share this property.
Before that, we prove a lemma pointing out several important features of the involutive structure of .
If x = a + b ∈, then
x^* x
= |a|^2
+ ( a b^* + b a^*) is a dual number
and x^* x = x x^*.
The first equality is easily proved by substituting x^* and x into (<ref>).
To prove the result is a dual number, we note first that the norm of a quaternion is a real number. In addition, q^* + q ∈ for every quaternion q and ( a b^*)^* = b a^*, which implies a b^* + b a^* is also real.
The second identity follows from a direct calculation.
If x = a + b ∈, x is invertible if and only if it is not nilpotent. In that case, x^-1 = x^*/x^* x.
If a = 0, we have x y = b c 1 for every y = c + d ∈.
Therefore, x is invertible only if a 0, that is, only if x is not nilpotent.
If a 0, it follows from <Ref> that the real coordinate of x^* x is positive.
Thus, by <Ref> x^* x is invertible.
Finally, taking y = x^*/x^* x one obtains that y x = 1.
The equality x y = 1 follows easily from the second equality in <Ref>.
In <cit.>, absolute value was also defined for dual quaternions. If x = a + b ∈, then
| x |
= |a| + a b^* + b a^*/2 |a| a 0
|b| a = 0.
It can be readily seen that when x is a dual number, (<ref>) and (<ref>) agree.
In <cit.>, the following properties of the absolute value of dual quaternions were proved:
Let x, y ∈.
* |x| is a dual number.
* If x is invertible, then |x| = √(q q^*).
* |x| = |x^*|.
* |x| ≥ 0 for all x and |x| = 0 if and only if x = 0.
* |x y| = |x| |y|.
* |x + y | ≤|x| + |y|.
§.§ Representing (3) on the quaternions
The algebra of quaternions, which we surveyed in <Ref>, has a close relationship with the special orthogonal group (3), the group of norm and orientation-preserving linear transformations of the Euclidean space ^3.
The subset _1{ q ∈| q | = 1 } is actually a subgroup of the multiplicative group of . This follows easily from the fact |q|^2 = q^* q = q q^*. In particular, this also implies that the inverse of every q ∈_1 is its conjugate q^*, making the group structure of _1 closed under involution and dependent on the involutive structure of the algebra of quaternions.
A well known result states that _1 is a double cover of (3) <cit.>.
We succinctly construct the covering map here.
We begin by identifying ^3≅𝕋 as vector spaces, where = { a + b + c a, b, c ∈}⊂.
Then, for every q ∈_1, define a map ^3→^3 by
φ(q) (x) = q x q^*, where x ∈^3 is treated as a quaternion in via the identification above.
We then have the following:
The covering map is a 2-to-1 surjective group homomorphism φ : _1→(3), such that φ^-1 (g) = { q, -q } for every g ∈(3).
This double cover allows one to represent elements of (3) on _1.
More specifically, one can represent (3) on _1+ = { q ∈_1 q + q^*≥ 0 } by choosing an appropriate q ∈φ^-1 (g).
§.§ Representing (3) on the dual quaternions
The special Euclidean group (3) is the group of rigid motions of ^3, i.e., rotations and translations of ^3, but not reflections.
It is a semidirect product (3) = (3) ⋉^3.
Its elements are most directly represented as pairs of the form (R, t ) ∈(3) ×^3 with the group operation defined as (R_1, t_1) ∘(R_2, t_2) = (R_1 R_2, t_1 + R_1 t_2). Here, ∘ can be thought of as the composition of two affine transformations.
Let _1 = { x ∈ x^* x = 1 } be the set of unit dual quaternions.
Let = { 1 + 1/2 t t ∈}, where we defined in <Ref>.
Note that _1⊂_1 and ⊂_1.
We state an analogue of <Ref>. It is a synopsis of <cit.> and its proof can be found there.
We have:
* _1 = ⋊_1.
* Let ψ: _1→(3) be defined by ψ(q, t) (x) = (φ(q), t ). Here, φ is the double cover defined in <Ref>, q ∈_1 and 1 + 1/2 t ∈ so t is identified with an element of ^3.
Then ψ is a 2-to-1 surjective group homomorphism such that ψ^-1 (g) = { - x, x} for all g ∈(3).
<Ref> offers a way to represent elements of (3) in much the same way we represent elements of (3) on the quaternions.
We represent (R, 𝐭) ∈(3) by x = q t ∈_1, where q is a unit quaternion with non-negative first coordinate representing the rotation R and 𝐭 is represented by t = 1 + 1/2 t', t' = t_1 + t_2 + t_3.
Its action on ^3 can be defined similarly to what we saw in the quaternion case.
Given 𝐬∈^3, let s = 1 + s', s'=t_1 + t_2 + t_3. Note the absence of a 1/2 factor in this embedding of ^3 into the dual quaternions, compared to how the translational part is represented on in <Ref>.
The action of (R, 𝐭) ∈(3) on 𝐬 is then s ↦ x s x^*. Indeed, we have:
x s x^* = (q + 1/2 t' q )
( 1 + s' )
(q^* + 1/2 q^* t'^*)
= q q^*
+(
1/2 t' q q^*
+ q s' q^*
+ 1/2 q q^* t'^*)
= 1
+ (
q s' q^*
+ t'
) .
In expanding the products in the second transition we used the fact ^2 = 0, which implied no products of two dual coordinates could appear in the result.
The dual coordinate of the result is exactly the action of (R,𝐭) on 𝐬, because q s' q^* is the action of R on 𝐬 by <Ref> and because of the manner in which we represented ^3 on quaternions when we constructed t' and s'.
Cui2023 proved that it is possible to project an almost arbitrary dual quaternion onto the unit dual.
Their result is stated here as follows:
Let x = a + b ∈ be non-zero.
* If a 0, then q/|q| is a unit dual quaternion solving
min_v ∈_1| v - x |^2.
* If a = 0, then any q' = a' + b' such that a' = b/|b| and b^* b' + b'^* b = 0 solves (<ref>).
This theorem exposes an aspect of the close tie between (3) and the algebra of dual quaternions.
We focus on its first part.
It can be interpreted as the idea that every invertible dual quaternion is essentially a unit dual quaternion multiplied by a dual number. Thus, in a sense, the multiplicative structure of the algebra of dual quaternions is almost entirely generated by the group of unit dual quaternions.
This property has a geometric interpretation via the absolute value function. Namely, dividing a dual quaternion by its absolute value yields a solution to the optimization problem (<ref>), a sort of projection function. This function itself is defined via the dual quaternion absolute value (<ref>), which itself intimately related to the algebraic structures on the algebra of dual quaternions as indicated in <Ref>.
Finally, we refer the interested reader to <Ref>, where we compare various representations of (3), focusing on features which are perhaps most salient in applied work.
§.§ Matrices of Dual Quaternions and Their Spectral Decomposition
Modules are an extensively studied generalization of the notion of a vector space to vectors and matrices with entries from an arbitrary algebra.
See <cit.> for an introduction to module theory. Here, we provide a synopsis of the necessary information.
A (right-)module [M] over an algebra [A] is a commutative group ( [M], + ) along with a map [M] ×[A] →[M], (m, x) ↦ m · x, which satisfies the following properties.
First, it is distributive in the sense that m · (x + y) = m · x + m · y and (m_1 + m_2) · x = m_1· x + m_2· x.
Second, the operation is associative, (m · x) · y = m · (x y).
We often refer to this map as the action of [A] on [M] and often omit the multiplication sign.
Oversimplifying to an extent, a module can be thought of as a vector space in which the underlying field was replaced by an arbitrary algebra.
We focus on two types of free modules of the algebra of dual quaternions, ^n and . Given a positive integer n, the free -module is ^n = {𝐦 = (m_1, …, m_n)^⊤ m_j∈, j=1,…,n}. It is acted upon by via entry-wise right-multiplication, i.e., (𝐦 x)_j = m_j x.
Addition is defined entry-wise.
Moreover, can be thought of as the module of all n × n matrices of dual quaternions.
As an -module, one can identify it with ^n^2, and so addition is also performed entry-wise and so is right-multiplication by a dual quaternion.
However, it actually is an algebra, not just a module, with the multiplication of matrices defined in the familiar way.
Similarly, one can define the product 𝐀𝐱 of a matrix 𝐀∈ and 𝐱∈^n in the usual way.
It is instructive to think of 𝐀 as a -linear operator on ^n in the sense that 𝐀( 𝐱 a + 𝐲 b ) = (𝐀𝐱) a + (𝐀𝐲) b for any 𝐱, 𝐲∈^n and a, b∈.
Several types of real and complex matrices have dual quaternions counterparts. The conjugate transpose of a matrix 𝐀∈ is (𝐀^*)_i j = 𝐀_j i^*.
This can easily be shown to be an involution on .
We say a matrix 𝐀∈ is Hermitian if 𝐀 = 𝐀^*.
It is unitary if 𝐀^*𝐀 = 𝐀𝐀^* = 𝐈_n, where 𝐈_n is the identity matrix, which is defined in usual way.
Deepening the analogy with real and complex matrices, a notion of eigenvector and eigenvalue exists for matrices of dual quaternions.
We say λ∈ is a (right-)eigenvalue of the matrix 𝐀∈ if there is 𝐱∈^n such that 𝐀𝐱 = 𝐱λ.
As the name suggests, the non-commutativity of means one can also define a left-eigenvalue of a matrix of dual quaternions, though we will not be presenting any results on these. Whenever we speak of eigenvalues and eigenvectors of dual quaternion matrices we will be referring to right-eigenvalues and right-eigenvectors.
We are now ready to state the spectral theorem for Hermitian matrices of dual quaternions <cit.>:
Let 𝐀∈.
If 𝐀 is Hermitian, then there are a unitary 𝐔∈ and a diagonal Σ∈ with dual number elements such that
𝐀 = 𝐔Σ𝐔^*.
The diagonal elements of Σ have the form σ_j = α_j + β_j∈ with α_1≥α_2≥…≥α_n. Here, σ_j is the eigenvalue of 𝐀 corresponding to the eigenvector 𝐮_j, the jth column of 𝐔.
This theorem essentially states that Hermitian matrices of dual quaternions have a full set of eigenvectors. Furthermore, in the dual quaternion case, as in the complex case, the eigenvalues turn out to be self-adjoint, that is, they satisfy σ^* = σ, with respect to the appropriate involution.
This highlights the striking similarities between this theorem and the well-known spectral theorem of complex Hermitian matrices.
§.§ Power Iteration for Hermitian Matrices of Dual Quaternion
In a recent paper, Cui2023 developed a dual quaternion analog of the power iteration, capable of approximating the top eigenvalue of a Hermitian dual quaternion matrix and its corresponding eigenvalue.
We clarify below the sense in which an eigenvalue is the top eigenvalue.
Let 𝐀∈ be a Hermitian matrix of dual quaternions.
Given some initializer 𝐯^0∈, the power iteration defined in <cit.> is
𝐲^(k)
= 𝐀𝐯^(k-1), λ^(k-1)
= ( 𝐯^(k-1))^*𝐲^(k), 𝐯^(k)
= 𝐲^(k)/𝐲^(k)_2.
Here, the dual quaternion norm used is defined as follows:
𝐱_2
= √(∑_j=1^n|x_j'|^2) 𝐱 = 𝐱' , 𝐱' ∈^n ,
√(∑_j=1^n| x_j|^2) .
For simplicity, we refer to 𝐱_2 as a norm, even though it is not a norm in the usual linear-algebraic sense.
Several addition definitions are required.
Following <cit.>, we write c_k = Õ_D(s^k) for some s < 1 if c_k = a_k + b_k∈ and for some polynomial h(k) we have a_k = O ( s^k h(k) ) and b_k = O (s^k h(k)).
Cui2023 proved the following result:
Let 𝐮_1, …, 𝐮_n be the eigenvectors of 𝐀 ordered so that their corresponding eigenvalues σ_j = α_j + β_j∈ satisfy |α_1| > |α_2| ≥…≥|α_n|.
If 𝐯^(0) = ∑_j=1^n𝐮_jγ_j with γ_j∈
such that γ_1 has a non-zero real coordinate, then the sequence defined by (<ref>) satisfies
𝐯^(k)
= 𝐮_1 c ( 1 + Õ_D( |α_2/α_1|^k) ), λ^(k)
= σ_1( 1 + Õ_D( |α_2/α_1|^2 k)),
where c = (α_1) γ_1/|γ_1|.
This theorem essentially shows that the power iteration (<ref>) converges to 𝐮_1 and its corresponding eigenvalue. At every iteration, 𝐯^(k) and λ^(k) are essentially the desired values multiplied by some dual number, which approaches 1 at an exponential rate.
In this and in the general form of the iteration (<ref>), the dual quaternion power iteration is strikingly similar to the well-established power iteration for real and complex matrices <cit.>.
§.§ Isometries of eigenspaces of matrices of dual quaternions
At this point, we have established the background material and notation required to point out two important facts.
First, the eigenspaces of a Hermitian matrix of dual quaternion are invariant to the action of unit dual quaternions.
Indeed, if x ∈ and 𝐱∈^n is an eigenvector of 𝐀∈ corresponding to eigenvalue λ∈, then
𝐀( 𝐱 x )
= 𝐱λ x
= (𝐱 x) λ.
Therefore, 𝐱 x is also an eigenvector corresponding to eigenvalue λ.
The last transition follows, because the x and λ commute, since the latter is in the center of .
The second fact relates to the isometries of ^n with respect to the dual quaternion norm defined in (<ref>).
Given 𝐱∈^n and an invertible x ∈, we have
𝐱 x _2
= 𝐱_2|x|.
This can be proved by considering both cases of (<ref>) separately and using <Ref>.
Therefore, 𝐱 =𝐱 x _2 if and only |x| = 1, which is equivalent to x^* x = 1, i.e., x is a unit dual quaternion. Therefore, the unit dual quaternions are the isometries of a free module of dual quaternions with respect to the norm structure induced by (<ref>).
Combining these two facts together, we conclude that for a fixed unit dual number q, the → map 𝐱↦𝐱 q has two important properties. First, it preserves the norm (<ref>). Second, it maps eigenvectors of Hermitian matrices of dual quaternions to eigenvectors corresponding to the same eigenvalue.
These two properties play a key role in the spectral approach to synchronization we develop in the next section.
§ THE SPECTRAL APPROACH TO SYNCHRONIZATION PROBLEMS VIA DUAL QUATERNIONS
In this section, we apply the dual quaternion representation of (3) to the problem of (3) synchronization.
In <Ref>, we describe group synchronization problems and noise types which were considered in the literature.
In <Ref>, we describe the spectral method, a prevalent solution to group synchronization problems.
As we describe in <Ref>, this approach can be generalized to the dual quaternions algebra and therefore can be used to address (3) synchronization problems.
We dedicate <Ref> to numerical experiments demonstrating that our proposed method indeed works on simulated data.
§.§ Group Synchronization: Problem Statement
Let G be a group.
Let g_1, …, g_n be elements of G and let g_i j = g_i g_j^-1 be their ratios, 1 ≤ i , j ≤ n.
A synchronization problem is an estimation problem in which one attempts to estimate { g_i 1 ≤ i ≤ n }, the absolute group elements, from their ratios { g_i j 1 ≤ i < j ≤ n }.
Often, the clean ratios are perturbed by some sort of noise.
Because g_1 g, …, g_n g generate the same set of clean measurements for every arbitrary g ∈ G, we can only hope to estimate g_1, …, g_n up to a global right-multiplication by an arbitrary element of G.
In order to describe the typical noise models, it is convenient to represent the group faithfully as some subgroup of the general linear group, the group of invertible matrices.
Thus, we assume in this section that one has an injective homomorphism ρ : G →GL_k ([F]), with [F] = or [F] = [C].
The absolute group elements and their ratios are mapped onto GL_k ([F]) via ρ in a consistent manner. Namely, ρ (g_i j) = ρ (g_i) ρ (g_j)^-1.
The elements of G are represented by elements of [F]^k × k, the k × k matrices with entries in [F].
We refer to [F]^k × k as the representation space and to the image of ρ as the representing subspace.
Throughout this subsection, we suppress the homomorphism ρ and assume the absolute group elements and their ratios are already faithfully represented in some matrix algebra.
Given a group represented in this manner, three kinds of noise sources are usually considered in the literature.
First, pertrubative noise can be applied to each clean ratio.
This has two forms.
Multiplicative noise perturbs the clean ratio, while still remaining in the group, so that we measure g_i j = g_i jε_i, j, with ε_i, j∈ G.
Additive noise perturbs the clean ratio within the representation space, so that we measure g_i j = g_i j + ε_i, j, with ε_i, j∈[F]^k × k.
We here assume that pertrubative noise is either multiplicative for all ratios, or additive for all ratios.
The last two kinds of noise are best viewed as applied to a set of measurements. This set of measurements may be assumed to be either a complete set of clean measurements { g_i j 1 ≤ i < j ≤ n }, a measurement set afflicted with pertrubative noise, {g_i j 1 ≤ i < j ≤ n }, or a subset of one of these.
Every set of measurements can be envisaged as an undirected graph of order n, in which each vertex is labeled with an absolute group element and the edges correspond to the measurements in the set. See <Ref> for an example.
The last two kinds of noise are best viewed as acting on the measurement graph.
The first, which we term selection noise, merely removes a subset of the edges of the measurement graph.
The second, which is often called corruptive or adversarial noise, replaces the labels of a subset of edges with a randomly generated labels which are independent from the clean group ratios.
In both kinds of noise, the affected subset of edges may be chosen randomly or deterministically.
The measurement graph has a subgraph formed by edges which were not corrupted.
We assume that this subgraph is connected, regardless of the method used to choose the subset of edges affected by selection or corruptive noise.
<ref> shows an example of a measurement graph with corruptive noise.
All combinations of the three kinds of noise were considered in the literature, both in numerical investigations of estimation methods and in their theoretical analysis <cit.>. The literature cited provides a mere illustrative selection.
§.§ Group Synchronization via the Spectral Method
Let g_1, …, g_n be elements of a group G.
We emphasize at the outset of this subsection that all elements of G are faithfully represented within some matrix algebra [F]^k × k.
Whenever perturbative noise is considered, it is always assumed to be multiplicative.
The spectral method relates synchronization problems to eigenproblems on a matrix, which is essentially a generalized adjacency matrix of the synchronization graph.
Let 𝐘∈[F]^n k × n k be a block matrix such that the (i, j)th block 𝐘_i j, i < j, is the measurement of the ratio g_i g_j^-1.
If the measurement is clean, perturbed (multiplicatively) or corrupted, we assign 𝐘_j i = 𝐘_i j^-1.
If the measurement is missing, we set 𝐘_i j = 𝐘_j i = 0_ k × k.
The diagonal blocks are 𝐘_i i = 𝐈_k for i=1, …, n.
Essentially, the block on the ith row of blocks and jth column of blocks is a measurement of the ratio of g_i g_j^-1.
Since these measurements label the edges of the synchronization graph, this is a generalization of the notion of the adjacency matrix of a graph.
In the absence of noise, it is evident that 𝐘 = 𝐠𝐠^-1.
Here, 𝐠∈[F]^n k × k is the column block matrix with g_i as its ith block and 𝐠^-1∈[F]^k × n k is the row block matrix with g_i^-1 as its ith block.
It is immediate that 𝐘𝐠 = n 𝐠 and that n is the only non-zero eigenvalue of 𝐘.
Thus, the columns of 𝐠 form a basis of the eigenspace of 𝐘 associated with n.
Furthermore, the kernel of 𝐘 is the orthogonal complement of the space spanned by the columns of (𝐠^-1)^⊤∈[F]^n k × k.
Thus, [F]^n is the direct sum of the kernel of 𝐘 and the eigenspace of 𝐘 associated with n.
Overall, in the absence of noise, the eigenspaces structure of the measurement matrix 𝐘 contains information on 𝐠, and therefore contains information on the absolute group elements.
In the presence of noise, the measurement matrix 𝐘 can be viewed as a disrupted form of an hypothetical, clean measurement matrix of the form we described in the previous paragraph.
The eigenspaces of the measurement matrix can also be viewed as a disruptions of the eigenspaces of this hypothetical matrix.
If the disruption is mild enough, the eigenspaces of the top k eigenvalues of 𝐘 is expected to preserve enough of the information of the eigenspace associated with the eigenvalue n of this hypothetical matrix.
By projecting these top eigenvectors of the measurement matrix onto the representation of G, it is possible to obtain an estimate of the absolute group elements. This step may also somewhat alleviate the disruption of the eigenspace caused by the noise.
The resulting estimation method can be formulated as a two step procedure:
* Find a set of unit eigenvectors of 𝐘 associated with its largest k eigenvectors. These can be placed in a matrix 𝐔∈^n k × k.
* Project these eigenvectors onto G.
Implementing this two step method is fairly straightforward using existing numerical algebra libraries.
The first step is can be easily implemented using all modern numerical eigenproblem solvers.
The second step, often referred to as “rounding” in the synchronization literature <cit.>, can be more involved, depending largely on G and its chosen representation.
A few examples will serve to illustrate the range in the complexity of this task.
If G = (2) is represented as unit complex numbers, the result of the first step is a unit vector of complex numbers.
Simply dividing each of its entries by its complex norm yields a vector of unit complex numbers.
This is possible, provided the disruption caused by the noise is mild enough so as to ensure all entries of the eigenvector are non-zero.
This rounding procedure was used to great effect in (2) synchronization, e.g. <cit.>.
If G = (k) is represented as real orthogonal k × k matrices, the result of the first step is a n k × k matrix which can be treated as a column block matrix with k × k blocks.
The projection is carried out block-wise.
The SVD of each block is calculated, yielding a decomposition 𝐁 = 𝐔Σ𝐕^⊤, where 𝐔 and 𝐕 are orthogonal matrices and Σ is a diagonal matrix.
The rounding step is completed by replacing every block with the image of the map 𝐁↦𝐔diag(1, …, 1, (𝐔𝐕^⊤)) 𝐕^⊤.
In <cit.>, G = (3) was represented on the real 4 × 4 matrices of the form
[𝐑 𝐭
0^⊤ 1
].
This matrix represents (R, 𝐭) ∈(3), where R is a rotation represented as 𝐑, a real 3 × 3 orthogonal matrix with determinant of 1.
Consider the matrix 𝐕∈^4 n × 4, the columns of which are the top eigenvectors of the measurement matrix which were calculated in the first step.
The rounding procedure considered by <cit.> begins by extracting every fourth row of 𝐕. Denoting the resulting ^n × 4 matrix by 𝐕_4, all solutions of 𝐕_4𝐮 = 0 are found by finding an orthonormal basis 𝐮_1, 𝐮_2, 𝐮_3 for the kernel of 𝐕_4. A unit vector solution for 𝐕_4𝐮_4 = (1, …, 1 )^⊤ is also found.
Since nothing guarantees that solutions to these equations even exist, they are approximated in the least squares sense.
Let 𝐔∈^4 × 4 be the matrix with columns 𝐮_1, 𝐮_2, 𝐮_3, 𝐮_4.
Let 𝐕' = 𝐕𝐔 and substitute every forth row of the resulting matrix with (0, 0, 0, 1).
Now, each of the 4 × 4 blocks of 𝐕' has almost the form (<ref>).
If this matrix is complex, zero out the imaginary part of every entry.
Finally, the top 3 × 3 submatrix of every block is projected onto (3) by utilizing its SVD in the manner described in <Ref>.
Non-linear transformations are employed in all three examples.
However, the simplicity of the rounding step in <Ref> for (2) is contrasted by the multi-step, complicated procedure shown in the <Ref> for (3), the focus of this paper.
This difference stems from inherent features of the representation of the group.
The group (2) is embedded within the complex plane in a way which respects the algebraic properties of the representation space.
First, the norm of a complex number is directly related to the conjugation operation, i.e., |z| = √(z z^*) for any z ∈[C]. The submanifold of the complex plane into which (2) is embedded, the unit circle, is entirely defined in terms of this conjugation operation, i.e., [T] = { z ∈[C] z z^* = 1 }.
Second, the embedding takes into consideration the *-algebra structure of the complex plane. The group inverse is the conjugation operation.
These features do not hold for the matrix representation (<ref>) of (3).
As we discuss in greater detail in <Ref>, this representation merely embeds (3) within the multiplicative group of the matrix algebra of real 4 × 4 matrices.
It does not rely or respect the other algebraic structures of the representation space. Merely transposing (<ref>) almost always yields a matrix which does not represent an element of (3).
We now discuss the nature of the rounding procedure.
On the surface, all it does is ensure that the output of the spectral method are elements of the group, but a closer look reveals a more complicated situation.
In the noiseless case, which is simplest, we saw that the measurement matrix has a single non-zero eigenvalue.
We consider now the linear isometries of its eigenspace with respect to the standard Euclidean norm.
These isometries form a compact group.
Numerical linear algebra tools ensure we shall obtain unit eigenvectors of the measurement matrix. These may be 1/n𝐠, where the 1/n term ensures these columns are unit eigenvectors. However, more often they are expected to be 1/n𝐠𝐒, where 𝐒 is an isometry of the eigenspace of the non-zero eigenvalue of the measurement matrix.
Let us consider the three examples above. In (2) synchronization (<Ref>), the group of isometries is (2) itself, represented by unit complex numbers.
Indeed, since the eigenspace is one-dimensional, every isometry has the form 𝐱↦𝐱 x for x ∈[C]. By substituting 𝐱 x into the standard Euclidean norm, it is easy to show that it is an isometry if and only if 𝐱 x = x|x|, which holds if and only if |x| = 1. Therefore, x ∈(2).
It follows that in the noiseless case, the spectral method recovers the absolute group elements up to a right-multiplication by an element of (2). And so, in the presence of noise, the rounding step alleviates a disruption in the eigenspace structure and nothing more.
This is not the case in the other two examples. There, the rounding has to perform other tasks, which are unrelated to the disruption of the eigenspaces caused by the noise and will be required even in its absence.
Again, it is instructive to consider the noiseless case first.
In (k) synchronization (<Ref>), the non-zero eigenspace is a k-dimensional subspace of ^n and its isometry group with respect to the standard Euclidean norm on ^n is (k), the group of orthogonal transformations of ^k. The group (k) is a proper subgroup of (k) and so the unit eigenvectors in the eigenspace can be separated into two disjoint populations. The first population is obtained by applying (k) to the columns of 1/n𝐠 from the right. The second can be obtained by applying elements of (k) ∖(k).
This means that being a solution to the synchronization problem implies being a basis of an eigenspace of the non-zero eigenvalue of the measurement matrix, but the converse does not hold.
We refer to this phenomena as the eigenspace-synchronization gap.
Therefore, in the presence of noise, the rounding procedure performs at least the following two tasks. It alleviates the disruption to the eigenspace structure and ensures that the resulting estimate of the absolute group elements is indeed the absolute group elements up to a multiplication by an arbitrary element of G from the right. The latter can no longer be taken for granted.
These issues are exacerbated considerably in (3) synchronization (<Ref>), where the group is non-compact. There, even in the absence of noise, the group (3) is not even a subgroup of the isometries of the eigenspace.
This is the case because the group of isometries is compact, while (3) is not.
In this case, the eigenspace-synchronization gap is not so much a gap as an abyss.
The domain of the rounding procedure is a compact manifold, matrices formed with four unit eigenvectors as columns. Its codomain is a non-compact manifold, block matrices with blocks of the form (<ref>).
The rounding procedure maps this compact domain to this non-compact codomain.
Thus, even in the absence of noise, for (3) synchronization it is another large step away from the neat correspondence exhibited in (2) synchronization (<Ref>) between eigenvectors of the measurement matrix and estimates of the absolute group elements. In the presence of noise, it does a great deal more than handling the eigenspace disruption.
In closing this subsection, we digress to note an important distinction between the way we introduced the spectral method and the way it is often discussed in the literature. In the literature on synchronization, the relationship between the eigenvectors of the noisy measurement matrix and an estimate of the absolute group elements is often arrived at using a optimization terminology.
For instance, the initial goal of both <cit.> and <cit.> is an estimate the absolute group elements obtained by solving a certain least squares problem over 𝐠∈ G^n. Here, G^n is the set of all column block matrices with blocks representing elements of G.
Since the domain of the problem is non-convex, this is a challenging optimization problem to solve in practice. This constraint is therefore changed to all matrices satisfying 𝐠^*𝐠 = n 𝐈_k.
In the case of <cit.>, which worked on (2) represented on the unit complex numbers, this change in the constraint is a relaxation of the constraint of the original least squares problem. That is, the feasible set of the original problem is replaced by a larger set.
However, this change is not a relaxation of a constraint in <cit.>, which worked on (3) with the matrix representation (<ref>).
Regardless of that, in both cases, the new least squares problem turns out to be easily solved by finding the eigenvectors of the measurement matrix.
We note this here merely to point out that the optimization perspective provides a useful heuristic for deriving practically useful estimation methods for synchronization problems.
However, it does not offer an immediate way to prove when and why these methods work.
We here treat the noisy measurement matrix as a disrupted version of a clean, complete measurement matrix, and seek to approximate the eigenspaces of the latter using the eigenspaces of the former.
In the optimization perspective, the very same eigenproblem is considered as a relaxed optimization problem, which one hopes approximates the solution of the non-convex problem.
This is a difference without much consequence to the governing mathematics of the spectral method. Regardless of the heuristic used to derive the method, considerable effort is required to come up with and prove theoretical guarantees on the performance of the spectral method.
Indeed, there is a rich literature of formal characterizations of when and how solutions of the eigenproblem provide good estimates of the absolute orientations, e.g., <cit.> with many mathematical tools, including random matrix theory and perturbation theory, being brought to bear on these questions.
§.§ A Spectral Approach to (3) Synchronization via Dual Quaternion Representation
In <Ref>, we discussed the contrast between the rounding step in (2) and (3) synchronization in <Ref> and <Ref>.
We identified two types of differences between them. First, their respective representations have different features. For instance, the representation of (2) agreed with involutive algebraic structure of the space it was embedded in, whereas the representation of (3) did not.
Second, in a sense, the rounding does more in the (3) case than in the (2) case.
We now utilize the material we surveyed in <Ref> to develop a spectral synchronization method for (3), which is more similar to the spectral method for (2) in both these respects.
We note in passing that the quaternions and the dual quaternions were used in the past to address synchronization problems <cit.>, but this is the first time an analog of the spectral method for synchronization was developed using dual quaternions.
All the necessary components of the spectral method are present over the dual quaternions.
Elements of (3) are represented as unit dual quaternions, as we surveyed in <Ref>.
This representation agrees with both the multiplication operation of the algebra and its involution, since the dual quaternion conjugate of unit dual quaternion is its inverse.
The generalized adjacency matrix of the synchronization graph is a matrix of dual quaternions, rather than a block matrix with real or complex entries.
The resulting matrix is Hermitian, in the dual quaternion sense.
By the spectral theorem for dual quaternion matrices, it has a eigenvector decomposition, as described in <Ref>.
Now, given a clean, complete set of measurements, the measurement matrix has the form 𝐘 = 𝐠𝐠^*, where 𝐠 = (g_1, …, g_n)^⊤ is a column vector of unit dual quaternions and 𝐠^* is its conjugate transpose.
In the presence of noise, we consider the dual quaternion eigenvector of the measurement matrix corresponding to its top eigenvalue as an approximation of 𝐠, the eigenvector of the clean measurement matrix.
Here, “top” is with respect to the order of eigenvalues established in <Ref>.
This eigenvector can be approximated using the power method we surveyed in <Ref>.
The rounding is carried out by applying the map defined in <Ref> to each entry this eigenvector. This rounding procedure stems directly from the close relationship between the algebraic structure and geometry of the dual quaternion algebra and the unit dual quaternions, as we discussed at the end of <Ref>.
If the noise is mild enough, all entries of the top eigenvector are expected to be invertible, in which case the projection amounts to applying the map q ↦q/|q| entry-wise to the eigenvector.
Overall, we obtain the following two-step algorithm:
* Find the eigenvector of 𝐘 corresponding to its largest eigenvalue, in the order established in <Ref>.
Use the power iteration we surveyed in <Ref>.
Denote this eigenvector by g = (g_1, …, g_n)^⊤.
* Carry out the substitution g_j→g_j/|g_j| on each of its entries (j=1,…, n).
As we demonstrate in <Ref>, this algorithm yields comparable performance to the state-of-the-art spectral approach to (3) synchronization described in <cit.>.
We emphasize the simplicity of the rounding step compared to the one employed by <cit.>, which we described in <Ref>. Its simplicity is owed entirely to the close relationship between (3) and the algebra of dual quaternions.
We also emphasize that here the rounding procedure does nothing more than handle the disruption the noise causes to the eigenspaces of the clean measurement matrix. Indeed, as we discussed in <Ref>, the isometry group of the eigenspaces of 𝐘 is actually the unit dual quaternions and as we discussed in <Ref>, each unit dual quaternion represents an element of (3). Thus, there is no eigenspace-synchronization gap, despite the non-compact setting. Our method enjoys similar performance to the current state-of-the-art in spectral synchronization for (3), yet reaps the benefits of embedding (3) in an algebra that is much closer to it in structure.
Taken together, the resulting spectral method is remarkably similar to the spectral method of (2) we showcased in <Ref>.
This is evident in two places. First, in the properties of the representing space and its relationship to the subset which represents the group. Second, in the underlying tasks performed by the rounding step.
This similarity is especially remarkable when we consider the fact that (2) is a compact, commutative group, while (3) is a non-compact and non-commutative group.
Finally, we note that Cui2023 addressed a synchronization problem with selection noise, though they do not use the word synchronization at all.
They proposed an iterative method, which is distinct from the spectral method we constructed here, both in its form and its theoretical underpinnings.
First, it is not the spectral method as we conceived it here. At every iteration, the iteration (<ref>) is used to approximate the eigenvector of some matrix of dual quaternions.
Second, their iterative method was motivated by the need to solve an explicit optimization problem over the dual quaternions, whereas we relied on the heuristic picture of the eigenspaces of a matrix being disrupted by noise.
§ NUMERICAL EXPERIMENTS
We run several numerical experiments demonstrating the utility of the dual quaternions in synchronization problems.
We begin in <Ref> by describing how the synthetic synchronization measurement matrices were simulated in our experiments and describe our experimental procedure.
In <Ref> and <Ref> we show and discuss the results of the synthetic data experiments. They demonstrate that using the dual quaternions to represent (3) yields comparable estimates to the current state-of-the-art spectral synchronization method <cit.>, while enjoying the simpler rounding procedure obtained by embedding (3) in the algebra of dual quaternions.
§.§ Synthetic Experiments Setup
We conducted several experiments on synthetic measurements matrices, modeled after the synthetic data experiments carried out in <cit.>, the current state-of-the-art in (3) synchronization.
Probability functions on (3) were defined using the angle-axis-translation representation discussed in <Ref>.
Ground truth elements g_1, …, g_n∈(3) were drawn from i.i.d. random variables.
The angles were independently and uniformly distributed over [0, 2 π ).
The axis was independently and uniformly distributed over S^2.
Translations of the form 𝐭 = (t_1, t_2, t_3)^⊤∈^3 were independently distributed with t_1, t_2 and t_3 being i.i.d. standard Gaussian random variables.
The measurement matrix 𝐘∈ follows the following model:
𝐘_i j
=
e_i j( s_i j g_i g_j^-1ε_i, j + (1 - s_i j) c_i j) 1 ≤ i < j ≤ n,
1 i = j ,
𝐘_j i^* 1 ≤ j < i ≤ n.
Here, (e_i j 1 ≤ i < j ≤ n ) are i.i.d Bernoulli random variables with parameter p and
(s_i j 1 ≤ i < j ≤ n ) are i.i.d. are the same with parameter q.
The former are indicators of existing measurements and the latter are indicators of entries which were not corrupted.
Denoting by ∘ the entry-wise product of two matrices of elements of (3), the matrices 𝐄 = (e_i j), 𝐒 = (s_ i j) and 𝐄∘𝐒 = (e_i js_i j) are all the adjacency matrices of random graphs following the Erdős-Rényi model with parameter p, q and p q, respectively.
The corrupted entries (c_i j 1 ≤ i < j ≤ n) are i.i.d. with an angle in degrees normally distributed with zero mean and variance σ_r^2, axis distributed uniformly over S^2 and translation with t_1, t_2, t_3 being i.i.d. Gaussian with zero mean and variance σ_t^2.
All random variables mentioned above were also independent of each other.
Given a ground truth 𝐠∈(3)^n and an estimate 𝐠∈(3)^n, we measure the quality of the estimate as follows.
Due to the symmetry in the observations, we have to align 𝐠 and 𝐠 in the following sense. We find a g ∈(3) such that a distance metric between 𝐠 and 𝐠 g is minimized, where 𝐠 g is the result of multiplying by g from the right every entry of 𝐠.
In <Ref>, we discuss the specifics, including the distance metric we used.
Here, we describe the procedure itself.
First, we assume that every element of (3) is represented by (q, 𝐭) with q ∈ being a unit quaternion with non-negative first coordinate as discussed in <Ref> and 𝐭∈^3.
Second, if g = (q, 𝐭),
𝐠 = ((q_1, 𝐭_1), …, (q_n, 𝐭_n)) and
𝐠 = ((q_1, 𝐭_1), …, (q_n, 𝐭_n)), we have
q
= s/|s|,
s
= ∑_j=1^nq_j^* q_j,
and
𝐭
= 1/n∑_j=1^nφ(q_j^*) ( 𝐭_j - 𝐭_j).
This alignment procedure was used on the estimate obtained by both methods employed in our experiments, our dual quaternion spectral synchronization and the spectral method of <cit.>, which used the matrix representation (<ref>).
We note that our alignment method is considerably simpler to apply than the alignment method utilized by <cit.>, whose numerical experiments inspired our own.
All experiments were repeated fifty times, each with freshly generated ground truth and noise realization with the specified experimental parameters.
From each estimate we extracted the entry-wise error, after alignment, for the rotations and translations, separately.
For the rotations, the error was calculated as
d(q_1, q_2)
= 2 arccos( 2 q_1q_2^2 - 1),
where q_1q_2 is the Euclidean inner product applied to q_1 and q_2 considered as elements of ^4 with the obvious embedding.
See <Ref> at the end of <Ref> for an explanation of this metric.
For the translations, we used the Euclidean distance,
d (𝐭_1, 𝐭_2)
= 𝐭_1 - 𝐭_2.
For each, the mean, minimum and maximum error were calculated and each of these measures was averaged over the fifty repeats. The plotted data for every quantity is always its mean over the repeats.
We described the approach of <cit.> in <Ref>.
Both estimation methods were implemented in Python.
Arrigoni2016 originally implemented their approach in MATLAB.
We took care to use equivalent functions to theirs in the Python ecosystem.
Code used to simulate the figures below is available at https://github.com/idohadi/dqsync-python/https://github.com/idohadi/dqsync-python/.
§.§ Response to Missing Entries and Perturbative Noise
In <Ref>, we show the results with increasing levels of multiplicative noise on the rotations and no noise on the translations.
In <Ref>, we show the results with increasing levels of multiplicative noise on the translations and no noise on the rotations.
Consistent with the semidirect product structure of (3), in <Ref> the estimate of the translational component is also affected by noise, whereas in <Ref> the quality of the estimate of the rotations is basically flat, unaffected by the noise.
When we vary the level of noise on both the rotational and translational part in <Ref>, the error rises on both components.
Overall, using the dual quaternion representation of (3) yields marginally better mean entry-wise error, especially in the estimate of the rotation and when more entries are missing (p = 0.05 vs. p = 0.3).
It is also evident that using the dual quaternions yields somewhat stabler estimates, as indicated by the smoother dual quaternion curves evident in <Ref>.
This may be a feature of our implementation, rather than the method, as these peaks are not evident in the plots shown in <cit.>.
All trends we discussed above are also evident in minimum and maximum entry-wise errors.
We conclude that the two methods perform comparably and that the dual quaternions can be used effectively to represent (3) in applied estimation problems.
§.§ Response to Corruptive Noise
In <Ref>, we used no perturbative noise but there were missing entries.
In <Ref>, there are no missing entries but there is perturbative noise.
The same trends are evident in both of them.
First, the rotation estimate is better using the matrix representation (<ref>), while the translation estimate is better using the dual quaternion representation.
Second, the differences in estimate quality are larger in <Ref> than in <Ref>.
Overall, these experiments indicate no clear advantage to either method, but do underscore the fact the dual quaternions can be used effectively to represent (3) in applied estimation problems.
§ COMPARISON OF REPRESENTATIONS OF (3)
We compare the properties of three representations of (3) which were used in applied mathematics work.
Our discussion is skewed towards properties which we think are central to how notions from linear algebra can be used to solve applied problems over (3).
The most straightforward way to represent elements of (3) is as affine transformations, a pair (R, 𝐭) ∈(3) ⋉^3.
These may come in several variants, differing in the way the rotation is represented.
Here we survey the axis-angle and rotation matrix representations.
In the axis-angle representation, a rotation R is represented by the axis 𝐯∈ S^2⊂^3 of the rotation and the angle θ (in radians) by which vectors are rotated around 𝐯.
Both can be concisely encapsulated in a rotation vector 𝐱 = θ𝐯∈^3, which are elements of B_3(0, 2 π), where B_3(𝐱, r ) = {𝐲∈^3𝐱 - 𝐲_2 < r }.
Every pair of a rotation vector and a translation can represent an element of (3) and one can easily extend that to pairs of the form ^3×^3 by taking the Euclidean norm of the rotation vector modulo 2 π.
Thus, we say that representation space, the space of all elements of the form of the given representation, is ^3×^3.
The representing subset, the subset of the representation space into which (3) is embedded, is B_3(0, 2 π) ×^3.
Furthermore, there is projection from the representation space to the representation subset, mapping every element of the representation space to the representing subset.
We refer to this representation as the axis-angle-translation representation.
From all this it follows in particular that in this representation the dimension of the representation space is 6, which matches the manifold dimension of (3), which makes it the most parsimonious representation we will survey.
However, there is no direct way to calculate the product of two elements of (3) in this representation. It is necessary to convert them into one of the other representations we will survey.
We do not know of any established way to construct something like an algebra over this representation space and consequently we know of no such way to construct matrix algebras and investigate their spectral properties.
The other (3) representation we consider are rotation matrices, real 3 × 3 orthogonal matrices with a determinant of 1.
This representation comes with a multiplicative structure of matrix multiplication and it is embedded in the matrix algebra of all real 3 × 3 matrices.
The representation space of (3) is then ^3 × 3×^3≅^12.
Its multiplicative and vector space structure can be identified with the set of all affine transformations of ^3, transformations of the form 𝐱↦ A 𝐱 + 𝐭, where A is a linear transformation and 𝐭∈^3.
This set has both an addition of and multiplication operations, defined by adding or composing, respectively, two affine transformations.
Despite this, it does not form a ring, because these two binary operations do not satisfy one of the distributive properties.
Borrowing an example from <cit.>, suppose that (A_1, 𝐭_1) and (0, 𝐭_2) are two affine transformations. Here, 0 is the zero linear transformation and 𝐭_2 0.
We have
(0, 𝐭_2) ∘( (A_1, 𝐭_1) + (A_1, 𝐭_1) )
= (0, 𝐭_2),
but
(0, 𝐭_2) ∘(A_1, 𝐭_1) + (0, 𝐭_2) ∘(A_1, 𝐭_1)
= 2 𝐭_2.
While not being a ring, the affine transformations form a different algebraic structure, that of a near-ring.
As far as we know, matrix spaces and spectral decompositions were not studied over near-rings in general and over the affine transformations in particular.
Furthermore, to the best of our knowledge, a projection of affine transformations onto the representing subset has not been worked out in the literature.
Finally, (3) has a group embedding within the matrix algebra ^4 × 4.
An element (R, 𝐭) ∈(3), where R is represented as 𝐑, real 3 × 3 orthogonal matrix with determinant of 1, can be represented as a block matrix
[𝐑 𝐭
0^⊤ 1
].
Rectangular block matrix where all blocks are of this form obviously belong to the matrix algebra ^4 n × 4 n, which definitely has a spectral decomposition theorem and many other matrix decompositions.
Furthermore, an invertible 𝐀∈^4 × 4 can also be projected onto the representing subset by a two step procedure.
First, finding a invertible matrix 𝐁∈^4 × 4 such that the matrix 𝐀𝐁 has (0, 0,0, 1) in its bottom row. This amounts to changing the coordinates of the matrix 𝐀 to one which satisfies this particular constraint on the bottom row.
Second, denoting by (𝐁𝐀)_3× 3 the upper left 3 × 3 submatrix of 𝐁𝐀, one takes its singular value decomposition (𝐁𝐀)_3× 3 = 𝐕𝐒𝐔^⊤, where 𝐒 is diagonal and 𝐔 and 𝐕 are orthogonal.
To obtain a matrix of the form (<ref>), replace (𝐁𝐀)_3× 3 with 𝐕diag(1, 1, (𝐕𝐔) ) 𝐕^⊤.
The great advantage of the matrix representation (<ref>) is the fact that its representation space is a matrix algebra.
This allows the use of the familiar linear algebraic toolkit and in particular allows one to define the projection we described above.
However, this is also the source of the greatest disadvantage.
The group (3) is embedded in the multiplicative group of ^4 × 4, and no more.
The other algebraic structures defined on ^4 × 4 are inconsistent with the embedding.
Merely transposing (<ref>) would generally take one out of the representing subspace, indicating the involution of ^4 × 4 plays no role in representing (3).
Furthermore, the relationship between the projection and group structure is hard to characterize.
This last problem comes to the fore in the situation discussed in <Ref>, where we describe the projection onto (3) of an entire column block matrix with ^4 × 4 blocks, which is used in an actual application <cit.>.
Perhaps underscoring the gap between (3) and the representation space in this case, is the fact (3) is a manifold of dimension 6, while the representation space is 16-dimensional.
Consider the unit dual quaternion representation of (3) we surveyed in <Ref>.
We argue that the representation space is very close to the structure of (3), yet similar enough to ordinary vector spaces to allow some of the same linear-algebraic notions that are so useful to the applied mathematician to be brought to bear on applied problems.
The representation space is very close to the structure of (3) in two ways.
First, by the fact nearly all dual quaternions are elements of (3) scaled by dual numbers, as shown by <Ref> of <cit.>.
Second, the inverse of an element of (3) represented on the unit dual quaternion is its dual quaternion conjugate, as we saw <Ref>. In this way, the involutive algebra structure of the dual quaternions is intimately tied with the group structure of (3).
Most importantly, despite the non-commutativity of this algebra, it is sufficiently simple to allow one to establish a spectral theorem as we saw in <Ref> and also the derivation of a numerical schemes to approximate eigenvectors, like we surveyed in <Ref>.
We suggest that other numerical schemes, like the Arnoldi iteration and other Krylov subspace methods, may also be generalizable to the dual quaternions.
Taken together, these properties indicate that the dual quaternion algebra balances well the difficulties incurred by working over non-commutative algebra with the advantages it offers.
§ ALIGNMENT OF TWO DUAL QUATERNION VECTORS
Let 𝐠 = (g_1, …, g_n)^⊤∈(3)^n and 𝐠 = (g_1, …, g_n)^⊤∈(3)^n.
Assume that elements of (3) are represented as
g_j = (q_j, 𝐭_j) and g_j = (q_j, 𝐭_j),
where q_j and q_j are unit quaternions with a non-negative first coordinate, representing a rotation as we discuss in <Ref>, and 𝐭_j and 𝐭_j are translations represented as vectors in ^3.
Recall <Ref> and the double cover φ defined there.
Aligning 𝐠 and 𝐠 amounts to finding g = (q, 𝐭) solving
min_g ∈(3){∑_j=1^n𝐪_j𝐪 - 𝐪_j_F^2
+ ∑_j=1^nφ (q_j) (𝐭) + 𝐭_j - 𝐭_j_2^2}.
Here, 𝐪_j, 𝐪_j and 𝐪 are real 3 × 3 orthogonal matrices with determinant of 1 representing the same elements of (3) as q_j, q_j and q.
These matrices are constructed by representing φ(q) in the standard basis of ^3.
Also, ·_2 is the Euclidean norm and ·_F is the Frobenius norm.
Solving (<ref>) amounts to finding g minimizing the sum of squared distances between corresponding rotations of 𝐠 and 𝐠 and a sum of squared distances between the translations of 𝐠 and 𝐠.
The two sums in (<ref>) are two decoupled optimization problems, since the left one depends only on q and the right one only on 𝐭.
We consider these two decoupled problems separately.
We begin with the translation sum.
Let F (𝐭) be the right sum in (<ref>).
We wish to find the minimum of F in ^3.
Because the Euclidean norm is preserved under orthogonal transformations and φ is a homomorphism, F can be written as
F (𝐭)
= ∑_j=1^n𝐭 + φ(q_j^*) (𝐭_j - 𝐭_j)^2.
Fermat's theorem then yields that its exterema is
𝐭
= 1/n∑_j=1^nφ(q_j^*) ( 𝐭_j - 𝐭_j),
exactly as in (<ref>).
That it is a minimum follows from the fact F is convex and defined on ^3, and therefore can have no local maxima.
The rotation sum is slightly more involved.
First, we note that for any𝐑_1 and 𝐑_2 two real 3 × 3 orthogonal matrices with determinant of 1 we have
𝐑_1 - 𝐑_2_F^2
= 𝐑_1_F^2
+ 𝐑_2_F^2
- 2 (𝐑_1𝐑_2^⊤)
= 3 - 2 ( 𝐑_1^⊤𝐑_2).
A minimizer of the left sum in (<ref>) is therefore a maximizer over the unit quaternions with non-negative first coordinate of
G_1 (𝐪)
= ∑_j=1^n( 𝐪^⊤𝐪_j^⊤𝐪_j).
Finally, let q_1 and q_2 are quaternions representing the same elements of (3) as 𝐑_1 and 𝐑_2, respectively. As we argue below, the following identity holds:
( 𝐑_1^⊤𝐑_2)
= 4 q_1q_2^2 - 1,
where ·· is the Euclidean inner product on ^4 applied to quaternions via their obvious identification with ^4.
Therefore, the maximizer of G_1 over the unit quaternions with non-negative first coordinate is the maximizer over the same domain of
G (q)
= ∑_j=1^nq_j qq_j.
We note that
q_1q_2
= 1/2( q_1^* q_2 + q_2^* q_1),
and so
q_1 qq_2
= 1/2( q^* q_1^* q_2 + q_2^* q_1 q )
= 1/2( q^*( q_1^* q_2) + ( q_1^* q_2)^* q )
= qq_1^* q_2.
Therefore, for any unit quaternion with non-negative first coordinate q we have:
G (q)
= ∑_j=1^nqq_j^* q_j
= q∑_j=1^nq_j^* q_j≤q_2∑_j=1^nq_j^* q_j_2
= ∑_j=1^nq_j^* q_j_2.
In the above we used the Cauchy-Schwartz inequality applied to the quaternions, again identified as elements of ^4.
Simple substitution into G shows that the q defined in (<ref>) is a unit quaternion with non-negative first coordinate which achieves this bound.
It remains to prove (<ref>).
It follows from the following lemma:
Let q be a unit quaternion.
Let 𝐑 be the real 3 × 3 orthogonal matrix with determinant of 1 formed by representing φ(q) using the standard basis of ^3.
If q = a + b + c + d, then
(𝐑)
= 4 a^2 - 1.
We obtain (<ref>) as a corollary of <Ref> due to a combination of two facts. First, 𝐑_1^⊤𝐑 in (<ref>) expressed in quaternions is q_1^* q_2. Second, as can be worked out from (<ref>), the first coordinate of the product q_1^* q_2 is q_1q_2.
Using the axis-angle representation of 𝐑, we consider it a rotation around axis 𝐫∈ S^2 by angle θ.
From <cit.>, (𝐑) = 2 cosθ + 1, while <cit.> yields a = cosθ/2.
Therefore, using the trigonometric identity cosθ = 2 cos^2θ/2 - 1, we obtain
4 a^2 - 1
= 4 cos^2θ/2 - 1
= 2 ( 2 cos^2θ/2 - 1) + 1
= 2 cosθ + 1,
as required.
From the identities used in the proof of <Ref>, it follows that the metric we defined on the unit quaternions in in <Ref> satisfies
d (q_1, q_2)
= 2 arccos( cosθ) = 2 θ_1 2,
where θ_1 2∈ [0, π ) and θ is the rotation angle in the angle-axis representation of the rotation q_1^* q_2.
§ DATA AVAILABILITY
The data underlying this article are available in the GitHub repository at <https://github.com/idohadi/dqsync-python/>.
§ FUNDING
This work was supported by the Binational Science Foundation [2019752]; Binational Science Foundation [2020159] to T.B.; the Israeli Science Foundation [1924/21] to T.B.; and the Deutsche Forschungsgemeinschaft [514588180] to N.S.
|
http://arxiv.org/abs/2307.04902v1 | 20230710210730 | An evolutionary game with environmental feedback and players' opinions | [
"E. M. Lorits",
"E. A. Gubar"
] | cs.GT | [
"cs.GT",
"91A22"
] |
An evolutionary game with environmental feedback and players' opinions
Lorits E. M., Gubar E. A.
======================================================================
-60pt
§ INTRODUCTION
Evolutionary games are a developing subfield of game theory <cit.>. Evolutionary games are used to model change in large but finite populations, where all agents have biological, social or economic characteristics that determine their behaviour. Furthermore, each agent is assumed to have no significant effect on the state of the population.
Evolutionary game theory is widely used in many scientific fields. For example, a book <cit.> was published in 2022 that collected many evolutionary models of real biological processes. In addition, evolutionary games
are used to simulate the interaction of large numbers of agents in a network <cit.>. In medicine, evolutionary games can be used, for example, to find methods to fight cancer <cit.> or to solve the problem of vaccinating the population <cit.>.
Papers <cit.> consider the population and the changes that occur in that population, taking into account the environment and the state of that environment. In addition, the effect of environmental feedback on a population has been studied extensively <cit.>. The paper explores the idea of how the state of the environment and agents' opinions about it affect the state of the system. The population, the environment and agents' opinions form a hierarchical structure, where a change in one parameter of the system responsible for the state of the environment, the population or agents' opinions, causes a change in the remaining elements of the system. On the one hand, the state of the environment depends on the prevalence of a particular type of behaviour in the population. On the other hand, the state of the population depends on the popularity of the opinions in the population. The popularity of agents' opinions depends on the state of the environment and the population. In this paper, we consider the state of the environment and the popularity of the opinions as control parameters on the population dynamics.
§ AN EVOLUTIONARY GAME
Consider a population of size N existing in a finite space. It is assumed that the state of the population changes as a result of random pairwise interactions between its agents. It is also assumed that the number of agents is large and that each individual agent has no significant effect on the population <cit.>. Another assumption is that the population has two types of behaviour that agents can follow. An agent's choice of the ith type of behaviour is similar to the choice of the ith pure strategy in a non-cooperative game. It leads to a partition of the population into two subgroups. The agents of each subgroup are programmed to use the same pure strategy. The state of the population is defined as a vector x_N (t) = (x_1 (t), x_2 (t)), where each component x_i(t) is the fraction of the population using the pure strategy i. This vector can be thought of as a mixed population strategy <cit.>. Let x(t) = x_1(t), then x_2(t) = 1 - x(t). The payoffs of the agents refer to the number of offspring (in biological systems) or the number of successors (in economic and social systems) that follow the net strategy i.
Over time, random pairwise meetings between agents occur in the population.
The outcomes of these encounters can be described by a bimatrix game <cit.>. Traditionally, in evolutionary games, it is customary to consider all processes on behalf of the first player, so everything below is formulated in terms of the first player. Let e^i be the vector corresponding to the ith net strategy of the player. The ith element is one and all others are zero. We introduce the function u(e^i,x_N) = e^i · Ax, i=1,…, n as the expected payoff of the agent with pure strategy i when facing a random opponent. This payoff depends on the population state vector x_N. Based on the payoff of the randomly chosen agent, the corresponding average payoff of the population is determined
u(x_N,x_N) = ∑_i∈ K x_iu(e^i,x_N), i=1,…, n.
The change in population composition corresponds to a change in the proportion of agents adhering to net strategy i. These changes are described by the replicator dynamics equation (<ref>), depending on the fractional distribution of players in the population x_N and the first player's payoff matrix A.
ẋ =x(1-x)(u(e^1,x_N) - u(e^2,x_N)).
In this paper, the population is assumed to be dependent on environmental feedbacks that affect the expected payoffs of agents. The resources available to the agents are considered as the environment. The state of the environment is described by the parameter n(t), n∈ [0, 1], where n = 0 (n = 1) when the environment is completely consumed (replenished). The change in the state of the environment is determined by the dynamics (<ref>) proposed in <cit.>. It depends on the change in the state of the population.
ṅ = n(1-n)(θ x + ψ (1-x)),
where the multiplier n(n - 1) corresponds to the logistic growth of the environmental parameter. Depending on the state of the population x_N, resources are replenished or depleted. Agents following the first pure strategy replenish resources n at a rate θ > 0, while agents following the second pure strategy cause resources to be depleted at a rate ψ = -1.
The player payoff matrix A_n establishes the relationship between the population and the state of the environment <cit.>.
A_n =
[ a_11^n a_12^n; a_21^n a_22^n ]
= nA_1 + (1-n)A_0
= n
[ a_11^1 a_12^1; a_21^1 a_22^1 ]
+ (1-n)
[ a_11^0 a_12^0; a_21^0 a_22^0 ].
When n = 1 (n = 0), the game is defined by the payoff matrix A_1 (A_0). The matrices A_1 and A_0 are set so that the non-cooperative game, which takes place between agents at random encounters in the population, has different Nash equilibrium positions in replenished and depleted environments.
An article <cit.> examines changes in population state using replicator dynamics, which takes into account feedback from the environment and depends on public opinion. Opinion reflects population agents' awareness of the state of the environment.
In contrast to this study, the current work assumes that each agent has its own personal opinion about the state of the environment, but does not have reliable information about it.
Consider the case where an agent can hold one of two opinions, m_1 or m_2, regardless of the strategy it chooses. We define the distribution of opinions in the population as a vector y_N (t) = (y_1 (t), y_2 (t)), where each component y_i (t) is the proportion of agents in the population holding opinion m_i. For convenience, we denote y_1 = y(t), y_2 (t) = 1 - y(t). The process of opinion distribution in a population can be described by mean dynamics, which allows changes in the population to be described by a proportional imitation rule <cit.>.
Since any imitation protocol is subject to noise, i.e. errors in estimating the opponent's expected payoff, it is assumed that an agent can have different levels of confidence in the opinions of opponents, depending on the opponent's strategy.
Let's introduce a matrix B, whose elements b_ij represent the degree of confidence of the agent with opinion m_i in the agent pursuing strategy j. Based on a pairwise imitation protocol <cit.>, an imitation protocol (<ref>) has been designed to describe changes of opinion popularity in the population.
Let us determine the expected playoff of the agent with opinion m_i to determine the pairwise imitation protocol.
Since an agent with opinion m_i can follow either the first or the second type of behaviour in the population, its average payoff is obtained by multiplying the proportions of agents using its strategy j by the expected payoff of the player with the corresponding type of behaviour: x_ju(e^j,x_N,A_n) given the coefficients of the confidence matrix B. Thus, the average payoff of the agent with opinion m_i is given by
-10pt
∑_j=1^2 x_ju(e^j,x_N,A_n)b_ij.
Thus, the expected playoff of an agent with an opinion of m_i can be written as
y_i∑_j=1^2 x_ju(e^j,x_N,A_n)b_ij.
Consequently, the imitation protocol is:
p_ij = [y_j∑_l=1^2 x_lu(e^l,x_N,A_n)b_jl - y_i∑_l=1^2 x_lu(e^l,x_N,A_n)b_il]_0^1.
Thus, we get the dynamics, which describes the popularity of the opinions in the population based on the mean dynamics (<ref>) and pairwise imitation protocol:
ẏ =(1-y)p_21-yp_12.
It is assumed that the state of the population depends on the popularity of opinions. Under this assumption, the first player's payoff matrix can be rewritten in the form A_y = yA_1 + (1-y)A_0, which is obtained from the matrix (<ref>) if the parameter n representing the state of the environment is replaced by the fraction of players y with an opinion m_1.
Thus, an evolutionary game with environment-opinion feedback can be represented as
{ẋ =x(1-x)(u(e^1,x_N,A_y) - u(e^2,x_N,A_y)),
ṅ =n(1-n)(θ x + ψ (1-x)),
ẏ =(1-y)p_21-yp_12.
.
§ EXAMPLE 1. HAWK-DOVE GAME
As mentioned in the previous chapter, random pairwise encounters of population agents are described by a bimatrix game of two individuals. In the numerical experiments, bimatrix games with a known structure were considered. For these games, Nash equilibrium positions are obtained theoretically. This allows us to estimate the effect of environmental feedback and the opinions of the agents on the population.
In the current experiment, the Hawk-Dove game is chosen as the base game describing the interaction of the agents. The parameter v – is the value of the resource, the parameter c – is the cost of resources.
Since the model takes into account changes in the population state as a function of agents' opinions, according to the formula (<ref>), we need to introduce two payoff matrices
A_0 =
[ v_0-c_0/2 v_0; 0 v_0/2 ],
A_1 =
[ v_1-c_1/2 v_1; 0 v_1/2 ].
The Hawk-Dove game is characterized by three Nash equilibria: two asymmetric equilibria in the pure strategies (e^1,e^2), (e^2,e^1) and one symmetric equilibrium in the mixed strategies (x_N,x_N), where x_N=(v/c, 1 - v/c) <cit.>. In this case the average population payoff is u(x_N,x_N) = v/2-v^2/2c.
In the numerical experiment, the following parameters are used for the system (<ref>):
-20pt
Figure <ref> shows the behaviour of the population as a function of the distribution of agents by opinion. It is observed that for initial values of the proportion of population agents holding opinion m_1, less than 0.5, the population reaches a stationary position where x_N =(0.33; 0.67) (Figure <ref>a)). On the other hand, for initial values of the proportion of population agents holding opinion m_1 of at least 0.5, the system reaches a stationary position where x_N = (0.7; 0.3) (Figure <ref>b)).
§ EXAMPLE 2. THE PRISONER'S DILEMMA
In the current experiment, the Prisoner's Dilemma in its economic interpretation is chosen as the base game describing the interaction of agents. In this game, the first strategy corresponds to the player's choice to cooperate and the second to the choice to defend.
Since the change in the state of the population as a function of the agents' opinions is considered in the model according to the formula (<ref>), it is necessary to introduce two playoff matrices
A_0 =
[ 3.5 1; 2 0.75 ],
A_1 =
[ 4 1; 4.5 1.25 ],
for whose elements the relations (<ref>) are true.
a_11^0 > a_21^0 , a_12^0 > a_22^0,
a_11^1 < a_21^1, a_12^1 < a_22^1.
The Prisoner's Dilemma game is characterized by the existence of a single Nash equilibrium state, and given the relations (<ref>), for the game given by the matrix A_1(A_0), this state is (e^2,e^2)((e^1,e^1)).
In the current numerical experiment, the parameters of the system (<ref>) take on values:
-20pt
As can be seen from the graph in Figure <ref>a), at the initial time moment, the proportion of cooperating agents decreases to zero as the environment begins to enrich. However, as the proportion of defending agents in the population increases, the environmental resources decrease to zero. After several oscillations, the system reaches an equilibrium state (e^1,e^1), all players have the opinion m_2, the environment is replenished, i.e. n=1.
Through a series of numerical experiments, it was found that the environment and the opinions of the agents have a significant impact on the stationary position of the population. In most cases, a change in the initial value of the population parameter, the environment, or the popularity of the opinions causes a change in the stationary position that the system reaches. The choice of the confidence matrix also plays an important role in the results of the simulation of changes in the population depending on the environment and the opinions of the agents.
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|
http://arxiv.org/abs/2307.04437v2 | 20230710092701 | HORTENSIA, a program package for the simulation of nonadiabatic autoionization dynamics in molecules | [
"Kevin Issler",
"Roland Mitrić",
"Jens Petersen"
] | physics.chem-ph | [
"physics.chem-ph",
"physics.comp-ph"
] |
AIP/123-QED
HORTENSIA]HORTENSIA, a program package for the simulation of nonadiabatic autoionization dynamics in molecules
Julius-Maximilians-Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
[email protected]
Julius-Maximilians-Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
[email protected]
Julius-Maximilians-Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
We present a program package for the simulation of ultrafast vibration-induced autoionization dynamics in molecular anions in the manifold of the adiabatic anionic states and the discretized ionization continuum. This program, called HORTENSIA (Hopping real-time trajectories for electron-ejection by nonadiabatic self-ionization in anions), is based on the nonadiabatic surface-hopping methodology, wherein nuclei are propagated as an ensemble along classical trajectories in the quantum-mechanical potential created by the electronic density of the molecular system. The electronic Schrödinger equation is numerically integrated along the trajectory, providing the time evolution of electronic state coefficients, from which switching probabilities into discrete electronic states are determined. In the case of a discretized continuum state, this hopping event is interpreted as the ejection on an electron. The derived diabatic and nonadiabatic couplings in the time-dependent electronic Schrödinger equation are calculated from anionic and neutral wavefunctions obtained from quantum chemical calculations with commercially available program packages interfaced with our program.
Based on this methodology, we demonstrate the simulation of autoionization electron kinetic energy spectra that are both time- and angle-resolved. In addition, the program yields data that can be interpreted easily with respect to geometric characteristics such as bonding distances and angles, which facilitates the detection of molecular configurations important for the autoionization process.
Moreover, useful extensions are included, namely generation tools for initial conditions and input files as well as for the evaluation of output files both through console commands and a graphical user interface.
[
Jens Petersen
August 12, 2023
===================
For submission:
Repository link: <https://github.com/mitric-lab/HORTENSIA_LATEST.git>
Licensing: MIT
Language: Python ≥ 3.8
§ INTRODUCTION
After generation of a temporary molecular anion through electron attachment, there are three possible competing relaxation mechanisms.<cit.>
These are a) radiative deactivation, assuming that there is a lower-lying anion state that is stable with respect to ionization, b) dissociative electron attachment, in which the captured electron induces geometric change in the molecule resulting in fragmentation into more stable products, a neutral and an anionic subsystem.
And lastly, c) autoionization, in which after a finite period of time the metastable state decays via electron ejection.
The process of dissociative electron attachment is observed for example in DNA, where capture of low-energy electrons leads to single and double strand breaks<cit.>, or in a variety of substances in nanoscale thin films<cit.>.
Prominent examples for autoionization include excited dipole- and quadrupole-bound anions with binding energies slightly below the ionization threshold<cit.>, intermolecular Coulombic decay at the FADH^- cofactor involved in DNA-photolesion repair<cit.> and autoionization induced by vibrational excitation in organic molecules<cit.>.
Generally the finite lifetime of a metastable state with respect to autoionization can vary strongly from only a few femtoseconds<cit.> up to milliseconds<cit.>.
Recently, several experiments have provided insights into the dynamics of such processes in dipole- and quadrupole-bound organic anions on a (sub-)picosecond timescale.<cit.>
Although the process of autoionization is well-known and -observed experimentally by a multitude of methods, as can be seen in the references given above, the theoretical description of autoionizing systems is challenging<cit.>, especially if one is interested in the mechanistic details of the intricate ultrafast relaxation dynamics.
Autoionization processes can follow different general mechanisms, depending on how energy is redistributed among the system's degrees of freedom. Besides a purely electronic variant, where already the electronic energy of the system lies above the ionization threshold and electron ejection may proceed via tunneling, there is also the possibility of a nonadiabatic mechanism in which rotational or vibrational energy of the nuclei is transformed into the kinetic energy of the ejected electron.
In the following, we focus on the case of vibrational autoionization. This process can thus be viewed as a nonadiabatic transition between a vibrationally excited bound N-electron system and continuum electronic states consisting of an N-1 electron molecular core and a free electron. Early theoretical treatments have focused on the computation of ionization rates<cit.> as well as on establishing propensity rules for the ionization transitions<cit.>. While a full dynamical treatment of vibrational autoionization is highly desirable, an entirely quantum-dynamical approach is computationally prohibitive. As an alternative, a mixed quantum-classical ansatz can be considered, further motivated by the success of this type of methodology in the description of bound-state nonadiabatic processes and the simulation of time-resolved spectroscopic signals.<cit.> Although to date there have been several implementations of mixed quantum-classical dynamics simulations for bound-state problems made publicly available<cit.>, no program addressing the simulation of vibration-induced autoionization processes has been published so far.
Therefore, in this work we present the program package implementing our approach to describe vibrational autoionization through quantum-classical dynamics in the framework of the surface-hopping methodology in the manifold of bound and continuum electronic states as described recently<cit.>.
Therein, nuclear motion is considered classically, while the electronic system is treated quantum-mechanically.
Nonadiabatic transitions between electronic states accompanied by change of the classical vibrational energy of the molecule describe the energy exchange between the two subsystems.
With this program package and the underlying methodology, one is able to gain insight into the geometric and electronic evolution in the course of the autoionization process as well as to calculate the time-, energy- and angle-distribution of the generated free electrons, which serve as experimental observables for monitoring autoionization dynamics.
We illustrate our program on the example of the 2-cyanopyrrolide anion, which bears a dipole-bound excited state slightly below the electron detachment threshold while the vibrationally excited states are metastable and decay via autoionization.<cit.>
In the following section a brief theoretical description of the method is given. In section <ref> an overview of the actual implementation is provided. The subsequent section <ref> details performance-related issues, namely quality of approximations in the theory and runtime and memory optimization within the program, as well as a dynamics simulation example for the 2-cyanopyrrolide anion. Finally in section <ref> a conclusion and outlook are given.
§ THEORY
Our methodological framework is based on the surface-hopping procedure as proposed by Tully<cit.>, in which the coupled electron-nuclear dynamics of molecular systems is approached in a quantum-classical fashion.
Specifically, the nuclei are propagated classically according to Newton's equations of motion,
MR̈(t)
=
𝐅_i(𝐑[t])
≡
-∇_R E_i(R[t]),
where the force 𝐅_i(𝐑[t]) is obtained as the negative gradient of the electronic potential energy surface (PES) E_i(R[t]). In the above equation, M denotes a diagonal matrix containing the nuclear masses.
For an ensemble of initial conditions, this leads to trajectories R(t) moving on the given PES.
Simultaneously, the electronic time-dependent Schrödinger equation
iħΨ̇(r;R[t])
=
Ĥ_elΨ(r;R[t])
,
with the electronic Hamiltonian Ĥ_el is solved.
The electronic wavefunction can be expanded into a set of orthonormal basis states, which in the case of autoionization includes bound states Φ_m' (denoted with a primed index) as well as continuum states Φ̃_m” (denoted with a double-primed index):
Ψ(r,R[t],t)
= ∑_m' c_m'(t) Φ_m'(r,R[t])
+
∑_m”∫ d^3k c̃_m”(k,t) Φ̃_m”(k,r,R[t]),
where k denotes the continuously varying wave vector of the free electron, while m” is the quantum number of the remaining neutral state.
We assume the wavefunctions Φ_m' and Φ̃_m” to be single Slater determinants (ground state) or an expansion of singly excited Slater determinants (excited state).
In the frame of the presented methodology we discretize the continuum states, leading to
∫ d^3k c̃_m”(k,t)
Φ̃_m”(k,r,R[t])
≈∑_i
(Δ V_k)^1/2c̃_m”(k_i,t)
(Δ V_k)^1/2Φ̃_m”(k_i,r,R[t])
≈∑_i
c_m”(k_i,t)
Φ_m”(k_i,r,R[t]),
where Δ V_k denotes the volume element in k-space and the discretized and continuum state expansion coefficients are related according to c_m”(k_i,t)=(Δ V_k)^1/2c̃_m”(k_i,t). The actual determination of the wave vectors and the implementation of the discretization procedure are explained in detail in the next chapter.
Insertion of Eq. (<ref>) into the time-dependent Schrödinger equation (<ref>), multiplication from the left by an eigenstate ⟨Φ_n| and evaluation of the arising terms leads to a set of coupled differential equations for the electronic state coefficients c_n:
ċ_n(t)
=
∑_j
[
-i/ħ H_nm(R[t]) - D_nm (R[t])
]
c_m(t),
with the matrix elements of the electronic Hamiltonian H_nm = ⟨Φ_n | H_el | Φ_m|$⟩ and the nonadiabatic couplingsD_nm = ⟨Φ_n | Φ̇_m|=⟩ Ṙ·⟨Φ_n | ∇_R | Φ_m|$⟩.
These can be divided into separate expressions for the bound and continuum states, resulting in the diabatic and nonadiabatic couplings between two bound anion states,
H_n'm' =
⟨Φ_n' | Ĥ | Φ_m'|⟩hij
D_n'm' =
⟨Φ_n' | Φ̇_m'|,⟩
and between a bound and a discretized continuum state,
H_n”m'(k_i)
=
(Δ V_k)^1/2⟨Φ̃_n”(k_i) | Ĥ | Φ_m'|⟩hik
D_n”m'(k_i)
=
⟨Φ_n”(k_i) | Φ̇_m'|=⟩
(Δ V_k)^1/2⟨Φ̃_n”(k_i) | Φ̇_m'|.⟩
In the above equations, the approximation to neglect the coupling terms between the continuum states has been introduced.
The discretized continuum states consist of an antisymmetrized product of a bound N-1 electron neutral state and a molecular scattering state of the free electron
Φ̃_n”(k_i)
=
A(
Φ^(n)_n”·ψ(k_i)
).
The simplest approximation to the free electron states in the presence of a neutral molecular core are plane waves
ψ(k_i)≈ Ne^ik_i·r
with a normalization constant N = (2π)^-3/2 to satisfy the orthonormality demanded in Eq. (<ref>).
Since this function would be completely independent on the electronic and nuclear configuration of the molecular core, which is a strong simplification, the plane waves are orthogonalized with respect to the anion's molecular orbitals (MOs) ϕ_m to include (at least to a certain degree) dependence on the molecular structure according to
ψ̃(k_i)
=
(2π)^-3/2 N_ortho(
e^ik_i·r
-
∑_m^occ⟨ϕ_m | e^ik_i·r|ϕ⟩_m
)
=
N_ortho(
ψ(k_i)
-
∑_m^occ⟨ϕ_m | ψ(k_i)| ⟩ϕ_m
),
with the normalization constant
N_ortho
=
(
1
-
∑_m^occ|
⟨ϕ_m | ψ(k_i)|⟩|^2
)^-1/2
arising from the orthogonalization.
Notably, the summation over m includes the occupied MOs in all 'relevant' Slater determinants of all considered electronic states, that is, we considered all determinants which are needed to sufficiently represent the ground state and full CIS wavefunction of the excited state.
Beginning from the highest contribution to a wavefunction, determinants are included until a specific percentage or a user-adjusted maximum number of configurations per electronic state is reached (95 % / 10 configurations in the case of vinylidene<cit.>).
Considering for now the special case where only the anion's ground state is included, the used MOs are simply the energetically lowest ones up to the highest-occupied molecular orbital (HOMO).
The overlap integral between a plane wave and an MO present in Eq. (<ref>), ⟨ϕ_m | ψ(k_i)|$⟩, can be computed analytically by expanding the MO into the Gaussian atomic orbital (AO) basis, with the integral involving a single AO|ν⟩given by
⟨ν | ψ(k)|=⟩
(2π)^-3/2∫ d^3𝐫 e^ik·rφ_ν(r)
=
(2α_ν)^-3/2exp(ik·A_ν -k^2/4α_ν)
×∏_j=x,y,z
(-i√(4α_ν))^-n_ν,j
H_n_ν,j(
k_j/√(4α_ν))
,
where theH_n_ν,jare the Hermite polynomials of ordern_ν,j.
§.§ Electronic coupling terms
There are anionic systems, for example the vinylidene anion<cit.>, that do not support a bound excited state, in which case the consideration of only the ground state and the continuum in the process of autoionization is sufficient.
Besides that, for example in molecules exhibiting dipole-bound excited states <cit.>, several bound anionic states and the interaction among them are relevant as well.
Nonetheless, to keep the formalism concise, if not noted otherwise we discuss in the following the electronic coupling terms for the special case of both anion and neutral molecule being in their respective electronic ground states, which in turn are represented by a single Slater determinant.
The generalization to excited states and/or multideterminantal wavefunctions is straightforward.<cit.>
We denote the bound anionic ground state wavefunction by|Φ_0⟩and the continuum wavefunctions by|Φ_i⟩, the latter being constructed as an antisymmetrized product of the neutral ground state and a free electron state function with wave vectork_i, similar to Eq. (<ref>).
§.§.§ Diabatic couplings
In the case of two adiabatic bound anion states, the coupling matrix elementsH_n'm'given in Eq. (<ref>) yield zero for alln' ≠m'since these states are orthonormal eigenstates of the electronic Hamiltonian.
On the other hand, since in our methodology the bound and continuum state wavefunctions are constructed using separate quantum-chemical calculations for the anion and neutral, and the free electron wavefunction is taken as a plane wave, the continuum state functions are crude approximations to the actual adiabatic eigenfunctions of the electronic Hamiltonian for theN-electron system and therefore, diabatic couplings between the bound and continuum electronic states arise.
As elaborated in detail in Ref. aid, according to Eq. (<ref>) and definingV_i0^dia(k_i)as
H_i0(k_i)
≡⟨Φ_i | Ĥ | Φ_0|≡⟩(Δ V_k)^1/2
V^dia_i0(k_i),
the diabatic coupling between a bound and a continuum state can be written in terms of the AO basis as
V^dia_i0(k_i)
= ∑_λμν[
A_λμν(
⟨𝐤_i λ || μν|-⟩∑_σ
B_σ⟨σλ || μν|⟩) +
A̅_λμν(
⟨𝐤_i λ | μν|-⟩∑_σ
B_σ⟨σλ | μν|⟩)
].
In this formula the Greek letters denote the AO basis functions,⟨𝐤_i λ| μν|$⟩ is an electron-electron repulsion integral and ⟨𝐤_i λ || μν|=⟩⟨k_i λ | μν|-⟩⟨k_i λ | νμ|$⟩ its antisymmetrized variant.
The prefactorsA_λμν,A̅_λμνandB_σcomprise AO expansion coefficients and overlap integrals and are defined as follows (assuming that the extra electron of the anion hasαspin):
A_λμν
= ∑_n^occ,α∑_q,p<q^occ,α
(-1)^n+p+q-1det 𝐒_in,pq
×(
c_λ^(n)
-
∑_u^occ,α
c_λ^(u)
S_nu)
c_μ^(p) c_ν^(q)
A̅_λμν
= ∑_n̅^occ,β∑_p^occ,α∑_q̅^occ,β
(-1)^n̅+p+q̅-1det 𝐒_in̅,pq̅
×(
c_λ^(n̅)
-
∑_u̅^occ,β
c_λ^(u̅)
S_n̅u̅)
c_μ^(p) c_ν^(q̅)
B_σ
= ∑_r^occ,α∑_ρ
c_σ^(r)
c_ρ^(r)⟨k_i | ρ|,⟩
where the indices (including their variants with an overbar)p,q,rrefer to anion MOs,n,uto neutral MOs, anddet 𝐒_in,pqdenotes the minor determinant of the overlap matrix between continuum and bound state orbitals where the rows of the free electron orbitalψ̃(𝐤_i)and neutral orbitalχ_nas well as the columns of anion orbitalsϕ_pandϕ_qhave been deleted.
For the full derivation of these equations the reader is referred to Ref. aid.
§.§.§ Nonadiabatic couplings
The nonadiabatic coupling terms as defined in Eqs. (<ref>) and (<ref>) are calculated using the finite-difference approximation for the time derivative, which leads to
D_i0(t)
=
⟨Φ_i(t) | d/dtΦ_0(t)|
⟩ ≈1/2Δ t(
⟨Φ_i(t-Δ t) | Φ_0(t)|-⟩⟨Φ_i(t) | Φ_0(t-Δ t)|⟩)
In the case of two anionic bound states, these terms are evaluated according to Refs. mitric2008, werner2008, werner2010.
One can simplify the arising terms by integrating over all but one electron coordinate. For the first term of Eq. (<ref>) this yields
⟨Φ_i(t') | Φ_0(t)|=⟩
N^-1/2⟨ψ̃(k_i,t') | ψ^D(t',t)|,⟩
where we have abbreviatedt'=t-Δtand have defined the one-electron functionψ^D(t',t), which is an analog to a molecular Dyson orbital with theN- andN-1- wavefunctions taken at different time steps and geometries.
Using Eqs. (<ref>) and (<ref>) the resulting nonadiabatic coupling terms read
D_i0(k_i,t)
=
(Δ V_k)^1/2 N_ortho/2 √(N)Δ t[
⟨ψ(k_i) | ψ^D(t',t)|-⟩⟨ψ(k_i) | ψ^D(t,t')| ⟩-
∑_n
⟨ψ(k_i) | ϕ_n(t)| ⟩⟨ϕ_n(t') | ψ^D(t',t)|⟩
+
∑_n
⟨ψ(k_i) | ϕ_n(t)| ⟩⟨ϕ_n(t) | ψ^D(t,t')|⟩].
§.§ Adiabatic ionization and electronic decay
The main focus of the above presented methodology lies on describing the nonadiabatic process of vibrational autoionization. However, in the course of the molecule's dynamical evolution instances can occur where the occupied anionic state becomes unbound as the result of changes in nuclear geometry.
In this case, ionization is possible as an exclusively adiabatic electronic process without coupling to the nuclear motion.
This process can be included approximately in our method by simulating the temporal spread of the ejected electron as a wavepacket evolving freely in space. As a quantitative measure, the electronic spatial extent, i.e., the expectation value of𝐫̂^2, is calculated as a function of time.
Specifically, once a time step is reached where the VDE has become negative, the highest-occupied orbital of the last bound geometry,ϕ(r, t_0), is used as the initial free electronic wavepacket.
In the case where one only considers the anionic ground state, this corresponds to the HOMO.
If also an excited state is involved, natural transition orbitals (NTOs)<cit.> are calculated and the highest-occupied and lowest-unoccupied NTO (HONTO and LUNTO) are used for the anionic ground and excited state, respectively.
Such an electronic wavepacket is then propagated in time and its spatial extent is evaluated according to
⟨𝐫̂^2|(⟩t)
=
⟨ϕ(𝐫,t) |𝐫̂^2 |ϕ(𝐫,t)|⟩
=
∑_μν c_μ c_ν⟨φ_μ(𝐫,t) | 𝐫̂^2 | φ_ν (𝐫,t)|.⟩
Hereφ_μ, νdenote the Gaussian atomic basis functions freely propagated in time:
φ_μ(𝐫,t) = ∫ d^3𝐫' K(𝐫,𝐫',t,0) φ_μ (𝐫',0)
with the free electron propagator
K(𝐫,𝐫',t,0)
=
𝐫 | e^-i𝐩̂^2 t/2m_eħ|𝐫'.
Using Cartesian Gaussian basis functions ofs,panddtype one obtains the following analytic expression for the electronic wavepacket:
φ_μ(𝐫,t)
=
N_l_xl_yl_ze^-α/1+iβ tr^2[
-Λiβ t/2α
(1+iβ t)^-5/2
+.
.
(1+iβ t)^-3/2 - ∑_j l_j∏_j=x,y,z (r_j - A_j)^l_j],
whereAis the spatial center of the respective basis function,l_idenotes the angular momentum quantum number for thei'th spatial direction andΛis a constant that is unity if one of thel_i = 2and zero if alll_i<2.
The AO integrals in Eq. (<ref>) are calculated with an implementation of the McMurchie-Davidson scheme<cit.>.
To relate the spatial extent in a simple way to the lifetime of the unbound state, an auxiliary spherically symmetric electron distribution is considered which within the initially determined radiusr_0=√(⟨r^2|(⟩t_0))contains a probability of 99%. Subsequently, with⟨r^2|$⟩ increasing with time, the probability within r_0 decreases, giving rise to a population decay curve which can be related to a time constant τ.
The latter is incorporated into the propagation of the electronic wavefunction given by Eq. (<ref>) by adding an imaginary component to the electronic state energy,
E^(a)→
E^(a)-iħ/2τ,
which leads to an exponential population decay due to adiabatic ionization in regions where the VDE is negative for the given electronic state.
§.§ Surface-hopping procedure
Solution of the set of Eqs. (<ref>) along a nuclear trajectory yields the time-dependent electronic state coefficients c_n(t).
Within the surface-hopping methodology, a switch from the occupied bound electronic state n to any other state m is determined by the hopping probability depending on the electronic state populations ρ_nn = |c_n|^2, which is
P_n→ m
=
-ρ̇_nn/ρ_nnρ̇_mm/∑_k ρ̇_kkΔ t
for ρ̇_nn < 0 and ρ̇_mm > 0 and zero in any other instance. In the above expression, the sum over k includes all states with ρ̇_kk>0.
In case a surface hop occurs, to ensure energy conservation the nuclear velocities are rescaled such that for kinetic energies T and electronic potential energies E_n of anion (a) and neutral (n) the following conditions are fulfilled:
T'^(a)
=
T^(a) +
E_n^(a) -
E_m^(a)
for a hop between anionic bound states and
T'^(n)
=
E_n^(a) +
T^(a) -
E_m^(n) -
E_el(k_i)
for a hop into the continuum (i.e. autoionization).
For a more detailed description of the hopping procedure the reader is referred to Ref. domckebook.
§ PROGRAM IMPLEMENTATION
In the following chapter a detailed account of how the theory is actually implemented in the program package will be provided.
For an easier understanding, in Fig. <ref> the program flow is displayed schematically, with a color code indicating the module handling the respective task.
Starting from the generation of an ensemble of nuclear coordinates R(t) and velocities Ṙ(t) at the time t = t_initial using the module in the folder (red), a first quantum-chemical calculation is performed by an external quantum-chemistry program - to date these include Gaussian09/Gaussian16 <cit.> and QChem <cit.> (blue) - which yields the forces from which the accelerations R̈(t) of the nuclei are computed.
The nuclei are then propagated by integration of Newton's equations of motion for one nuclear time step using the module (orange).
With the new nuclear coordinates R(t + Δ t), a new set of quantum-chemical calculations can be performed, yielding the new energy gradients necessary for the evaluation of the velocities Ṙ(t + Δ t).
With the quantum-chemical calculations at t and t + Δ t, one is now able to construct the electronic continuum states as well as the coupling matrices of the diabatic and nonadiabatic couplings using the module (green).
From this point, the electronic state coefficients c(t) are propagated in parallel to the nuclear dynamics by integrating the electronic Schrödinger equation, yielding c(t + Δ t).
These are utilized to compute hopping probabilities from the occupied bound state to all other (bound and continuum) states.
The switching between the states is induced stochastically according to the respective hopping probabilities given in Eq. (<ref>).
After writing the results into the various output files time is shifted to t = t + Δ t, thereby completing one time step.
To make this initial overview more specific, in the following the underlying algorithms are explained in more detail.
§.§ Electronic structure calculation
All electronic structure and energy gradient calculations can be performed by using any Kohn-Sham (TD)-DFT level of theory provided within the Gaussian09, Gaussian16 or QChem program packages.
The AO basis set needs to be defined explicitly in a separate input file, thus also allowing for additional augmentation of basis sets, which is of utmost importance when describing molecular anions.<cit.>
The and modules provide an interface to the external programs by creating input files and calling the respective programs. The and modules contain classes that parse the external output files and organize the data into the form needed in the program.
§.§ Generation of initial conditions
The initial nuclear coordinates and velocities are determined by stochastic sampling of an appropriate probability distribution function for the harmonic normal modes of the system.
These can be computed from the electronic Hessian matrix at an optimized geometry of the studied molecule.
For molecules in the vibrational ground state as well as for a thermal ensemble of molecules, the Wigner function
ρ_W({Q_i,P_i})=1/(πħ)^N∏_i=1^N α_i(T) exp(-α_i(T)/ħω_i(P_i^2+ω_i^2Q_i^2))
with
α_i(T)
=
tanh(ħω_i/2k_BT)
is employed, where {Q_i,P_i} denote the normal coordinates and momenta, ω_i is the angular frequency of normal mode ν_i and T the thermodynamic temperature.
Besides these cases, in experiments investigating vibration-induced autoionization another type of initial conditions is often important in which one or more normal vibrations of the system are excited by laser irradiation.
In principle, the respective initial conditions could be also generated by using a Wigner function. However, Wigner functions for excited vibrational states can assume negative values and can thus not be directly identified with a probability distribution.
A possible approach might be to regard the positive and negative parts of the Wigner function separately as probability distributions and to run a "positive" and a "negative" ensemble of initial conditions, the final properties of the system then being obtained by appropriate averaging.
As a more efficient alternative, which gets on with only one single ensemble, we employ a positive definite probability distribution constructed from the excited-vibrational state wavefunctions in position and momentum space,
ρ^(i)_υ(Q_i,P_i)=|χ^(i)_υ(Q_i)|^2|χ̃^(i)_υ(P_i)|^2,
where χ^(i)_υ(Q_i) and χ̃^(i)_υ(P_i) are the harmonic oscillator wavefunctions for quantum state υ of normal mode ν_i in position and momentum space, respectively.
§.§ Nuclear dynamics
Given Newton's equations of motion (<ref>), the nuclei are propagated by numerical solution using the velocity Verlet algorithm <cit.> for a user-defined time step.
Within this algorithm, the nuclear coordinates at t+Δ t are obtained from a Taylor series expansion around the coordinates at t:
R(t + Δ t)
≈R(t) +
Ṙ(t)Δ t +
1/2 M^-1F(t) Δ t^2,
where in the last term the acceleration has been formulated using the force F given by the electronic potential energy gradient (cf. Eq. (<ref>)).
With the new nuclear coordinates, the force at t + Δ t can be evaluated, giving rise to the new nuclear velocities
Ṙ(t + Δ t)
=
Ṙ(t) +
Δ t/2 M^-1[
F(t) + F(t + Δ t)
]
.
Due to the approximative nature of the algorithm above and the accuracy of the calculated energy gradients, it is possible that the velocities develop small overall translational or rotational components although the initial conditions were determined with these degrees of freedom set at rest.
These numerical inaccuracies are detected, in the case of translational velocity by the shift of the center of mass away from the origin of the coordinate system, in the case of rotation by the calculation of the angular velocity according to
ω_rot = I^ -1L
with the moment of inertia I and the angular momentum L.
The translational and rotational portions of the nuclear velocities are then subtracted from the total velocity and the remaining vibrational part is rescaled to ensure energy conservation.
After each nuclear dynamics step, the new nuclear coordinates and velocities are written into separate output files, the coordinates in a format of consecutive xyz files which can be visualized easily by external software (for example with the VMD program package <cit.>, which is warmly recommended).
§.§ Electronic dynamics
Since the evaluation of electronic coupling terms in Eq. (<ref>) is, apart from the external quantum-chemistry calculations, the computationally most expensive step in the dynamics, several approximations need to be implemented, which will be discussed in the following
§.§.§ Calculation of coupling terms
Before calculating the coupling terms, the discretization procedure for the generation of wave vectors needed to construct the continuum state wavefunctions will be discussed.
To uniformly discretize angular orientation and kinetic energy of ejected electrons, it is natural to discretize angular and energetic distribution separately.
Since the kinetic energy of a plane wave is
E_kin(k_i) = ħ^2 |k_i|^2/2 m_e
and therefore proportional to the length of the wave vector squared, this length is discretized such that the desired energy range is covered evenly.
For a given energy, the vector orientations are approximately evenly distributed according to the Fibonacci sphere algorithm <cit.>.
The volume elements Δ V_k needed for calculating the bound-continuum couplings in Eqs. (<ref>) and (<ref>) are constructed as the difference of spherical caps around the corresponding wave vectors with the base diameter as an average over the six nearest points on the sphere surrounding the vector.
In the diabatic coupling terms in the AO basis (Eq. (<ref>)) two types of four-center integrals are present: (i) such involving four Gaussian-type atomic orbitals (GTOs), ⟨σλ | μν|$⟩. These are evaluated by using the library <cit.> within the PySCF program package <cit.>.
(ii) integrals involving a plane wave of wave vector𝐤_iand three GTOs,⟨𝐤_i λ| μν|$⟩.
These terms can in principle be calculated analytically as outlined, e.g., in Ref. colle1987, but this is computationally unfeasible for the present purpose since an immense number of plane waves has to be included for a proper discretization of the ionization continuum. Instead, the plane waves are approximated by their Taylor expansion around the center of basis function |μ⟩, R_μ.
As will be discussed in the Performance Section later on, for sufficient accuracy in the approximation it is necessary to include not only the zero'th order term (assuming the plane wave to be constant in the vicinity of the molecule), but also the first-order term, resulting in the approximation
e^i k·r =
e^i k·R_μe^i k· (r - R_μ)
≈e^i k·R_μ[
1 + i k· (r - R_μ)
].
This leads to two terms for the two-electron integrals as follows:
⟨𝐤_i λ | μν|≈⟩e^i k·R_μ[
⟨λ | μν|+⟩
i k⟨λ | μ̃ν|⟩].
In the above expression, |μ̃⟩ is an AO basis function with an angular momentum quantum number by one higher than |μ⟩ while having the same Gaussian exponent.
This heavily reduces the amount of two-electron integrals to be computed from n_AO^3 n_PW to n_AO^2 [n_AO + n'_AO], with n_AO being the total number of AO basis functions, n'_AO the total number of basis functions with increased quantum number and n_PW the total number of plane waves. For instance, in the case of vinylidene in Ref. aid, this amounts to a reduction by a factor of ∼30000.
These terms are again evaluated using the PySCF module.
The prefactors A, A̅ and B present in Eq. (<ref>) are straightforwardly implemented in Python according to Eqs. (<ref>), (<ref>) and (<ref>).
Evaluation of the Dyson orbitals needed for the calculation of the nonadiabatic couplings is implemented as described before in Ref. humeniuk2013 for arbitrary basis sets for the anion and the neutral molecule.
After construction of the Dyson orbitals from all bound anionic states to the neutral ground state the nonadiabatic coupling terms are then calculated according to Eq. (<ref>). To ensure that the wavefunctions of bound states do not switch their arbitrary signs (which can happen, since the external quantum-chemistry calculations are independent of each other), the overlap of electronic wavefunctions of all bound states are tracked throughout the trajectories and accounted for in all formulae involving the respective states.
§.§.§ Calculation of electronic state coefficients
The electronic degrees of freedom are propagated by solving the time-dependent Schrödinger equation (<ref>) in the manifold of all considered bound anion and continuum electronic states using Adams' method as implemented in the class of Python's module <cit.> with a user-defined integration time step. For increased computational stability the equations are beforehand transformed into the interaction picture, introducing the new electronic state coefficients
a_n(t)
=
c_n(t) e^i/ħ H_nn t.
Inserting this into Eq. (<ref>) leads to the actually implemented electronic equation of motion
ȧ_n(t)
=
∑_m
[
-i/ħH̃_nm - D_nm]
a_m(t)
e^-i/ħ (H_mm - H_nn) t
where H̃_nm denotes the Hamiltonian matrix of the system with zeros on the diagonal.
§.§.§ Hopping procedure
Hopping probabilities are directly evaluated according to Eq. (<ref>) from the state coefficients: A random number between 0 and 1 is generated using the function in the module and hopping is conducted accordingly.
Once a trajectory hops into a continuum state, it could in principle be straightforwardly continued on the potential energy surface of the neutral molecule.
Although this can be quite insightful if one is interested in the subsequent geometric changes of the ionized system, we follow a different approach and stop the trajectories after electron detachment since our focus is set on the actual autoionization process.
This allows us to implement a modification of the surface-hopping procedure that leads to a great improvement of the hopping statistics. The idea is to divide a single trajectory into 'sub-trajectories', i.e. to evaluate if a trajectory hops a number n_subtraj of times (see Fig. <ref>).
Every time a sub-trajectory hops into the continuum, n_subtraj is reduced by one and once it reaches zero, the underlying nuclear dynamics is stopped.
It has to be noted that this procedure is only followed for hops into the continuum, while for hops between bound anionic states only a single hopping event per trajectory and time step is possible due to the need to continue the nuclear dynamics on an unambiguously determined potential energy surface.
§.§ Graphical user interface
Our program package comes with a graphical user interface (GUI) for the input generation as well as an analysis tool for trajectories.
An example of the former is displayed in Fig. <ref>.
In the input generator, which is started with
[language=bash]
hortensia –gui
in addition to all relevant settings for the actual simulation, the user may find options for the generation of a complete folder structure for the trajectories as well as bash submit scripts to be used with the Slurm Workload Manager<cit.>.
Furthermore, the above mentioned Wigner ensemble scripts can be used and initial conditions can be generated. Therefore it is highly recommended to use the GUI feature.
Additionally, through the command
hortensia –analysis
one can open the analysis tool which is able to read output files and visualize them in a sub-window using the program package <cit.>.
§.§ Installation
The most convenient way to install the program package is downloading or cloning the https://github.com/mitric-lab/HORTENSIA_LATEST.gitrepository on our Github page<cit.>. In the main folder, execute
[language=bash]
python cysetup.py build_ext –inplace pip install .
to first compile the Cython modules and then install the program. The program package requires (and will automatically pip install)
*
* - for faster summation of large arrays, mainly in the calculation of the two-center integrals in Eqs. (<ref>) and (<ref>)
* - mainly in the integration of the electronic Schrödinger equation as outlined in subsection <ref>
* - for the calculation of the two-electron integrals in Eqs. (<ref>) and (<ref>)
* - for the parallelization of diabatic couplings
* - for the plots in the sub-window of the analysis tool described before
and all dependencies thereof.
Using the command
[language=bash]
pip uninstall hortensia_latest
will uninstall the program package.
§ DISCUSSION
In this section we will quantify aspects of the program related to overall performance.
This includes the quality of approximations within the methodology as well as optimization of time consumption and computational resources.
Moreover the exemplary autoionization dynamics of the 2-cyanopyrrolide anion is discussed.
§.§ Accuracy of k-vector discretization and integral approximations
The accuracy of the Fibonacci sphere algorithm for angular discretization in k-space is illustrated in Fig. <ref> by the covered surface area of a unit sphere using a given number of distributed points.
The total surface area (orange graph) is presented with the relative error|A_fib-A_sphere|/A_sphere(green graph) to the exact surface area4π≈ 12.566(blue line).
The approximated area rapidly converges to a value of∼12.243, which corresponds to a relative error of∼2.575 %.
Since in the coverage of k-vector lengths no additional approximation is introduced and for their respective volume elements the k-space is divided energetically evenly (thus covered exactly with respect to vector length), the error in the surface area for specific vector lengths equates to the overall error of the volume elements.
Therefore the sum of these volume elements results in a total volume that deviates by less than 3 % from the actual sphere for arbitrary numbers of vector orientationsn_s ≥ 30and lengthsn_E(giving a total number of wave vectorsn_k = n_E · n_s). |
http://arxiv.org/abs/2307.05868v1 | 20230712014655 | Photon-induced droplet-like bound states in one-dimensional qubit array | [
"J. Talukdar",
"D. Blume"
] | quant-ph | [
"quant-ph",
"cond-mat.other"
] |
Homer L. Dodge Department of Physics and Astronomy,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Center for Quantum Research and Technology,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Homer L. Dodge Department of Physics and Astronomy,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
Center for Quantum Research and Technology,
The University of Oklahoma,
440 W. Brooks Street,
Norman,
Oklahoma 73019, USA
We consider an array of N_e non-interacting qubits or emitters that are coupled to a one-dimensional cavity array with tunneling energy J and non-linearity of strength U. The number of cavities is assumed to be larger than the number of qubits. Working in the two-excitation manifold, we focus on the bandgap regime where the energy of two excited qubits is off-resonant
with the two-photon bound state band. A two-step adiabatic elimination of the photonic degrees of freedom gives rise to a one-dimensional spin Hamiltonian with effective interactions; specifically, the Hamiltonian features constrained single-qubit hopping and pair hopping interactions not only between nearest neighbors but also between next-to-nearest and next-to-next-to-nearest spins. For a regularly arranged qubit array, we identify parameter combinations for which the system supports novel droplet-like bound states whose characteristics depend critically on the pair hopping. The droplet-like states can be probed dynamically. The bound states identified in our work for off-resonance conditions are distinct from localized hybridized states that emerge for on-resonance conditions.
Photon-induced droplet-like bound states in one-dimensional qubit array
D. Blume
August 12, 2023
=======================================================================
§ INTRODUCTION
Qubits or, more generally, few-level emitters coupled to a cavity array provide a platform with which to investigate fundamental aspects of matter-light interactions. Topics of interest include the generation of photon-mediated entanglement between non-interacting separated qubits <cit.>, of ultrastrong matter-light interactions <cit.>, of broad matter-light hybrid bound states <cit.>, and of effective photon-photon interactions <cit.>. Photonic baths have been realized using nanophotonic wave guides <cit.>, superconducting resonators <cit.>, and plasmonic waveguides <cit.>. Qubit realizations include Rydberg atoms <cit.>, quantum dots <cit.>, and transmon qubits <cit.>.
It was recently shown that the addition of a Kerr-like non-linearity to the tight-binding Hamiltonian, which accounts for the tunnel-coupling of the single-mode cavities, leads to intriguing and qualitatively novel phenomena if the energy of two excited qubits is tuned to be in resonance with the two-photon bound state band that exists due to the Kerr-like non-linearity <cit.>. For two qubits initialized in their excited state, e.g., the non-trivial mode structure of the bath, i.e., the cavity array with non-linearity, was shown to support emission dynamics that ranges from exponential decay to fractional populations to Rabi oscillations <cit.>. For many qubits, supercorrelated radiance was predicted <cit.>. This work instead investigates the off-resonant
or band-gap
regime <cit.>
within the framework of Schrödinger quantum mechanics. To reduce the high-dimensional Hilbert space to a physically intuitive and numerically more tractable model, effective constrained single-qubit and pair hopping interactions are derived through a two-step procedure that adiabatically eliminates single- and two-photon processes. The resulting effective one-dimensional spin Hamiltonian, which lives in the two-excitation manifold (i.e., two flipped spins), is shown to capture the key features of the full Hamiltonian.
The effective constrained single- and two-qubit hopping interactions, which are derived under the assumption that the coupling strength g between an emitter and a cavity is small compared to the tunneling energy J, are directly proportional to g^2 and g^4, respectively. Even though the scaling of the effective interactions with g suggests that the single-qubit hopping dominates over the two-qubit hopping, we identify a parameter regime where the latter, which depends on the non-linearity U, impacts the eigenstate characteristics appreciably. Specifically, the pair hopping interaction favors localization of excited qubits in or near the middle of the
qubit array, giving rise to a new class of droplet-like bound states. These bound states are distinct from two-string bound states that exist, e.g., in the XXX spin Hamiltonian that is solvable via the Bethe ansatz <cit.>. Unlike Hamiltonian that are tractable via the Bethe ansatz, our emergent one-dimensional spin model features non-negligible nearest-neighbor, next-to-nearest-neighbor, and next-to-next-to-nearest-neighbor interactions. It is shown that the radiation dynamics, if initiated from an initial state that contains two qubit excitations but no photons, depends strongly on how the two qubit excitations are distributed among all possible two-qubit excitation eigenkets. A fully symmetric initial state is shown to induce oscillatory dynamics between the
droplet-like
ground
state and a scattering state.
Dependence of the dynamics on the initial state is, of course, a well known phenomenon that has, e.g., been exploited in the study of phase transitions and critical points as well as in sensing applications.
The remainder of this article is organized as follows. Section <ref> introduces the system Hamiltonian and the reduction of the Hilbert space to the qubit degrees of freedom. Section <ref> shows that the effective qubit Hamiltonian supports a new class of liquid-like
or droplet-like bound states. Section <ref> illustrates that these droplet-like states can be probed dynamically. Last, a summary and outlook are provided in Sec. <ref>.
§ DERIVATION OF EFFECTIVE QUBIT HAMILTONIAN
Section <ref> introduces the total Hamiltonian Ĥ of the matter-light hybrid system. Focusing on the band gap regime of the photonic lattice, Sec. <ref> derives the effective spin Hamiltonian Ĥ_spin.
§.§ Total Hamiltonian Ĥ
The total Hamiltonian Ĥ reads
Ĥ = Ĥ_qubit + Ĥ_bath + Ĥ_qubit-bath,
where Ĥ_qubit is the Hamiltonian of the uncoupled qubits, Ĥ_bath the bath Hamiltonian, and Ĥ_qubit-bath the qubit-bath coupling Hamiltonian. The qubit system consists of N_e qubits with a transition energy of ħω_e between the ground state |g⟩_j and the excited state |e⟩_j of the jth qubit (see purple ovals and rectangular box in top-left corner in Fig. <ref>). We are interested in the regime where the qubits form a regularly arranged finite lattice (N_e finite and much greater than 1). The qubit Hamiltonian Ĥ_qubit is given by
Ĥ_qubit=
ħω_e/2∑_j=1^N_e
(σ̂_j^z + Î_j),
where σ̂_j^z=|e⟩_j⟨e|-|g⟩_j⟨g| and Î_j^z=|e⟩_j⟨e|+|g⟩_j⟨g|.
The bath Hamiltonian Ĥ_bath is a one-dimensional tight-binding Hamiltonian with non-linearity U,
Ĥ_bath = ħω_c ∑_n=1^Nâ_n^†â_n
-J ∑_n=1^N(
â_n^†â_n+1 + â_n+1^†â_n)
+U/2∑_n=1^Nâ_n^†â_n^†â_nâ_n,
where â^†_n and â_n, respectively, create and destroy a photon at the nth cavity (blue box in Fig. <ref>). In our calculations, the number of cavities N is chosen such that the results are independent of N; we find that N=501 is sufficiently large for the N_e considered. In Eq. (<ref>), ħω_c is the single-mode photon energy, J (J>0) denotes the tunneling energy of the tunnel coupled cavities, and U is the non-linear onsite interaction. The Kerr-like non-linearity in Eq. (<ref>) corresponds to effectively repulsively interacting photon pairs (U > 0) or effectively attractively interacting photon pairs (U < 0). In our work, we consider a negative U, which gives rise to two-photon bound states ψ_K,b with center-of-mass wave vector K and energy E_K,b, in addition to the two-photon scattering continuum (blue and dark green regions in Fig. <ref>) <cit.>. The black line in Fig. <ref> shows a sketch of a two-photon bound state wave function ψ_K,b that extends over several lattice sites. Accounting for all allowed center-of-mass wave vectors K, the two-photon bound states give rise to an energy band (green and dark green regions in Fig. <ref>). For large values of the onsite interaction strength |U| (|U|/J>4), the two-photon bound state band does not overlap with the two-photon scattering continuum. For |U|/J=1, as considered in this paper, the upper part of the two-photon bound state band overlaps with the lower part of the two-photon scattering continuum (the overlap region is shown in dark green in Fig. <ref>). The difference between the energy 2ħω_e of the two-qubit excited state and the K=0 two-photon bound state energy E_0,b, which coincides with the bottom of the two-photon bound state band, defines the detuning δ,
δ=2ħω_e-E_0,b.
The band gap regime, which is the focus of the present work, is characterized by negative detunings δ.
The qubits are coupled to the photons through the system-bath or qubit-bath Hamiltonian Ĥ_qubit-bath,
Ĥ_qubit-bath=
g ∑_j=1^N_e(
â_n_jσ̂_j^+ + â_n_j^†σ̂_j^-
),
where σ̂_j^+ is the raising operator (σ̂_j^+=|e⟩_j⟨g|) and σ̂_j^- the lowering operator (σ̂_j^-=|g⟩_j⟨e|) of the jth qubit. The label n_j can take any value between 1 and N. In this work, the qubits are assumed to be arranged in a regular pattern with spacing x, where x/a=n_j-n_j-1.
Related works considered regularly placed impurity qubits coupled to an atomic array <cit.>.
Our figures concentrate on x/a=1. For reference, a larger qubit spacing x/a=2 as well as the case where the qubits are all coupled to the same cavity (x/a=0) are discussed in the text. Since the counter rotating terms are excluded in Eq. (<ref>), our treatment is restricted to the weak coupling regime, i.e., g ≪ J. The requirement that single- and two-photon processes are off-resonant [|(ħω_c-2J)-ħω_e|>g and |δ| > g] can, for negative δ as considered in this work, be combined
into
one equation, namely
|U|>4J√((1+g/4J)^2-1).
For fixed U/J, Eq. (<ref>) puts an upper limit on g/J. Conversely, for fixed g/J, Eq. (<ref>) puts a lower limit on |U|/J.
The total Hamiltonian conserves the number of total excitations (sum of qubit and photonic excitations) <cit.>. As a consequence, the Hilbert spaces with 0, 1, 2, … total excitations are decoupled. This work focuses on the two-excitation manifold.
§.§ Effective spin Hamiltonian Ĥ_spin
As mentioned above, we focus on negative detunings such that the energy of two excited qubits is in resonance with the band gap. We find that the band gap physics in the two-excitation manifold is well described by the spin Hamiltonian Ĥ_spin, which is derived by adiabatically eliminating the photon degrees of freedom in a two-step process (see Appendix <ref> for details). We emphasize that the approach taken here is distinct from the master equation approach pursued in Ref. <cit.>. The first step is, in spirit, identical to prior work <cit.>. Neglecting the two-photon scattering continuum and adiabatically eliminating the single-photon states, effective constrained single-qubit hopping interactions of strength W_jl (see Ĥ_single below), effective interactions between states with two and no qubit excitations [F_K,b in Eq. (<ref>)], and effective interactions between two two-photon bound states with wave vector K and K' [G_K,K' in Eq. (<ref>)] arise. While the latter two interactions were discussed in Refs. <cit.>, the effective qubit hopping interaction was not. The reason is that Refs. <cit.> focused on N_e=2 (Ĥ_single vanishes for N_e=2). The hopping Hamiltonian Ĥ_single reads
Ĥ_single=1/2∑_i,j,l=1^N_e(W_jlσ̂^+_iσ̂^+_jσ̂^-_iσ̂^-_l+
W_ilσ̂^+_iσ̂^+_jσ̂^-_lσ̂^-_j ).
Since the triple sum includes terms where two or three of the indices are equal, the order of the operators in Eq. (<ref>) is important. As discussed in more detail below, Ĥ_single describes constrained single-qubit hopping
or constrained flip-flop interactions. We find that the effective interactions G_K,K' contribute negligibly to the band gap physics considered in this
work; thus, they are set to zero.
Calculations that treat the full Hamiltonian Ĥ show that the photonic contribution to the eigenstates is smaller than 10% for the parameter combinations considered in this work. This motivates
our
second approximation, namely, the adiabatic elimination of the states B̂_K^†|g,⋯,g, vac⟩, i.e., basis kets that describe a photon pair with wave vector K, with the qubits in the ground state. Step two yields the effective spin Hamiltonian Ĥ_spin (see Appendix <ref> for details),
Ĥ_spin= Ĥ_single+Ĥ_pair,
where
Ĥ_pair=∑_i=1^N_e-1∑_ j=i+1^N_e∑_l=1^N_e-1∑_h=l+1^N_eY_ij,lhσ̂^+_iσ̂^+_jσ̂^-_lσ̂^-_h.
The effective four-qubit (or two-qubit hopping) interactions Y_ij,lh emerge from the interactions F_K,b (see below). As might be expected naively, W_jl and Y_ij,lh are directly proportional to g^2 and g^4, respectively, since they emerge as a consequence of the first and second adiabatic elimination steps, respectively. The effective spin Hamiltonian Ĥ_spin is independent of the photonic degrees of freedom. The characteristics of the cavity array and the geometric arrangement of the qubits (i.e., the value of x) enter through the interaction strengths W_jl and Y_ij,lh.
We now highlight selected properties of the single- and two-qubit hopping interactions. Figure <ref>(a) illustrates the constrained single-qubit hopping interaction W_jlσ̂^+_iσ̂^+_jσ̂^-_iσ̂^-_l. The term “constrained” is used since the hopping of the excitation from qubit l to qubit j (σ̂^+_jσ̂^-_l piece) depends on the number of excitations at qubit i (σ̂^+_iσ̂^-_i piece; in this example, we assume i≠ j and i≠ l). If qubit i is excited, hopping from qubit l to qubit j occurs with strength W_jl. If, in contrast, qubit i is not excited, hopping from qubit l to qubit j does not take place. We refer to the excited qubit i as a spectator.
We emphasize that
our treatment does not assume that the system is in the Markovian regime.
After the first adiabatic elimination, basis kets with two excited qubits are coupled to each other via Ĥ_single if they contain a common excited qubit. The second adiabatic elimination leaves Ĥ_single unchanged. Thus, in the Hilbert space spanned by the N_e(N_e-1)/2 two-excitation qubit states, Ĥ_single couples each basis ket that contains two excited qubits to 2(N_e-2) other basis kets as well as to itself. While we
refer
to W(0) as onsite hopping interaction, it is also known as “self interaction”
or “self energy”
(see, e.g., Ref. <cit.>).
The strength W_jl,
W_jl=W(0) exp(-|n_j-n_l|a/L_0),
of the constrained single-qubit hopping interaction falls off exponentially as a function of |n_j-n_l|a, i.e., the difference between the cavities n_j and n_l that the qubits j and l are coupled to. The onsite hopping energy W(0) and length L_0 read
W(0)=-2J(g/2J)^2/√((Δ/2J)^2-1)
and
L_0=-a/ln(Δ/2J-√((Δ/2J)^2-1)),
respectively, where
Δ=ħ(ω_c-ω_e)=1/2(-δ+4J√(1+(U/16J)^2)).
Figure <ref> shows the onsite hopping energy W(0) and length L_0 for fixed g/J and U/J as a function of the dimensionless detuning δ/J. It can be seen that W(0)/J is negative and that the magnitude of W(0) increases with decreasing |δ/J|. Larger |W(0)| (note, Fig. <ref> shows W(0) as opposed to |W(0)|) are accompanied by larger L_0. For the detuning considered in this work (|δ/J|≪ 1), W_jl is—for x/a=1—appreciable not only for nearest neighbor hopping but also for next-to-nearest and next-to-next-to-nearest neighbor hopping.
Next, we discuss the effective pair hopping interaction Y_ij,lh. Figure <ref>(b) illustrates Y_ij,lhσ̂^+_iσ̂^+_jσ̂^-_lσ̂^-_h, which annihilates excitations at the lth and hth qubit and creates excitations at the jth and ith qubit. The effective pair hopping interaction Y_ij,lh is given by
Y_ij,lh=-g^4/NJ^2∑_KF_K,b(n_i,n_j)F_K,b^*(n_l,n_h)/Δ_K,b,
where F_K,b is given in Eq. (<ref>). As W_jl, Y_ij,lh is negative. The pair hopping interaction Ĥ_pair couples each excited qubit pair to all other excited qubit pairs. Figure <ref> shows the interaction Y_ij,lh for x/a=1 as functions of the pairs (i,j) and (l,h) for N_e=60. Specifically, the indices that specify the states σ̂_i^+σ̂_j^+|g,⋯,g⟩ are organized based on the separation between the excited qubits, i.e., |j-i|. In a qubit array with N_e qubits, there are N_e-1 basis states with a separation of |j-i|=1, N_e-2 basis states with a separation of |j-i|=2, and so on. The “lower left block" corresponds to |j-i|=|h-l|=1 [the pairs (i,j) and (l,h) both take the values (1,2), (2,3),⋯, (59,60)]. The “upper right block” corresponds to |j-i|=|h-l|=9 [the pairs (i,j) and (l,h) both take the values (1,10), (2,11),⋯, (51,60)]. Note that Fig. <ref> only considers a subset of pairs, i.e., |j-i|≤ 9 and |h-l|≤ 9. Within each block, the interaction is most negative along the diagonal and falls off approximately Lorentzian as one moves away from the diagonal. Moreover, starting with the block in the lower left corner, the interactions on the diagonal are less negative as one moves to blocks characterized by larger separations.
A key characteristic of the interaction Y_ij,lh is that it is—within each block—constant along the diagonal, along the off-diagonal, and so on. The fall-off of the interactions as one moves away from the diagonal within each block indicates that the Y_ij,lh interaction depends on the actual locations of the involved qubits in the spin chain. This implies that it is energetically more favorable for two excitations to be located in the middle of the chain than at the edge of the chain since the pair can hop to the left and to the right when located at the center and only to one side when located at the edge. This location dependence is critical for the formation of the droplet-like states discussed in the next section.
§ STATIONARY SOLUTIONS
Since we are working in the regime where g/J≪ 1, it might be expected naively that the constrained single-qubit hopping term Ĥ_single, which is directly proportional to g^2, dominates over the pair hopping term Ĥ_pair, which is directly proportional to g^4. While this is, indeed, the case in an appreciable portion of the parameter space, we show that there exists a parameter window in which the pair hopping interaction qualitatively changes the system characteristics.
It is noted that a fourth-order two-photon virtual process, which is proportional to g^4, was observed experimentally in transmon qubits coupled to a photonic crystal <cit.>.
Specifically, this section shows that the Y-term has a “pinning effect” that leads to the emergence of liquid-
or droplet-like
bound states.
Droplet states are self-bound and incompressible, and their excitation spectrum can be divided into compressional and surface modes <cit.>. We will show that the states referred to as droplet-like in this work are incompressible (their size is not solely set by the extent of the emitter array but by the entirety of system parameters). Moreover, the ground state is accompanied by a sequence of excitations that resemble compressional modes.
While our analysis is based on the approximate spin Hamiltonian Ĥ_spin, we checked that this Hamiltonian captures the key features of the full system Hamiltonian Ĥ qualitatively and in many cases even quantitatively correctly. The main advantage of using Ĥ_spin comes from the fact that it allows for a transparent interpretation of the results, in addition to being an interesting model in its own right.
We start by setting Ĥ_pair=0. We find it useful to compare Ĥ_single to the unconstrained one-qubit hopping Hamiltonian Ĥ_single, where Ĥ_single=2∑_i,j=1^N_eW_ijσ̂_i^+σ̂_j^-. This Hamiltonian emerges (without the factor of 2) when one works in the single-excitation manifold and adiabatically eliminates the single-photon states <cit.>. Ĥ_single differs from Ĥ_single because of the presence of the “spectator", i.e., the constraint makes the Hamiltonian Ĥ_single considered in our work unique. To highlight the differences, red and blue circles in Fig. <ref> show the eigenenergies of Ĥ_single and Ĥ_single, respectively, for N_e=60, g/J=1/50, U/J=-1, δ/J=-1/50, and x/a=1. The constraint introduces an upshift of the eigenenergies for all eigenstates. The upshift is larger for the more negative eigenenergies (measured relative to the bottom E_0,b of the two-photon bound state band) than the less negative eigenenergies. Interestingly, both Ĥ_single and Ĥ_single support step like pattern, with each plateau containing close to N_e eigenstates for the energetically lowest lying states. For the higher excited states, the steps are less pronounced.
Reference <cit.> referred to the energy band formed by the qubit dominated states as a metaband.
While we observe, similarly to Ref. <cit.>, that the width of the band decreases with increasing x, it is important to point out that that work considered qubit-array physics in the single-excitation manifold on resonance (and not off-resonance as in our case) and for significantly stronger coupling strengths (g/J of order 1).
For comparison, the black circles in Fig. <ref> show the eigenenergies for Ĥ_spin. It can be seen that Ĥ_pair appreciably impacts the 10 or so energetically lowest lying states and less so the higher-lying states. Importantly, the energies of the lowest few eigenstates of Ĥ_spin are pushed down due to the presence of the Ĥ_pair term. The downshift of the energies is associated with significant changes of the character of the eigenstates, i.e., a change from delocalized scattering states to localized bound states. We refer to this as “pinning” (see below for details). The energy spectrum shown in Fig. <ref> is unique to a qubit spacing of x/a=1. For larger spacings, but otherwise identical parameters, the hopping energies are smaller and the step-like pattern is washed out. Moreover, the most strongly bound states are less separated from the other states than for x/a=1 (i.e., Ĥ_pair introduces a smaller downshift for the ground state for x/a=2 than for x/a=1). For x=0, there exist three degenerate
energy levels:
for the same N_e, g/J, U/J, and δ/J as considered in Fig. <ref>, the x/a=0 spectrum for Ĥ_spin contains a single state with energy E-E_0,b=-0.2256 J, N_e-1 states with energy E-E_0,b=-0.0629J, and N_e(N_e-3)/2 states with energy E-E_0,b=δ=-0.02J.
To understand the influence of Ĥ_pair on the eigenspectrum, we use first-order non-degenerate perturbation theory. Treating Ĥ_pair as a perturbation, the first-order correction E^(1)_n to the eigenenergy E^(0)_n of the nth
droplet-like eigenket
|ϕ_n^(0)⟩ of Ĥ_single is given by
E^(1)_n=⟨ϕ^(0)_n|Ĥ_pair|ϕ^(0)_n⟩.
Figure <ref>(a) considers the six energetically lowest-lying droplet-like states (n=1-6). These droplet-like states correspond to state numbers 1, 2, 3, 4, 7, and 10.
Figure <ref>(a) shows that the perturbation energies (the open blue circles show E_n^(0)+E_n^(1)) lie below the zeroth-order energies E_n^(0) (green open triangles),
i.e., the stronger binding of Ĥ_spin compared to Ĥ_single due to Ĥ_pair is captured qualitatively in first-order perturbation theory. Higher-order corrections, which account for the mixing of the unperturbed states |ϕ^(0)_n⟩ play a larger role for the ground state (n=1) than for the excited droplet-like states (n=2-6). Figure <ref>(b) focuses on the energy of the lowest-lying droplet-like state and shows
that
energy as a function of the detuning. The detuning marked by an arrow is identical to the detuning used in Fig. <ref>(a). For large to moderate detunings, the results from the perturbation calculation (blue open circles) agree well with the exact diagonalization of Ĥ_spin (black solid circles). For relatively small detunings, however,
deviations are visible. While the first-order perturbative energy
improves upon
the
unperturbed energy, higher-order corrections play an increasingly more important role.
To characterize the eigenstates |ϕ_E⟩ of Ĥ_spin, we expand them in terms of the basis
kets
σ^+_iσ^+_j|g,⋯,g⟩,
|ϕ_E⟩=∑_i=1^N_e-1∑_j=i+1^N_ec^(E)_i,jσ^+_iσ^+_j|g,⋯,g⟩,
and analyze the expansion coefficients c^(E)_i,j as well as the pair correlation function P_pair(α), which measures the likelihood that the two excitations are located at qubits that are separated by α. The corresponding operator is given by
P̂_pair(α)=∑_i=1^N_e-ασ̂^+_iσ̂^+_i+α|g,⋯,g⟩⟨g,⋯,g|σ̂^-_iσ̂^-_i+α,
where α takes the values 1, 2, ⋯, N_e-1. For example, if α=1, the excitations are located at neighboring spins. In terms of the expansion coefficients, the pair correlation function for the eigenstate |ϕ_E⟩ is given by P_pair(α)=∑_i=1^N_e-α|c^(E)_i,i+α|^2.
Figures <ref>(a) and <ref>(b) show P_pair(α) for the ground state for N_e=60, g/J=1/50, U/J=-1, and x/a=1 for two different detunings, namely δ/J=-1/50 and -3/20. The blue dotted lines are obtained using Ĥ_spin. The full Hamiltonian Ĥ_full (black solid lines) yields results that are quite similar to those for Ĥ_spin, thus providing evidence that Ĥ_spin yields faithful results. For small |δ/J| [Fig. <ref>(a)], the pair correlation function peaks at α=1 and is essentially zero for α≫ 1. This indicates that the two excited qubits want to stay together. The fall-off of P_pair(α) suggests that the ground state corresponds to a bound state. This interpretation is confirmed by calculations for larger arrays (larger N_e) with otherwise identical parameters. We find that P_pair(α) for the ground state remains essentially unchanged when N_e is increased, i.e., the size of the ground state is independent of
N_e,
thereby justifying the classification as a self-bound state.
For larger |δ/J| [Fig. <ref>(b)], in contrast, the pair correlation function peaks at α≈ 10 for Ĥ_full and Ĥ_spin. This indicates that the two excited qubits have a tendency to spread out over the entire array. This interpretation is supported by the fact that the fall-off of the pair correlation function moves to larger α for larger N_e but otherwise identical parameters. Correspondingly, we classify the ground state considered in Fig. <ref>(b) as unbound. The inclusion of Ĥ_pair in the effective spin Hamiltonian Ĥ_spin (blue dotted line) is crucial. A comparison of the blue dotted line [P_pair(α) for Ĥ_spin] and red dashed line [P_pair(α) for Ĥ_single] reveals that Ĥ_pair has a pinning effect: it enhances, as already alluded to in Sec. <ref>, the probability to find excitations located at qubits that are close to each other. The effect is very prominent in Fig. <ref>(a), where the red line is much broader than the blue line.
If Ĥ_pair is neglected and N_e is increased, the red line in Fig. <ref> does not maintain its size, as is the case for Ĥ_spin, but increases. This unequivocally shows that Ĥ_pair is responsible for the emergence of self-bound states.
Figure <ref> shows the real part of the coefficients c_i,j^(n) for the four energetically lowest lying droplet-like bound states (n=1-4) for N_e=60, g/J=1/50, U/J=-1, δ/J=-1/50, and x/a=1; the imaginary part is equal to zero.
The droplet-like states shown in Fig. <ref> correspond to the state
numbers
1, 2, 3, and 4.
Figure <ref> employs relative and center-of-mass coordinates r and R, respectively, of the two excited qubits, r=|j-i| and R=(i+j)/2.
The white area characterized by r≥ 2R for R<N_e/2 and r≥ 2(N_e-R) for R≥ N_e/2 is unphysical as there is a constraint of i<j on the eigencoefficients due to the bosonic character or, equivalently, the exchange symmetry of the excitations. The small white dots, which exist in the physical i<j portions in Fig. <ref>, result from the transformation from the (i, j) spin indices to the (R, r) coordinates. In Figs. <ref>(a)-<ref>(d), the magnitude of the coefficients c^(n)_i,j decreases with increasing r for fixed R. Along the R coordinate, the number of nodes increases from zero for the ground state [n=1 in Fig. <ref>(a)] to three for the third excited droplet-like state [n=4 in Fig. <ref>(d)]. The nodes are to a very good approximation parametrized by R_node≈constant, i.e., they are, on the scale of Fig. <ref>, independent of r.
In what follows, we use a variational ansatz to understand the length scale that governs the droplet-like states and the number of droplet-like states that are supported by a qubit array of size N_e. Since Fig. <ref> suggests that the expansion coefficients of the nth droplet-like eigenstate decouple when plotted as functions of the relative coordinate r and the center-of-mass coordinate R, we introduce the product ansatz
c^(n)_r,R= Q^(n)(R)q(r).
Here, the function Q^(n)(R),
Q^(n)(R)=√(2/N_e)sin(nπ/N_eR),
corresponds to the nth particle in the box wave function and the function q(r),
q(r)=2√(L_r^3/π a^3)[1/(r-1)^2+(L_r/a)^2],
to an n-independent Lorentzian with characteristic length L_r. The length L_r is treated as a variational parameter. By construction, the variational states with different n are orthogonal.
Figure <ref>(a) compares the variational energies (red open squares) of the six droplet-like states that are supported by the qubit array for N_e=60, g/J=1/50, U/J=-1, δ/J=-1/50, and x/a=1 with those obtained by diagonalizing Ĥ_spin (black solid circles). We see that the variational energies agree extremely well with the exact eigenenergies of Ĥ_spin. In Fig. <ref>(b), the energy of the ground droplet-like state is shown as a function of δ/J for the same N_e, g/J, U/J, and x/a as used in Fig. <ref>(a). For large to moderate, in magnitude, detunings, the energies from the variational calculation (red open squares) agree well with the exact eigenenergies of Ĥ_spin (black solid circles). For small detunings, small deviations are visible. The variational calculation not only predicts the eigenenergy accurately but also the corresponding eigenstates. As an example, the green open circles in the inset of Fig. <ref>(a) show the pair correlation function obtained by the variational treatment; it agrees well with the results obtained for Ĥ_spin (blue dotted line). The number of droplet-like states supported by the qubit array is approximately equal to aN_e/L_r. Intuitively, this can be understood as follows. The system develops additional nodes along the R direction till the spacing between the nodes is comparable to the size of the droplet-like
state
along the r direction. For g/J=1/50, U/J=-1, δ/J=-1/50, and x/a=1, the variational ground state energy is minimized for L_r≈ 10a. The qubit array with N_e=60 supports
six
droplet-like states, in agreement with the estimate aN_e/L_r≈ 6.
As the qubit array spacing x is changed from
a to 2a,
the number of droplet-like bound states decreases from six to four. For x=3a, droplet-like bound states are no longer supported.
Similar results are found for other parameter combinations.
We note that Ĥ_spin also supports more highly excited modes, which have nodes along the r-coordinate. The variational treatment of these energetically higher-lying droplet-like states is beyond the scope of this work.
§ DYNAMICS
This section discusses the dynamics for negative δ (band-gap regime) for two different initial states in the two-excitation manifold, namely the partially symmetric state |PS⟩,
|PS⟩=1/√(N_e-1)∑_i=1^N_e-1σ_i^+σ_i+1^+|g,⋯,g⟩,
and the fully symmetric state |FS⟩,
|FS⟩=√(2)/√(N_e(N_e-1))∑_i=1^N_e-1∑_j=i+1^N_eσ_i^+σ_j^+|g,⋯,g⟩.
The fully symmetric state is a superposition of all basis kets (all basis kets contribute with an expansion coefficient √(2)/√(N_e(N_e-1))). The partially symmetric state, in contrast, only considers basis kets for which the excited qubits are nearest neighbors.
Figures <ref>(a) and <ref>(b) show the decomposition of the states |PS⟩ and |FS⟩, respectively, into the energy eigenstates |ϕ_E⟩ of Ĥ_spin for N_e=60, g/J=1/50, U/J=-1, δ/J=-1/50, and x/a=1. The state |PS⟩ has finite overlap with a large number of eigenstates from all over the eigenspectrum. The ground state contributes about 10% and the other states 3% or less. For the state |FS⟩ [Fig. <ref>(b)], in contrast, there are two energy eigenstates that dominate and together contribute 89% [52.6%, red square in Fig. <ref>(b), and 36.4%, blue triangle in Fig. <ref>(b)]. The lowest eigenstate, which contributes 52.6%, has
droplet-like
character while the excited eigenstate, which contributes 36.4%, has scattering
characteristics. Since the fully symmetric initial state is dominated by two eigenstates, the dynamics is expected to feature Rabi-like two-state oscillation dynamics. The dynamics for the partially symmetric state, in contrast, is expected to display features of dephasing, at least over certain time scales, due to the
superposition
of many energy eigenstates.
Figure <ref> shows the time dependence of the probability that two excitations belong to nearest neighbor qubits, i.e., qubits that are separated by
α=1 (black solid line), to qubits that are separated by α=6 (red dashed line), and to qubits that are separated by α=21 (blue dotted line).
These observables are for the same parameters as those used in Fig. <ref>. The time evolution of P_pair(α,t) for the initial states |PS⟩ [Fig. <ref>(a)] and |FS⟩ [Fig. <ref>(b)] is—as already anticipated based on the initial state decomposition—distinct. In Fig. <ref>(a), P_pair(α,t) for α=1 decays with damped oscillations. The damping or decay are attributed to the fact that a large number of eigenstates contribute to the initial state with comparable weight, giving rise to dephasing. In Fig. <ref>(b), P_pair(α,t) oscillates with nearly undamped amplitude for all α considered. The slight “distortions” of the oscillations are caused by dephasing effects of the eigenstates that contribute to the fully symmetric initial state with a small weight, i.e., less than 5%. The oscillation period of t≈ 2000ħ/J corresponds to an energy of 0.0031J. This energy agrees with the difference in energies of the two eigenstates that have the largest overlap with the initial state
|FS⟩
[red
square and blue triangle in Fig. <ref>(b)].
Figure <ref> shows the spin-spin correlation function P_corr(i,j,t),
P_corr(i,j,t)=⟨ψ(t)|σ̂^+_iσ̂^+_j|g,⋯,g⟩×
⟨g,⋯,g|σ̂^-_iσ̂^-_j|ψ(t)⟩
at eight different times ranging from zero in Fig. <ref>(a) to Jt/ħ=7500 in Fig. <ref>(h) for N_e=60, g/J=1/50, U/J=-1, δ/J=-1/50, and x/a=1. The initial state is |FS⟩. The plots in the left column are for Jt/ħ=0, 2220, 4440, and 6540. Comparison with Fig. <ref>(b) shows that P_pair(α=1,t) takes on a local minimum at these times. The plots in the right column of Fig. <ref>, in contrast, are such that P_pair(α=1,t) takes on a local maximum. We can see that P_corr(i,j,t) is mostly concentrated around the middle of the diagonal in the right column while it is much more spread out in the left column. The observation that the spin-spin correlations alternate between being more localized and being more spread out can be readily explained by the fact that the initial state |FS⟩ is dominated by contributions from the ground droplet-like state and a delocalized scattering state [red square and blue triangle Fig. <ref>(b)]. This suggests that the droplet-like ground state can be probed by initializing the qubit array in the fully symmetric state |FS⟩.
The calculations presented consider the ideal case scenario, where the excited state qubit has an infinite lifetime, the photon loss from the cavities is ignored, and imperfections—such as, e.g., a finite spread Δ J of the tunneling energies J and a finite spread Δω_c of the cavity frequencies ω_c that exist to a varying degree in experiment—are neglected. To observe the oscillations displayed in Fig. <ref>(b), the time scales associated with spontaneous qubit decay, photon losses, and “dephasing" due to the spread of system parameters must be larger than about 10^4 ħ/J. In the following discussion, we assume that the excited state lifetime of the qubit is longer than the time scale for photon losses.
A finite “bare" photon lifetime of ħκ^-1 leads to a characteristic decay time (Γ_c)^-1 that scales, for |δ_0| ≪ 2J, as
Γ_c = p_phκ / ħ, where p_ph≈ g^2 J^-1/2δ_0^-3/2/4 and
δ_0 denotes the detuning in the single-excitation manifold,
δ_0 = (ħω_c - 2J) - ħω_e <cit.>. Physically, the multiplicative factor p_ph can be understood as arising from the admixture of the photonic degrees of freedom to the hybridized bound state in the single-excitation manifold.
Rewriting δ_0 in terms of the detuning δ in the two-excitation manifold, we find
p_ph= g^2/4 √(J)[ 1/2( -δ +√(U^2 + 16 J^2))-2J ]^-3/2.
For δ/J=-1/50 and -3/20, as used in this paper, p_ph is equal to 5 × 10^-3 and 2 × 10^-3, respectively.
To observe multiple oscillations, (Γ_c)^-1 must be much larger than 10^4ħ/J; the equal sign holds for κ/J=2 × 10^-2 and 5 × 10^-2, respectively.
Superconducting circuit experiments have realized an eight cavity system with U/h=-255 MHz, J/h=5-20 MHz, and κ/h=5 kHz <cit.>. This translates to
κ/J=2.5 × 10^-4 to 10^-3, i.e., experiments are already operating in a regime where the photon lifetime is sufficiently long to observe the predicted phenomena.
For fixed spreads Δ J and Δω_c, one may attempt to increase δ such that the spreads become, if measured as a multiple of the detuning, smaller. Since a larger δ corresponds to a smaller photon contribution p_ph and hence a longer time scale for the photon losses,
there is some room to optimize the parameters for a specific experimental set-up.
While challenging, we conclude that the theory predictions put forward in this paper can be tested in state-of-the-art experiments.
§ CONCLUSION
This paper discussed the time-independent and time-dependent behaviors of a qubit array coupled to a non-linear photonic waveguide. Our interest was in the regime where the two-qubit transition energy lies in the band gap below the two-photon bound state band that is supported by the one-dimensional waveguide. We focused our attention on the two-excitation manifold. Even though the qubits are not interacting with each other, effective interactions—mediated by the waveguide—are introduced between qubits as a result of a two-step adiabatic elimination process. The resulting effective spin Hamiltonian, which was shown to accurately reproduce the key characteristics of the full Hamiltonian, features constrained single-qubit hopping and pair hopping interactions. The emergence of the latter critically depends on the presence of the Kerr-like non-linearity U. The effective spin Hamiltonian was shown to support a new class of
droplet-like bound
states that arise due to the pair hopping interaction. These droplet-like states extend over many qubit lattice sites and can be probed dynamically. For the fully symmetric initial state, the populations were found to oscillate back and forth between a
droplet-like
bound
state and a delocalized scattering state. While most of our discussion focused on N_e=60, g/J=1/50, U/J=-1, and δ/J=-1/50, we emphasize that the characteristics discussed in this paper are also observed for other parameter
combinations.
For fixed g/J, δ/J, N_e, and x/a, we find that the number of droplet-like states supported by the qubit array decreases as U/J becomes more negative. As |U|/J increases, the two-photon bound state becomes more localized and hence the overall strength of
the
pair hopping interaction becomes less negative.
Whether
or not
droplet-like bound states exist also depends on the qubit array spacing x. If the separation between two neighboring qubits
is increased, the number of droplet-like
states
supported by the array decreases.
The giant droplet-like bound states discovered in this work provide an intriguing example of utilizing structured baths to engineer effective spin-spin interactions that support quantum states with non-trivial correlations.
The droplet-like states considered in this paper, which emerge in the two-excitation sub-space, are distinct from the two-excitation scattering states considered in Ref. <cit.> in the absence of the non-linearity U. They are also distinct from hydrid qubit-photon states that emerge in the single-excitation manifold <cit.>.
Possible extensions may focus on topological wave guides <cit.>, higher-dimensional baths, superlattice-type arrangements of the qubits, qubits with multiple transition frequencies <cit.>, multi-level emitters, qubits with multiple point contacts <cit.>, and higher-excitation manifolds <cit.>. In all these scenarios, it will be interesting to explore the interplay between constrained single-qubit and pair hopping interactions.
§ ACKNOWLEDGEMENT
Support by the National Science Foundation through grant number PHY-2110158 is gratefully acknowledged. This work used the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma (OU).
§ DERIVATION OF Ĥ_SPIN
Starting with the full Hamiltonian Ĥ, this appendix derives the effective spin Hamiltonian Ĥ_spin. The adiabatic elimination procedure discussed in this appendix is illustrated in Fig. <ref>.
Time-dependent wave packet: Throughout, we assume that the two-photon scattering states can be neglected. This is justified since we are working in a regime where the two-qubit transition energy is far detuned from the two-photon scattering continuum. Under this approximation, the wavepacket |ψ(t)⟩ in the two-excitation manifold can be written as <cit.>
|ψ(t) ⟩ = exp(- 2ω_e t)
[
∑_i=1^N_e-1∑_j=i+1^N_ed_ij(t) σ^+_i σ^+_j|g,⋯,g, ⟩ + ∑_i=1^N_e∑_k c_ik(t) σ^+_i â_k^† |g,⋯,g, ⟩ +
∑_K c_K,b(t) B̂_K^† |g,⋯,g, ⟩],
where d_ij(t), c_ik(t), and c_K,b(t) denote expansion coefficients.
The operator
B̂_K^†
creates
a two-photon bound state with momentum K,
|ψ_K,b⟩=B̂^†|vac⟩. Inserting Eq. (<ref>) into the time-dependent Schrödinger equation, we obtain a set of coupled differential equations
ħḋ_ij(t) = g/√(N)∑_k [ exp( k a n_i) c_j k(t) + exp( k a n_j) c_i k(t)],
ħċ_i k(t) =
Δ_k
c_i k(t)
+g/√(N)∑_j=1,j≠ i^N_eexp( - k a n_j) d_ij(t) +
g/N∑_K M_b(k, n_i,K) c_K,b(t),
and
ħċ_K,b(t) =
Δ_K,b
c_K,b(t) + g/N∑_i=1^N_e∑_k [M_b(k ,n_i,K)]^* c_i k(t),
where i=min(i,j) and j=max(i,j). The energy detunings Δ_k and Δ_K,b are given by
Δ_k=E_k-ħω_e
and
Δ_K,b=E_K,b-2 ħω_e,
where E_k denotes the energy of a single photon with wave vector k. The matrix elements M_b(k, n,K) are defined as <cit.>
M_b(k, n,K)=√(2)×
∑_m exp[ m (k-K/2)a
+ n (K-k)a ] ψ_K,b(m),
where ψ_K,b(m)=⟨ ma|ψ_K,b⟩ is the two-photon bound state wave function (ma denotes the relative distance between the two photons). Stationary and time-dependent solutions to the Schrödinger equation for Ĥ_full are obtained through exact diagonalization, excluding the basis kets that span the two-photon scattering continuum. To characterize the distribution of the excited qubits, we monitor the pair correlation function P_pair(α,t),
P_pair(α,t)=⟨ψ(t)|∑_i=1^N_e-ασ̂^+_iσ̂^+_i+α|g,⋯,g,vac⟩×
⟨g,⋯,g,vac|σ̂^-_iσ̂^-_i+α|ψ(t)⟩,
as well as the spin-spin correlation function P_corr(i,j,t),
P_corr(i,j,t)=⟨ψ(t)|σ̂^+_iσ̂^+_j|g,⋯,g,vac⟩×
⟨g,⋯,g,vac|σ̂^-_iσ̂^-_j|ψ(t)⟩.
In what follows, we introduce several approximations that eliminate the photonic degrees of freedom from the problem and, in turn, introduce effective interactions between groups of qubits.
Adiabatic elimination of the single-photon states: Assuming that the changes of c_i k(t) with time can be neglected, i.e., ċ_i k(t)=0 in Eq. (<ref>), the single-photon states σ̂^+_j|g,⋯,g,k⟩ can be adiabatically eliminated. This approximation breaks down when g is too large or the single-qubit transition energy is too close to the single-photon band. The resulting differential equations read
ħḋ_ij(t) = ∑_l=1,l≠ j^N_eW_ild_lj(t)+ ∑_l=1,l≠ i^N_eW_ljd_il'(t) + g^2/√(N)J∑_K F_K,b(n_i,n_j)c_K,b(t)
and
ħċ_K,b(t) = Δ_K,b c_K,b(t) +g^2/√(N)J∑_i=1^N_e∑_j=i+1^N_e F^*_K,b(n_i,n_j)d_ij(t)+g^2/NJ∑_K'G_KK'(n)c_K',b(t),
where l=min(l, j), j=max(l, j), i=min(l, i), l'=max(l, i), and n=(n_1, n_2,⋯, n_N_e). It can be seen that the adiabatic elimination of the single-photon states introduces three effective interactions, namely F_K,b, G_KK', and W_jl. The effective interaction F_K,b(n_i,n_j) between the states σ̂_i^+σ̂_j^+|g,⋯,g,vac⟩ and B̂_K^†|g,⋯,g,vac⟩ is given by <cit.>
F_K,b(n_i,n_j)=
-∑_kJ/NΔ_k(exp( - k a n_i )
[M_b(k, n_j,K)]^*+
exp(- k a n_j) [M_b(k, n_i,K)]^*).
The effective interaction G_KK'(n) between the states B̂_K^†|g,⋯,g,vac⟩ and B̂_K'^†|g,⋯,g,vac⟩ is given by <cit.>
G_KK'(n)=-
∑_j=1^N_e∑_k
J/NΔ_k
[M_b(k,n_j,K)]^* ×
M_b(k, n_j,K').
The interactions F_K,b and G_KK' have been discussed extensively in the context of the two-qubit system (Ĥ with N_e=2) <cit.>. The effective interaction W_jl, in contrast, does not exist for N_e=2; it critically depends on having more than two qubits coupled to the cavity array. The functional form of W_jl is given in Eq. (<ref>) of the main text.
Equations (<ref>)–(<ref>) correspond to the effective Hamiltonian Ĥ_adia,0,
Ĥ_adia,0 = Ĥ_single+ g^2/J√(N)∑_i=1^N_e∑_j=i+1^N_e∑_K[F_K,b(n_i,n_j) σ̂^+_iσ̂^+_j B̂_K + F^*_K,b(n_i,n_j) σ̂^-_iσ̂^-_j B̂^†_K]+
∑_K Δ_K B̂_K^†B̂_K + g^2/NJ∑_K∑_K'G_KK'(n)B̂_K^†B̂_K',
where Ĥ_single is given in Eq. (<ref>) of the main text. For N_e=2, Refs. <cit.> found that the effective interaction G_KK' plays a non-negligible role only when the transition energy 2ħω_e of two qubits is in or nearly in resonance with the bottom of the two-photon bound state band. Since G_KK' plays, in general, a negligible role away from the bottom of the band, it is useful to define the effective Hamiltonian Ĥ_adia,1 by setting G_KK' in Ĥ_adia,0 to zero. The effective Hamiltonians Ĥ_adia,0 and Ĥ_adia,1 live in the (N_e(N_e-1)/2+N)–dimensional Hilbert space that is spanned by the states σ̂^+_iσ̂^+_j|g,⋯,g, vac⟩ and B̂^†_K|g,⋯,g,vac⟩ with wave vector K.
Adiabatic elimination of the two-photon bound states: For the band gap physics considered in this paper, the energy 2ħω_e of two excited qubits is not in resonance with the two-photon bound state band. Consequently, we adiabatically eliminate the two-photon bound states, i.e., we set the left hand side of Eq. (<ref>) to zero. Using this to eliminate c_K,b(t) from Eq. (<ref>), the resulting set of coupled
equations—setting G_KK'=0—reads
ħḋ_ij(t) = ∑_l=1,l≠ j^N_eW_ild_lj(t)+ ∑_l=1,l≠ i^N_eW_ljd_il'(t) +
∑_l=1^N_e-1∑_h=l+1^N_eY_ij,lhd_lh(t).
Equation (<ref>) corresponds to the effective spin Hamiltonian Ĥ_spin given in Eq. (<ref>) of the main text, which lives in the N_e(N_e-1)/2–dimensional Hilbert spanned by the qubit states σ̂^+_iσ̂^+_j|g,⋯,g⟩.
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|
http://arxiv.org/abs/2307.04179v1 | 20230709140458 | IANS: Intelligibility-aware Null-steering Beamforming for Dual-Microphone Arrays | [
"Wen-Yuan Ting",
"Syu-Siang Wang",
"Yu Tsao",
"Borching Su"
] | eess.AS | [
"eess.AS",
"eess.SP"
] |
[
Diancong Jin
August 12, 2023
===================
Beamforming techniques are popular in speech-related applications due to their effective spatial filtering capabilities. Nonetheless, conventional beamforming techniques generally depend heavily on either the target's direction-of-arrival (DOA), relative transfer function (RTF) or covariance matrix. This paper presents a new approach, the intelligibility-aware null-steering (IANS) beamforming framework, which uses the STOI-Net intelligibility prediction model to improve speech intelligibility without prior knowledge of the speech signal parameters mentioned earlier. The IANS framework combines a null-steering beamformer (NSBF) to generate a set of beamformed outputs, and STOI-Net, to determine the optimal result. Experimental results indicate that IANS can produce intelligibility-enhanced signals using a small dual-microphone array. The results are comparable to those obtained by null-steering beamformers with given knowledge of DOAs.
STOI, STOI-Net, null-steering, beamforming, microphone arrays
§ INTRODUCTION
Microphone arrays are commonly used in numerous speech-related applications including hearing aids and teleconferencing to isolate the desired signals that are often degraded
by ambient noise and other types of interference <cit.>.
Multi-channel speech enhancement (MCSE) techniques have been extensively studied to extract the desired signals
<cit.>.
Beamforming algorithms are usually a crucial component of these methods, as they utilize spatial diversity from multiple recordings to perform spectral and spatial filtering on multiple channel inputs, generating a speech-enhanced output
<cit.>.
For example, the delay-and-sum beamformer
<cit.> uses the geometry of the array and direction-of-arrival (DOA) information to parameterize the spatial-spectral filter.
The minimum variance distortionless response (MVDR) method <cit.>
minimizes the power of the noise signal while maintaining a distortionless response for the target signal, utilizing the knowledge of the covariance matrices and DOA or relative transfer function (RTF).
Additionally, null-steering beamformers (NSBF) have been proposed to
filter out signals from specific directions
<cit.>.
Conventional beamforming algorithms typically depend highly on an accurate DOA or RTF estimate to obtain the spatial information of the target signals.
Over the past few decades, multiple DOA estimation algorithms have been proposed in
<cit.>.
For DOA estimation algorithms specialized for multiple speech signals, the work in <cit.> used the coherence test and sparsity property of speech
to estimate accurate DOAs using clustering-based methods.
In addition to a direct DOA estimation approach
, time difference of arrival (TDOA) estimation methods <cit.> are also commonly used to localize the target signal.
One popular category is the application of the steered response power phase transform <cit.>, which scans over a predefined spatial region to parameterize the cross-correlation functions using each candidate location of the source, and then adopts a maximum likelihood estimator to estimate the TDOA.
In addition to these methods, the work in <cit.> discussed covariance subtraction and covariance whitening methods to obtain RTF estimates of the speech signal using well-estimated covariance matrices from noise-only and speech-noise frames.
Although these approaches have great potential to provide accurate spatial information, they typically rely heavily on multiple assumptions. In the case of <cit.>, the authors assumed accurate estimates of the noise covariance matrices for each time-frequency index.
If the noise covariance matrices
contain spatial statistics of the speech signal, the beamformers might not be aware of such errors and attenuate the corresponding signals without regard to how this might impact the intelligibility of speech signals.
Meanwhile, it is also worth noting that,
neural beamformers, such as <cit.>, have been proposed to perform state-of-the-art MCSE.
For these NN-based approaches, it is usually necessary to construct a dataset containing diverse utterances received by a microphone array in multiple scenarios.
In addition,
these neural beamformers are usually optimized over a large number of parameters, which makes each parameter hard to interpret.
In the field of speech processing,
a well-known metric for intelligibility is the short-time objective intelligibility (STOI) <cit.>.
The STOI function estimates the intelligibility of signals through
a series of signal-processing stages, including silence-segment elimination, feature extraction in the time-frequency (TF) domain, one-third octave band processing, feature normalization, and intelligibility mapping. In this process, the deteriorated sound signal and the corresponding clean reference signal are used simultaneously to compute the final score.
In this paper, the STOI function will be denoted as ℎ_STOI: ℝ^N × K×ℝ^N × K→ [0, 1] which is defined as the mapping from the magnitude of a pair of N × K short-time Fourier transform (STFT) matrices to the interval [0, 1], where N is the number of time frames and K is the number of frequency bins per frame. For simplicity, we will omit the steps such as silence-segment elimination for our description of the STOI function.
In addition, unlike metrics such as the speech intelligibility index in <cit.>, STOI is known for its reliable intelligibility evaluation of signals processed in the TF domain, where most acoustic beamforming systems perform MCSE. However, a clean reference is typically inaccessible. Therefore,
the authors in <cit.> proposed STOI-Net, a non-intrusive intelligibility assessment model that predicts STOI scores based only on the noise-corrupted waveforms.
In light of the heavy dependence of beamformers on the estimation of the DOAs or RTFs of the speech signals,
we propose a new optimization framework for an intelligibility-enhancing beamformer
without relying on the previously mentioned speech parameters.
Instead, we explicitly consider intelligibility as an optimization objective.
Works such as <cit.> have also incorporated the notion of intelligibility into the design of beamformers. We will perform intelligibility-based optimization within a set of null-steering beamformers. Hence, we call this intelligibility-aware null-steering (IANS) beamforming.
For the IANS beamforming process, an NSBF algorithm is first applied to generate a set of candidate signals via null-steering. The generated signals are then passed through a pre-trained STOI-Net to predict the associated STOI scores. IANS then outputs the utterance corresponding to the highest intelligibility score. Contrary to the previously mentioned neural beamformers, the proposed IANS algorithm doesn't require additional multi-channel training data.
Moreover, the IANS optimization problem only optimizes one parameter whose optimal value is interpretable. Furthermore, advanced single-channel SE methods, such as <cit.>, can be incorporated with IANS for downstream applications.
The remainder of this paper is organized as follows. Section <ref> discusses the signal model and related works including filter-and-sum beamformers, null-steering beamformers and STOI-Net.
Next, we will present our IANS optimization problem in Section <ref>.
In Section <ref>, the IANS algorithm will be discussed in detail.
Section <ref> presents the experimental setup and results. Finally, Section <ref> concludes the paper and discusses future works.
§ BACKGROUND AND RELATED WORKS
§.§ Signal model
In this study, the considered signal model comprises a
speech signal, s(t), and an interference signal, i(t), propagating in a room with a sound speed of c, received by a dual-microphone array at angles of θ_s and θ_i, respectively. The angles are
measured with respect to the first (reference) microphone, with 0^∘ being the endfire direction. The microphone array has a small spacing of ℓ, and we assume that the sound sources are stationary in space. We denote the room impulse responses (RIRs) for s(t) and i(t) with respect to the m^th microphone as
g^(m)_s(t) and g^(m)_i(t), respectively. The received signal at the m^th microphone can be expressed as the following:
x^(m)(t) = g^(m)_s(t) ∗ s(t) + g^(m)_i(t) ∗ i(t).
After obtaining the received signals x^(1)(t) and x^(2)(t), We can then apply the STFT to derive their corresponding N × K STFT matrices 𝐗^(1) and 𝐗^(2). Subsequently, we can define the received signal vector 𝐱[n, k] as follows:
𝐱[n, k] = [𝐗^(1)_n, k, 𝐗^(2)_n, k]^T.
Here, 𝐗^(m)_n, k represents the (n, k)^th element of 𝐗^(m), where
n=1, 2, ⋯, N and k=1, 2, ⋯, K.
§.§ Filter-and-sum beamformers
Filter-and-sum beamformers <cit.>
are a set of beamformers that perform the filter-and-sum operation to enhance the signal of interest. This process can be represented in the TF-domain as
Y[n, k] = 𝐰^H[n, k] 𝐱[n, k],
where 𝐰[n, k] is the weight vector for 𝐱[n, k], and Y[n, k] is the resulting TF component of the beamformed signal. We will denote this set of beamformers as ℱ_FSBF.
§.§ Null-steering beamformers
Within ℱ_FSBF, there is a subset of beamformers capable of nulling out signals coming from a particular direction ϕ while maintaining a (nearly) distortionless response at θ_d. We call this set the null-steering beamformer set, ℱ_NSBF.
We first define two vectors, the distortionless response steering vector 𝐚^(θ_d)[k] = [1, e^-j ω_k ℓ/ccosθ_d]^T and the null-response steering vector
𝐚^(ϕ)[k] = [1, e^-j ω_k ℓ/ccosϕ]^T, where ω_k is the frequency value at the k^th frequency bin. Each 𝐚^(ϕ)[k] is associated with a projection matrix Φ[k] defined in the following,
Φ[k] =𝐈 - 𝐚^(ϕ)[k](𝐚^(ϕ)[k])^H/||𝐚^(ϕ)[k]||^2
=𝐈 - 𝐚^(ϕ)[k](𝐚^(ϕ)[k])^H/2,
where (·)^H denotes the Hermitian transpose operator. Here, Φ[k] projects vectors into the subspace orthogonal to the span of 𝐚^(ϕ)[k].
In this paper, ℱ_NSBF is defined as a beamformer set
with time-invariant weight vectors defined as
𝐰[k] = Φ[k]𝐚^(θ_d)[k]/max((𝐚^(θ_d)[k])^HΦ[k]𝐚^(θ_d)[k], ϵ),
where ϵ is a small number to avoid 0 division. We note that without the max(· , ϵ), Eq. (<ref>) has been studied in <cit.> in the context of the MVDR beamformer.
It is also worth pointing out that in the context of beamformers such as the linearly-constrained minimum variance beamformer <cit.>, null responses are usually set as explicit constraints to a noise power minimization problem.
§.§ STOI-Net
In <cit.>, the authors proposed STOI-Net, a non-intrusive speech intelligibility assessment model that predicts the STOI scores of speech signals both frame- and utterance-wise using feature extraction and score calculation functions.
For the feature extraction, the STFT is applied to convert the peak-normalized time-domain waveform of interest into a sequence of frame-wise magnitude spectra in the frequency domain. These frames are then passed through 12 fully convolutional neural network layers to extract the acoustic representations. Next, the score-calculation function maps the extracted features to an intelligibility score. Specifically, frame-level scores are generated after applying 1) bidirectional long short-term memory, 2) an attention layer, and 3) fully connected nonlinear mapping functions to the extracted features. The final intelligibility score of the entire utterance is then obtained by applying a global averaging algorithm to all frame-level scores. It is worth noting that STOI-Net is not limited to a single neural network architecture. In <cit.>, two model architectures were used: one with an attention layer and the other without it. In the remainder of this paper, we will denote the STOI-Net model as a function ℎ_STOI-Net: ℝ^N × K→ℝ.
§ PROBLEM FORMULATION
Contrary to most optimal beamformers for speech enhancement (e.g., MVDR) where optimal weights are derived for each TF bin based on well-estimated DOAs, RTFs and covariance matrices, we propose an optimization problem based on the intelligibility of the entire utterance of the received speech signals.
Our primary goal is to identify a function 𝑓: ℂ^N × K×ℂ^N × K→ℂ^N × K within a function set ℱ that takes the received signals 𝐗^(1) and 𝐗^(2) as input
and maps them to an STFT matrix
with the maximum STOI score.
As we aim to perform optimization without having to train a new NN that learns from a dataset containing microphone array recordings from various scenarios, we will use a simpler function set ℱ=ℱ_NSBF
where we can limit all potential values of θ_d and ϕ on a discrete grid.
However, the number of feasible solutions grows quadratically with the resolution for the θ_d and ϕ axes on this grid. Hence, we only perform grid search for ϕ on a grid 𝒢 while fixing θ_d to an arbitrary angle ψ∈ [0^∘, 180^∘]. The grid 𝒢 is an ordered set containing P angles ranging from 0^∘ to 180^∘.
Thus, the number of possible beamformers we are considering here is P.
Our optimization problem can now be described as the following:
maximize_ϕ∈𝒢 ℎ_STOI(|𝑓_θ_d=ψ, ϕ(𝐗^(1), 𝐗^(2))|, |𝐒|)
subject to 𝑓_θ_d=ψ, ϕ∈ℱ_NSBF,
where 𝐒 represents the STFT matrix of the clean speech signal and |·| denotes the element-wise magnitude extraction for a matrix.
We refer to this problem as the STOI Null-steering (STOI-NS) problem, as it employs null-steering to optimize the true STOI function. We will denote the optimal beamformer and null angle for this problem as 𝑓_STOI-NS^⋆ and ϕ_STOI-NS^⋆, respectively.
However, 𝐒 is never accessible in practical scenarios. Therefore, using the pre-trained STOI-Net model ℎ_STOI-Net, we modify the optimization problem in (<ref>) as follows:
maximize_ϕ∈𝒢 ℎ_STOI-Net(|𝑓_θ_d=ψ, ϕ(𝐗^(1), 𝐗^(2))|)
subject to 𝑓_θ_d=ψ, ϕ∈ℱ_NSBF
Since STOI-Net was trained to estimate the STOI score of a signal, we consider this the Intelligibility-aware Null-steering (IANS) problem.
The IANS problem is now feasible without the clean reference 𝐒. The optimal beamformer and null angle for this problem are denoted as
𝑓_IANS^⋆ and ϕ_IANS^⋆, respectively.
It is clear that the STOI score of the output obtained from using the beamformer 𝑓_STOI-NS^⋆ is a natural upper bound of that using 𝑓_IANS^⋆ as we will show in
Section <ref>.
This optimization framework was inspired by works such as <cit.>,
where the authors trained speech enhancement systems by incorporating speech quality prediction neural networks <cit.> into the loss function.
Notably, contrary to conventional beliefs where it is generally thought to be necessary for ψ to be close to θ_s in order to perform speech enhancement, our method as we will show later in Section <ref> is no longer constrained to this requirement. Therefore, we do not regard ψ as an estimate of θ_s.
This also implies that, in the context of the STOI-NS and IANS optimization problem, we never guarantee the distortionless property of speech as in the MVDR beamformer. However, as we will show later in Section <ref>, intelligibility enhancement is still possible using the optimal null angles
ϕ^⋆_STOI-NS and ϕ^⋆_IANS.
These two angles
can be interpreted as optimal null angles chosen to minimize the
impact of
the interference signal, speech distortion and RIRs on intelligibility, while maintaining a nearly distortionless response at ψ.
Since there is a chance that ψ=θ_i,
it is advisable to perform the IANS algorithm twice using two different θ_d values (e.g., ψ and ψ + 80^∘ in this study).
It is worth noting that dual-microphone array beamformers usually correspond to beampatterns exhibiting a large main lobe and side lobe owing to the limited degrees of freedom. In other words, we can use this property to construct a directive null, as in Eq. (<ref>), while preserving a certain amount of gain for signals coming from all angles, except those within the vicinity of the null.
We also note that small microphone arrays tend to have frequency-invariant beampatterns as explained in <cit.>, which can also be an advantage since beamformers that are sensitive to frequency variations tend to produce more unpredictable results.
§ THE IANS ALGORITHM
This section describes the IANS algorithm which solves the IANS optimization problem in (<ref>). The algorithm consists of two stages: the NSBF stage and the STOI-Net stage as shown in Fig. <ref>,
where results from the first stage will be passed on to the second stage. The following subsections will provide more detailed explanations about the IANS algorithm.
§.§ Stage 1: NSBF
The initial step of the IANS algorithm involves applying the STFT on the two signals x^(1)(t) and x^(2)(t) to obtain 𝐗^(1) and 𝐗^(2).
We then generate a set 𝒴_(STFT) containing P STFT matrices {𝐘^(1), ⋯𝐘^(P)} by sending the pair (𝐗^(1), 𝐗^(2)) into P NSBF beamformers {𝑓_θ_d =ψ, ϕ=𝒢_1, ⋯ ,𝑓_θ_d =ψ, ϕ=𝒢_P}⊂ℱ_NSBF. If ψ = 𝒢_p, where p∈{1, 2, ⋯, P}, we let 𝐘^(p)=𝐗^(1) instead of using
𝑓_θ_d =ψ, ϕ=ψ.
Note that parallel computing can be used since each computation of the elements of 𝒴_(STFT) is independent of each other. It is also worth pointing out that the time-invariant weight vectors in Eq. (<ref>) can be computed and stored beforehand to save time.
Since we will later send these into STOI-Net, we apply the inverse-STFT operation (iSTFT) on each element in 𝒴_(STFT) to perform peak normalization in the time domain. We denote this set as 𝒴_(time)'. We do this to match the training conditions of STOI-Net as we mentioned in Subsection <ref>.
§.§ Stage 2: STOI-Net
Following the peak normalization, we perform STFT on each element in 𝒴_(time)' to convert them back to the TF domain and extract their corresponding magnitude components. We denote the resulting set as 𝒴_(STFT)”.
We then pass each element in 𝒴_(STFT)” into STOI-Net to predict their utterance-based STOI score. These scores are then stored in a STOI-Net score vector α. The optimal null angle ϕ_IANS^⋆ for 𝑓_IANS^⋆ can be obtained as
ϕ_IANS^⋆ = 𝒢_argmax (α).
Moreover, in the case where we have access to the clean reference signal 𝐒, we can replace STOI-Net with the real STOI function in this stage and obtain a STOI score vector β. Therefore, the value of
ϕ_STOI-NS^⋆ for 𝑓_STOI-NS^⋆ can be expressed as
ϕ_STOI-NS^⋆ = 𝒢_argmax (β).
In this study, the pre-trained STOI-Net model without the attention layer was directly obtained from the previous research[https://github.com/dhimasryan/STOI-Net] without any modifications, such as adaptation, retraining, or fine-tuning, for the MCSE task.
§ EXPERIMENTAL RESULTS AND ANALYSIS
§.§ Experimental setup
In this study, the Pyroomacoustics package <cit.> was utilized to simulate the signal model in Eq. (<ref>) using the image source method <cit.> with the following parameters. The simulated room has dimensions of [5m, 6m, 4m] with the RT60 parameter set to 150ms and the sound speed c set to 343m/s. The center of the microphone array is located at [2.5m, 3m, 1m]. The distance between the two microphones is set to ℓ=8mm with the array being parallel to the x-axis and the reference microphone being the microphone on the right.
The speech DOA θ_s is set to 90^∘, while the interference DOA θ_i can be one of four predefined directions: 22.5^∘, 67.5^∘, 112.5^∘, or 157.5^∘.
IANS then uses 512-point Hamming windows with
50% overlap to process the incoming signals.
The set 𝒢 is a uniform
grid over the interval [0^∘, 180^∘] with an angular resolution of 2^∘ (i.e., 91 angular values).
Additionally, IANS was evaluated using two values for ψ: 0^∘, representing the largest angle difference from θ_s, and 80^∘, which is relatively closer to θ_s.
Note that the values of ℓ and c are assumed known to the IANS algorithm. Additionally, the value of ϵ=1.11 × 10^-16.
Our experiments can be classified into two parts.
The first part uses an English dataset, namely, the Wall Street Journal <cit.> eval92 evaluation set.
From eval92, we first selected two male and two female speakers, and chose one utterance from each speaker to form the source signal. Babble and car noises in the NOISEUS corpus <cit.> and pink noise in NOISEX-92 <cit.> were applied as the interference. Five signal-to-interference ratios (SIRs), namely, -10, -5, 0, 5, and 10 dB, were used to create noisy utterances. The SIRs were mixed with respect to the first microphone as suggested by the Pyroomacoustics documentation.
Therefore, 240 source interference pairs (four clean utterances, three noises, four interference angles, and five SIRs) were used to form the English testing set (denoted as “WSJ”).
For the second part of experiments, we used a Mandarin dataset, namely, the Taiwan Mandarin Hearing in Noise Test <cit.> corpus, comprising 320 sentences. Two male speakers and two female speakers were selected from the dataset. One utterance, recorded in a noise-free environment, was selected from each speaker as the speech source.
For the interference signal, we chose three noise signals from the DEMAND <cit.> dataset: “tmetro", “pstation", and “npark."
Like in WSJ,
the aforementioned SIRs were used to create 240 source-interference pairs (four clean utterances, three noises, four interference angles, and five SIRs) for the Mandarin testing set (denoted as “TMHINT”). It is worth noting that STOI-Net was previously trained on the training set of the original Wall Street Journal dataset. The eval92 set was used to evaluate the generalization performance of STOI-Net.
Therefore, the English and Mandarin datasets in this study correspond to the matched and mismatched languages for STOI-Net, respectively. All single-channel recordings were sampled at 16kHz.
The experimental performance was evaluated in terms of STOI
and the wideband extension of the perceptual evaluation of speech quality (PESQ) <cit.> metric.
§.§ Evaluation results
For both testing sets, we labeled the enhanced results using the IANS algorithm with
θ_d=ψ as “IANS_ψ.”
Noisy utterances (labeled as “Noisy") received by the first microphone were used as the baseline.
Moreover, we also compared our IANS result with two additional systems. The first system
is the STOI-NS system which optimizes the STOI-NS problem given the clean reference 𝐒 for all utterances. The optimization procedure was detailed in Section <ref>.
Like in “IANS_ψ", we represent the results from STOI-NS with
θ_d=ψ as “STOI-NS_ψ.” For the second system, NSBF was performed by setting θ_d=θ_s and ϕ=θ_i. This system has the advantage of knowing the true DOAs of the speech and interference signals. Therefore, we label the corresponding results as “T-NSBF."
For the WSJ evaluation set, we list the average STOI and PESQ scores for all 240 utterances of “Noisy”, “IANS_0^∘”, “STOI-NS_0^∘”, and “T-NSBF” in Table <ref>. From the table, we can see that the STOI and PESQ scores for “IANS_0^∘" are higher than those for “Noisy", indicating an improvement in the intelligibility and quality of noisy speech signals from the English dataset using the proposed approach.
Table <ref> lists the STOI and PESQ scores of “Noisy”, “IANS_0^∘”, “STOI-NS_0^∘”, and “T-NSBF” associated with the TMHINT database. From the table, the improved metric performances from “Noisy" to “IANS_0^∘" confirm that the proposed IANS method can effectively enhance the intelligibility and sound quality of
noise-corrupted utterances. Notably, in both Tables <ref> and <ref>, STOI-NS_0^∘ has the highest STOI and PESQ scores on average, indicating that in these two experiments, if we properly choose the null angle to be ϕ_STOI-NS^⋆, we generate results with even higher intelligibility and quality than the results from null-steering beamforming where we had the prior knowledge of the DOAs of the speech and interference signals.
One potential factor that may have influenced this outcome is the non-anechoic nature of the room, resulting in signals propagating through multiple pathways. Hence, nulling the angle θ_i may not be the optimal choice for STOI.
Next, we will further investigate how different values of ψ affects the performance of “IANS_ψ” and “STOI-NS_ψ.” Specifically, we compared the results obtained using ψ = 0^∘ and ψ = 80^∘.
These results are presented in Tables <ref> and <ref>, which correspond to the WSJ and TMHINT databases, respectively.
From both tables, when comparing “IANS_0^∘" with “IANS_80^∘" and “STOI-NS_0^∘" with “STOI-NS_80^∘", we can see that, even though the results corresponding to ψ = 80^∘ yield higher PESQ scores, there is essentially no difference in STOI.
This implies that the large 90^∘ difference between ψ and θ_s has an insignificant effect on the ability of the IANS algorithm to generate intelligibility-enhanced results in our experiments.
Finally, we present an additional analysis of α and β in a particular scenario (Scenario A) to gain further insight into the similarities between the IANS and STOI-NS problems. The scenario consists of a female speaker from the WSJ dataset being interfered by the babble noise coming from a 22.5^∘ angle (i.e., θ_i=22.5^∘) with the SIR set to 0 dB. We let ψ=0^∘ for both IANS and STOI-NS, which means that the STOI value in β corresponding to ϕ=0^∘ is the STOI score of the unprocessed signal at the reference microphone as we explained in Subsection <ref>. The score values in both α and β are represented by the two curves depicted in Fig. <ref>.
The x-axis represents the values of ϕ in degrees, whereas the y-axis represents the values of α and β.
From the figure, we can see that the two curves have similar characteristics. Specifically, the lowest values for α and β both correspond to ϕ=θ_s=90^∘. Since both 𝑓_IANS^⋆ and 𝑓_STOI-NS^⋆ output results corresponding to the largest value in their respective score vectors, this suggests that they are both effective in preventing the speech signal from being severely attenuated in Scenario A.
Moreover, maximum values of α and β occur at ϕ_IANS^⋆=12^∘ and ϕ_STOI-NS^⋆=16^∘, respectively.
The corresponding STOI scores for
𝑓_IANS^⋆ and 𝑓_STOI-NS^⋆ are 0.902 (i.e., β_argmax(α)) and 0.903 (i.e., max(β)), respectively, which are at least 0.219 points higher than the STOI score of “Noisy" at 0.683, indicating the effectiveness in STOI enhancement of our IANS algorithm.
§ CONCLUSION AND FUTURE WORK
In this paper, we proposed a novel intelligibility-based optimization problem (i.e., the IANS problem) along with its corresponding enhancement system, the IANS beamformer.
The system determines the optimal output speech with the highest intelligibility scores by combining the NSBF and STOI-Net modules, where NSBF processes the input recordings and STOI-Net provides STOI predictions.
We conducted experiments using cross-lingual datasets (Mandarin and English). The experimental results show that the proposed IANS system can effectively map the input signals to intelligibility and quality enhanced speech. It was also demonstrated that IANS produces robust performance regardless of whether the distortionless response is steered near the direction of the speech source.
In the future, we will evaluate the combination of beamforming systems with different evaluation modules, such as quality and mean opinion score assessment models <cit.>, and test our system in more complex noisy environments.
In addition, we will integrate single-channel speech enhancement methods with IANS to further enhance speech signals.
IEEEbib
|
http://arxiv.org/abs/2307.04234v1 | 20230709172453 | Extreme N-emitters at high-redshift: signatures of supermassive stars and globular cluster or black hole formation in action? | [
"R. Marques-Chaves",
"D. Schaerer",
"A. Kuruvanthodi",
"D. Korber",
"N. Prantzos",
"C. Charbonnel",
"A. Weibel",
"Y. I. Izotov",
"M. Messa",
"G. Brammer",
"M. Dessauges-Zavadsky",
"P. Oesch"
] | astro-ph.GA | [
"astro-ph.GA"
] |
Observatoire de Genève, Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland
CNRS, IRAP, 14 Avenue E. Belin, 31400 Toulouse, France
Institut d'Astrophysique de Paris, UMR 7095 CNRS, Sorbonne Université, 98bis, Bd Arago, 75014 Paris, France
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 14-b Metrolohichna str., Kyiv, 03143, Ukraine
The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden
Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark
Marques-Chaves et al.
Extreme N-emitters at high-redshift
Recent JWST spectroscopic observations of the z=10.6 galaxy GN-z11 have revealed a very peculiar UV spectrum showing intense emission lines of nitrogen, which are generally not detected in galaxy spectra. This observation indicates a super-solar N/O abundance ratio at low metallicity, resembling only the abundances seen in globular cluster (GC) stars. This discovery suggests that we might be seeing proto-GCs in formation or possibly even signatures of supermassive stars.
To examine if other objects with strong N iv and/or Niii emission lines (N-emitters, hereafter) exist and to better understand their origin and nature, we have examined available JWST spectra and data from the literature.
Using the NIRSpec/JWST observations from CEERS we found an extreme N-emitter, at z=8.6782 showing intense and emission.
From the observed rest-UV and optical lines we conclude that it is compatible with photoionization from stars and we determine accurate abundances for C, N, O, and Ne, relative to H.
We also (re-)analyze other N-emitters from the literature, including three lensed objects at z=2.3-3.5 (the Sunburst cluster, SMACS2031, and Lynx arc) and a low-redshift compact galaxy, Mrk 996. We compare the observed abundance ratios to observations from normal star-forming galaxies, predicted wind yields from massive stars and predictions from supermassive stars (SMS with ∼ 10^4-10^5 ).
For we find a highly supersolar ratio log( N/O)=-0.18 ± 0.11, and abundances of log( C/O)= -0.75 ± 0.11 and log( Ne/O)= -0.63 ± 0.07, which are normal compared to other galaxies at the low metallicity (= 7.70 ± 0.18) of this galaxy. The three lensed N-emitters also show strongly enhanced N/O ratios and two of them normal C/O.
The high N/O abundances can be reproduced by massive star winds assuming a special timing and essentially no dilution with the ambient ISM. Alternatively, these N/O ratios can be explained by mixing the ejecta of SMS with comparable amounts of unenriched ISM. Massive star ejecta (from WR stars) are needed to explain the galaxies with enhanced C/O (Lynx arc, Mrk 996). On the other hand, SMS in the “conveyer-belt model” put forward to explain globular clusters, predict a high N/O and small changes in C/O, compatible with , the Sunburst cluster, SMACS2031, and GN-z11.
Based on the chemical abundances, possible enrichment scenarios and other properties, such as their compactness and high ISM density, we discuss which objects could contain proto-GCs. We suggest that this is the case for , SMACS2031, and the Sunburst cluster. Enrichment in the Lynx arc and Mrk 996 is likely due to normal massive stars (WR), which implies that the star-forming regions in these objects cannot become GCs. Finally, we propose that some N-emitters enriched by SMS could also have formed intermediate mass black holes, and we suggest that this might be the case for GN-z11.
Our observations and analysis reinforce the suggested link between some N-emitters and proto-GC formation, which is supported both by empirical evidence and quantitative models. Furthermore, the observations provide possible evidence for the presence of supermassive stars in the early Universe (z>8) and at z ∼ 2-3. Our analysis also suggests that the origin and nature of the N-emitters is diverse, including also objects like GN-z11 which possibly host an AGN.
Extreme N-emitters at high-redshift: signatures of supermassive stars and globular cluster or black hole formation in action ?
R. Marques-Chaves1,
D. Schaerer1,2,
A. Kuruvanthodi1,
D. Korber1,
N. Prantzos3,
C. Charbonnel1,2,
A. Weibel1,
Y. I. Izotov4,
M. Messa1,5,
G. Brammer6,
M. Dessauges-Zavadsky1,
P. Oesch1,6
Received date; accepted date
========================================================================================================================================================================================================
§ INTRODUCTION
Long known as the most distant spectroscopically-confirmed galaxy <cit.>, GN-z11 has recently lead to new exciting and intriguing results, after the first spectra of this galaxy were obtained with the JWST. Indeed, the JWST/NIRSpec observations of <cit.> allowed to confirm a very high redshift of this source (z=10.60) and showed the presence of hydrogen, carbon, oxygen, magnesium, and neon emission lines in the rest-UV and rest-optical spectrum, often seen in star-forming galaxies at low-redshift and detected at z ∼ 4-8 in other JWST spectra <cit.>.
Most surprisingly, however, the spectrum of GN-z11 revealed the presence of strong and lines <cit.>, which are very rarely detected in galaxies <cit.>. Furthermore, the object is found to be very compact <cit.>, which could indicate the presence of massive compact star clusters or point to an active galactic nucleus (AGN) <cit.>.
The discovery of the peculiar emission line spectrum has triggered a series of papers discussing in particular their origin and the nature of GN-z11.
<cit.> first suggested that the strong N emission lines may imply an unusually high N/O abundance. They also discussed whether the emission would be powered by star formation or photoionization from an AGN, without reaching clear conclusions on this issue. The quantitative analysis of the emission line spectrum of GN-z11 by <cit.> confirmed the high N/O abundance, with a lower limit of four times solar, finding also possibly a less extreme C/O ratio, and a metallicity (O/H), which is sub-solar, although not well constrained.
Using a suite of photoionization models, <cit.> inferred the N/O abundance with a lower uncertainty and constrained the metallicity to = 7.84^+0.06_-0.05, confirming in particular a large overabundance of N/O ≈ 3 × solar.
The finding of an exceptionally high N/O abundance at low metallicity (typically ten times the normal N/O value at this O/H) has triggered different speculations about the sources and processes explaining this enrichment. The scenarii discussed include enrichment from massive stars winds (WR stars) or AGB stars, i.e. relatively “classical scenarii”, or more “exotic” options such as pollution from PopIII star-formation, tidal disruption of stars from encounters with black holes, ejecta from very massive stars formed through collisions in dense clusters, and supermassive stars <cit.>. Supermassive stars, for example, have been invoked by <cit.> and <cit.> since very strong enrichment of N and low metallicity is difficult to explain and requires fairly fined-tuned conditions with classical scenarios <cit.>. Furthermore, such stars (with masses 1000 ) have been proposed to form by runaway collisions in very dense stellar clusters, and they could explain the long-standing problem of multiple stellar populations and peculiar abundance patterns observed in globular clusters (GC), as discussed by <cit.> and <cit.>. If correct, this would probably represent the first observational evidence of supermassive stars, which are also of great interest, for example for understanding the seeds of supermassive black holes
<cit.>
.
Not only the abundance ratios observed in GN-z11 resemble those of GC stars. Its compactness and high ISM density also indicate conditions expected in young very massive clusters, which could be proto-GCs <cit.>.
GN-z11 might thus also be the first high-redshift object where the long sought-for peculiar abundance patterns characterizing GCs are observed
<cit.>.
These exciting and surprising findings obviously pose the question of the uniqueness of GN-z11, beg for more examples, and call for a better understanding of similar objects, if they exist.
Indeed, although very rare, other galaxies showing emission lines of or in the UV (referred to as N-emitters subsequently) are known, as pointed out by <cit.> and found in the compilation of <cit.>. Apart from objects clearly identified as AGN, the Lynx arc, a lensed z=3.36 galaxy identified for and emission is probably the first N-emitter studied in detail <cit.>. From photoionization modeling <cit.> derive a high N/O ratio and sub-solar metallicity.
Another strongly lensed object at z=2.37, the multiply-imaged compact star cluster in the Sunburst arc which has extensively been studied in recent years <cit.>, shows emission, as shown in the high S/N spectrum of <cit.>. <cit.> have shown that N/O is also elevated (∼ 4 × solar) at a metallicity ∼ 1/5 solar.
Finally, in the low-redshift Universe, Mrk 996 uniquely stands out as the only galaxy showing strong emission in the UV <cit.>, and this blue compact dwarf galaxy has long been known as very peculiar, showing e.g. a high electron density, the presence of strong emission lines from WR stars in the optical, and a high N/O abundance, at least in its core <cit.>.
Here we present a detailed analysis of the z=8.68 galaxy observed with NIRSpec/JWST by the public CEERS survey <cit.>. This object has previously been studied by several authors <cit.>, but none of these have analysed the carbon and nitrogen abundance and its rest-UV spectrum.
Only very recently, <cit.> have analyzed the UV spectrum in detail. Similarly to GN-z11, this galaxy exhibits a very peculiar rest-UV spectrum, making it clearly an N-emitter. Showing numerous emission lines of H, C, N, O, Ne, and the auroral line, it allows us to accurately determine the chemical abundances of these elements and offers thus a unique opportunity to study the second N-emitter in the early Universe and to enlarge the sample of these rare objects.
We also analyze the other known N-emitters and compare their properties to those of and GN-z11. Finally, we confront the observed abundance patterns with predictions from normal massive stars and with predicted enrichment patterns from supermassive stars.
The paper is structured as follows. In Sect. <ref> we describe the observational data, reduction, and measurements used in this work. We then discuss the nature of the ionizing source of (Sect. <ref>). The chemical abundances and other physical properties of are derived in Sect. <ref>. In Sect. <ref> we compare the abundance ratios of to other N-emitters and normal star-forming galaxies, and we present different chemical enrichment scenarios to explain them. We also discuss the possible link between and proto-GCs. The main results of our work are summarized in Sect. <ref>. Throughout this work, we assume concordance cosmology with Ω_ m = 0.274, Ω_Λ = 0.726, and H_0 = 70 Mpc^-1.
§ CEERS-1019: A NEW STRONG N EMITTER AT HIGH REDSHIFT
CEERS-1019 (α, δ [J2000] = 215.0354^∘, 52.8907^∘) was initially identified as a z_ phot≃ 8.6 dropout galaxy by <cit.> and spectroscopically confirmed at z_ spec = 8.683 by <cit.> through strong Lyα emission (see also and ). It is one of the most distant Lyα emitter known and is thought to reside in an over-dense region and ionized bubble boosting substantially its Lyα transmission <cit.>. <cit.> also report a tentative detection of N v λ 1240 (4.6σ), suggesting a hard ionizing spectrum of this source.
Recently, much deeper spectroscopy of CEERS-1019 was reported and analyzed by <cit.>, <cit.>, and <cit.> using NIRSpec, along with NIRCam and MIRI imaging. Although with some discrepancies, these works derived important physical properties of CEERS-1019 such as its stellar mass (log(M_⋆/M_⊙≃ 8.7-10.1), gas-phase metallicities (≃ 7.6-8.0), ionizing indicators (e.g., O32 ≃ 13-18), among others. Interestingly, <cit.> reported a tentative (2.5σ) detection of a broad component in Hβ that could be related to AGN activity (the presence of an AGN will be further discussed in Section <ref>). Here, we re-analyze the available JWST data of CEERS-1019.
§.§ JWST NIRSpec and NIRCam observations
JWST/NIRSpec spectra are available for CEERS-1019 as part of the Cosmic Evolution Early Release Science (CEERS[<https://ceers.github.io/>]; ) program. These observations include both low-resolution PRISM and medium-resolution grating (G140M/F100LP, G235M/F170LP, and G395M/F290LP), providing spectral resolution of R≃ 100 and R≃ 1000, respectively, and a spectral coverage ≃ 1-5μm. Standard 3-shutter slits and a 3-point nodding pattern were used. The total exposure time for each medium-resolution grating was 3107 seconds, split into three individual exposures of 14 groups each. Deeper observations were obtained with the low-resolution PRISM, with a total exposure time of 6214 seconds. Both PRISM and medium-resolution observations were obtained with an aperture position angle PA ≃ 89.32 deg (see Figure <ref>).
Data reduction was performed using the official JWST pipeline [<https://jwst-pipeline.readthedocs.io/>] for Level 1 data products and msaexp[<https://github.com/gbrammer/msaexp>] for Levels 2 and 3. Bias and dark current are subtracted followed by the correction of the 1/f noise and the “snowball” events. We use the calibration reference data system (CRDS) context jwst_1063.pmap to correct spectra for flat-field and implement the wavelength and photometric calibrations. 2D spectra of each slitlet are then drizzle-combined and the background is subtracted following the three-shutter dither pattern.
Finally, 1D spectra are extracted using the inverse-variance weighted kernel following <cit.>. Figure <ref> shows the NIRSpec spectra of CEERS-1019.
CEERS-1019 was also observed with JWST/NIRCam with the F115W, F150W, F200W, F277W, F356W, F410M, and F444W filters with exposure times of ∼ 3000 seconds <cit.>.
NIRCam images were reduced using the grizli reduction pipeline <cit.>, which includes procedures for masking the “snowball” artifacts and minimizing the impact of 1/f noise. Photometry of is performed using SExtractor <cit.> in dual mode.
For each NIRCam filter, we use the point-spread functions (PSFs) provided by G. Brammer within the grizli PSF library,[ <https://github.com/gbrammer/grizli-psf-library> ] which are based on models from webbpsf <cit.>.
Images are then PSF-matched to F444W, which has the largest PSF within the NIRCam filters. We measure the flux of in each filter using a circular aperture of 0.16^'' radius (4 pix) and apply an aperture correction derived in F444W using the "FLUX_AUTO" measured in a Kron-like aperture with default Kron parameters. Then, we scale all fluxes to total fluxes based on the encircled energy of the circularized Kron aperture on the F444W PSF from webbpsf (see Weibel et al. in prep. for more details) As shown in the bottom left panel of Figure <ref>, shows a complex morphology with three compact clumps.
§.§ Emission line measurements
As shown in Figure <ref>, presents intense nebular emission in the rest-frame UV and optical. As a first step, we determine the systemic redshift of using well-detected (≥ 10σ) and uncontaminated (i.e., not blended) emission lines detected in the G395M spectrum. Using the centroids of , , , and we derive the mean value and scatter of z_ sys=8.6782 ± 0.0006.
Several rest-frame UV lines are detected with high significance (≥ 5σ) in the deep PRISM spectrum, such as [Unless stated otherwise, refers to the sum of the forbidden [N iv] λ 1483 and the semi-forbidden N iv] λ 1486 lines, which are not resolved in the Prism spectrum.], , , and . This contrasts with the shallower medium-resolution G140M spectrum that shows only Lyα and N iv] at ≥ 3σ. Thus we use the much higher signal-to-noise ratio (S/N) PRISM spectrum to measure the fluxes of the rest-frame UV lines.
We fit simultaneously several Gaussian profiles to account for the emission of N iv], C iv, O iii], N iii], and C iii], and a power-law in the form of f_λ∝λ^β to fit the continuum level between 1.3-2.1 μm (λ_0≃ 1300-2200Å). Since these lines are not resolved in the PRISM spectrum,[N iv] λ 1486 presents an observed line width FWHM = 394 ± 95 km s^-1 in the medium-resolution G140M spectrum.] we fixed the line widths of each line to the expected instrumental resolution at their corresponding wavelengths (R≃ 30-45)[<https://jwst-docs.stsci.edu/jwst-near-infrared-spectrograph/nirspec-instrumentation/nirspec-dispersers-and-filters>]. We repeat the fit 500 times while bootstrapping the spectrum according to its 1σ error, and consider the standard deviation of each parameter as its 1σ uncertainty. Table <ref> summarizes our flux measurements.
Along with Lyα, N iv] is found to be the strongest emission line in the rest-UV, stronger than C iv and C iii] by a factor ≃ 1.8 and ≃ 1.5, respectively. We also infer a steep UV slope of β_ UV^ spec = -2.11 ± 0.09 from the spectrum, which is consistent with the photometric one (β_ UV^ phot = -2.11 ± 0.15) using the apparent magnitudes in the F150W and F200W filters (F150W=25.25± 0.08 and F200W=25.29 ± 0.07).
Flux measurements of rest-optical lines are obtained using the G395M spectrum, which presents a similar depth as the PRISM spectrum but with a much higher resolution.
Optical lines are fitted separately over relatively narrow spectral windows (100 Å, rest-frame) and a constant is assumed for the continuum level. The width of the lines is set as a free parameter.
In total, we detect up to ten optical emission lines with high significance (Table <ref>), including Balmer lines that are useful for the determination of dust attenuation.
To account for wavelength-dependent slit losses and absolute flux calibration, we derive the synthetic photometry of NIRSpec spectra (PRISM and gratings) through each NIRCam filter bandpass and matched it to that obtained from observed photometry. In this process, we use a wavelength-dependent polynomial function yielding scaling factors for the slit-loss correction ranging from approximately 2.0 (F150W) to 3.6 (F444W).
Using fluxes and equivalent widths of the detected Balmer lines , , and Hδ, we iteratively derive the dust attenuation E(B-V) = 0.12±0.11 using the <cit.> attenuation curve and following the methodology of <cit.>, which accounts for the internal extinction and underlying hydrogen stellar absorption.
Other important lines, such as those that are sensitive to the electron temperature (T_ e, ) and density (n_e, N iv] λλ 1483,1486 and λλ 3727,3729) are also detected and are analyzed in more detail in Section <ref>. For the N iv] and doublets we fit two Gaussian profiles with similar widths and use the expected separation between the two transitions. We find line ratios of F_1483/F_1486 = 0.50±0.22 and F_3727/F_3729 = 0.98±0.27 for the N iv] and doublets, respectively.
We also check for the presence of spectral features that are usually associated with Wolf-Rayet (WR) stars. The so-called blue bump around 4600–4700Å, encompassing the emission from N iii λ 4640, C iii λ 4650, and , is detected neither in the G395M nor the PRISM spectra. We derive a 3σ upper limit relative to Hβ of He ii/Hβ≤ 0.26.
Similarly, the rest-UV line is not detected. Despite its low resolution, the PRISM spectrum clearly suggests no emission at the expected position of He ii, while the close O iii] emission is well detected (see Figure <ref>).
§ THE NATURE OF THE IONIZING SOURCE: STAR FORMATION VERSUS AGN
We now discuss the nature of the ionizing source of , building upon the recent findings by <cit.> and <cit.>, who suggest a possible AGN activity. In their study, <cit.> reported the detection of N v λ 1242 emission with an integrated flux of (2.8±0.6)× 10^-18 erg s^-1 cm^-2 with a narrow profile FWHM <90 km s^-1 (unresolved in the MOSFIRE spectrum). However, the G140M spectrum does not exhibit any significant emission around the expected position of N v λλ 1238,1242 (Figure <ref>, top left).
By considering the flux uncertainty around 1.2μm from the G140M error spectrum and assuming an unresolved line width of FWHM = 352 km s^-1, we infer a 3σ limit of 1.44×10^-18 erg s^-1 cm^-2. This limit stands well below the reported value of <cit.>. Furthermore, according to <cit.>, N v λ 1238 is expected to be twice as strong as N v λ 1242 under standard conditions. Hence, considering the reported flux of <cit.> for N v λ 1242, we would expect 11.6σ and 5.8σ detections for N v λ 1238 and λ 1242, respectively. These limits, however, are incompatible with our observations.
<cit.> reported a 2.5σ detection of a broad (≃ 1200 km s^-1) component in Hβ using the medium-resolution NIRSpec G395M spectrum. This broad component is not seen in stronger, forbidden lines like [O iii] λλ 4960,5008,
from which they suggest conditions similar to the broad line region (BLR) of an AGN. Using our own reduction of the G395M spectrum and a dual-component Gaussian profile to Hβ, we find a 2.2σ detection for the broad component (Figure <ref>, top middle). Clearly, deeper observations of Hβ (or Hα with MIRI) are needed to unambiguously confirm the presence and nature of the broad component in Hβ, as already discussed and suggested by <cit.>. Indeed, if a single Gaussian profile is used to fit the Hβ profile, a good fit is also found without penalizing significantly the residuals (Figure <ref>, top right). In this case, we find FWHM (Hβ) = 452±68 km s^-1 which differs only by 1.2σ from the nominal FWHM = 369±16 km s^-1 obtained for the much brighter [O iii] λ 5008 line.
If the existence of this broad component can be confirmed and attributed to the BRL, it would be expected that high-ionization semi-forbidden lines such as N iv], C iv, or C iii], which probe high-density regimes (n_ crit≳ 10^9 cm^-3), would display similar broad Doppler widths as observed in type-1 AGNs <cit.>.
However, these lines appear narrow in , especially N iv] which exhibits a high-significance detection and an intrinsic FWHM ≃ 160 km s^-1 after correcting for instrumental broadening.
Thus, our results suggest that the aforementioned semi-forbidden lines unlikely originate from the broad line region.
Instead, the properties of these lines, such as the narrow widths and the N iv] line ratio F_1483/F_1486 = 0.50±0.22 (implying densities n_e ≈ 10^4-5 , see Section <ref>), are consistent with narrow line regions of AGN or H ii regions. In the following, we discuss these two scenarios.
The lower panels of Figure <ref> present several diagnostic diagrams using different UV nebular lines: C iii]/He ii versus C iv/He ii, [O iii]/He ii, and N v/N iv]. Photoionization models of star-forming galaxies from <cit.> and narrow-line regions of AGN from <cit.> are shown in blue and red, respectively. In the right panel of Figure <ref> we show models of star-forming galaxies from the updated BOND grid using Cloudy <cit.>, which also includes N iv] and is available from the 3MdB[<https://sites.google.com/site/mexicanmillionmodels/>] <cit.>. These models encompass a wide range of parameters, including the ionizing parameter (-4.0 ≤log U ≤ -1.0), hydrogen number density (10^2≤ n_ H / cm^3≤ 10^4), and the power law index of the ionizing spectrum (-2.0 ≤α≤ -1.2). We have selected models with metallicities within the range 0.05 ≤ Z/Z_⊙≤ 0.20, which corresponds to the inferred metallicity for (12+log(O/H) =7.70±0.18, as indicated in Table <ref>). As illustrated in this figure, the position of (indicated by the blue circle) aligns with the predictions of star-forming models in all diagnostic diagrams. Clearly, the absence of He ii and N v, which probe energies >54 eV and >77 eV, respectively, places far away from the region occupied by AGN models. It is worth noting that <cit.> suggested recently that the high N iv]/N iii] ratio observed in is hardly reproduced by star formation models, pointing to an AGN contribution. However, the 3MdB photoionization models used here do predict very high ratios even well above the observed N iv]/N iii] =5.1±2.2, although requiring fairly high ionization parameters (log(U)≲ -2).
Other spectral features observed in , such as the intense N iv] emission compared to other UV lines (N iv]/C iv ≃ 1.8, N iv]/C iii] ≃ 1.5, N iv]/N v ≥ 2.6), and narrow profiles (FWHM ≃ 160 km s^-1 for N iv]) differ from those observed in AGNs, even those showing unusually strong Nitrogen lines <cit.>. The so-called Nitrogen-loud QSOs exhibit much weaker N iv] compared to other lines (e.g., N iv]/C iv≃ 0.02-0.38, , ) and, as expected, they present very broad Doppler widths (FWHM ≃ 1500-6000 km s^-1, ). Similarly, some type-2 AGNs also present N iv] emission <cit.>, but notably weaker compared to other high-ionization lines (N iv]/C iv ≃ 0.15, N iv]/C iii] ≃ 0.34, or N iv]/N v ≃ 0.30; ). An exception may be GS-14, a type 1.8 AGN at z≃ 5.55 recently analyzed by <cit.>. GS-14 exhibits broad components in Hydrogen and Helium lines (FWHM ≃ 3400 km s^-1, ) as well as narrow N iv] emission (FWHM ≃ 430 km s^-1, , ), but it also shows clear nebular emission in N v λ 1240 and O vi λ 1033 <cit.> which are not detected in .
In contrast, the spectrum of resembles those of other, yet also rare star-forming galaxies with intense emission in Nitrogen lines. Examples such as the Lynx arc <cit.>, SMACS-2031 <cit.>, Mrk 996 <cit.>, and the Sunburst cluster show narrow and prominent N iv] and/or [N iii] lines suggestive of high electron temperatures and densities like (see Section <ref>) and without any hint of AGN activity. The bottom panels of Figure <ref> also show the location of these strong N-emitters, all consistent with star-forming models like . The case of GN-z11, another strong N-emitter reported by <cit.>, appears to be ambiguous, consistent with both models of AGN and star formation, as already discussed in <cit.> and <cit.>.
In conclusion, our results suggest that, regardless of the presence of an AGN whose confirmation awaits deeper data, the high-ionization lines observed in are consistent with stellar photoionization.
§ OBSERVATIONAL AND DERIVED PHYSICAL PROPERTIES OF
§.§ ISM properties and element abundances
The rich set of emission lines detected from the rest-frame UV-to-optical spectrum allows us to determine
the electron temperature and density in the gas and the detailed abundances of numerous elements including
H, C, N, O, and Ne. The derived quantities are summarized in Table <ref>.
§.§ Electron temperature
To derive physical conditions and element abundances we follow the prescriptions
of <cit.>.
Briefly, these authors adopt the classical three zone model of the
H ii region with electron temperatures T_ e(O iii) for the high-ionization zone,
and T_ e(O ii) for the low-ionization zone.
The intermediate-ionization zone is not used here, since no such lines are detected.
The electron temperature T_ e(O iii) is derived both from the ratio of [O iii] line
fluxes λ4363/λ(4959+5007) and from the UV-to-optical line ratio of λ1660/λ5007.
The former ratio (rest-optical) is determined from the medium-resolution spectrum, the latter from the
PRISM spectrum.
In both cases we obtain T_e ≈ 18000 K, consistent within 1 σ, and with uncertainties between 1151 and 3252 K.
Subsequently, we adopt the electron temperature from the optical line ratios (T_e=18849 ± 3252 K) with the larger uncertainty, which is primarily due to the low S/N detection of .
The electron temperature in the low-ionization region is derived from relations obtained from the photoionization models of <cit.>.
§.§ Electron density
Several density indicators exist in the observed spectral range, but few can be used here in practice.
In the UV, the , , and doublets are density estimators.
However, the PRISM spectrum is of insufficient resolution to resolve any of these doublet lines.
is not detected, and has too low S/N in the medium-resolution spectrum.
Although of fairly low S/N, the doublet is detected with a ratio λ1483/λ1487 =0.50 ± 0.22 which indicates a fairly high electron density of n_e ≈ 10^4-5 <cit.>.
In the optical, the doublet is clearly detected, but not resolved from the medium-resolution spectra. Our measured line ratio λ3727/λ3729 = 0.98 ± 0.23 is consistent within the uncertainties with that obtained by <cit.> (0.639 ± 0.255), and compatible with n_e > 10^3 cm^-3 <cit.>.
The two density estimates could indicate a density gradient between the low and high ionization regions, but are also compatible with a single, relatively high density of n_e ≈ 10^4-5 , whose origin we discuss below.
In any case, the most important point to take away from this is that the electron density, although high, is lower than the critical densities of all the relevant emission lines used for the subsequent abundance determinations.
This holds for the (semi-)forbidden lines of [O iii] at 1666, 4363, 4959, 5007 (with critical densities n_ crit≥ 6.9 × 10^5 ), the two components of the doublet (n_ crit = 8.7 × 10^4 for 1907 and 10^9 for 1909), (n_ crit = 2 × 10^15 ), (a multiplet whose components have n_ crit≥ 10^9 ), (n_ crit = 3 × 10^9 ), and (n_ crit = 1 × 10^8 )
<cit.>.
Only the doublet, whose components have relatively low critical densities of n_ crit = 1 (4) × 10^3 for 3728 (3726), is therefore affected by the high density inferred for , whereas all other lines can safely be used to determine abundances, to which we now proceed.
§.§ Ionic and total metal abundances
The electron temperature T_ e(O iii) is used to obtain abundances of ions O^2+, N^3+, N^2+, C^3+, C^2+,
and Ne^2+; the temperature in the low-ionization region, T_ e(O ii), to derive the ionic abundance of O^+.
Ionic abundances are derived following <cit.> for the optical lines, and comparing different methods for the UV lines.
For C, N, and O, the observations provide two ionization stages, hence the ionic abundances will be
close to the total abundances, and we neglect further ionization corrections.
For Ne^2+ we use the ionization correction factor (ICF) following <cit.>. The results are listed in Table <ref>.
We derive a total oxygen abundance of = 7.70 ± 0.18, which is dominated by the ionic abundance of O^2+/H^+ (see Table <ref>).
Given the high density, could be decreased and hence the O^+/H^+ abundance underestimated.
However, in view of the high excitation observed from lines with high critical densities, it is likely that O^2+ is the dominant ionization stage over the majority of the region and hence the determination of O/H close to the correct value.
With available line detections the N/O abundance can be determined in different ways. First we use only the UV lines to compute the ionic abundance ratio (N^2++N^3+)/O^2+ using the expressions from <cit.> (V+04) and <cit.> (H+02), assuming the low-density regime. Then we determine N/H from the UV and optical line ratio (N and ) and use O/H determined from the optical lines. Both methods, marked as "UV only" and "UV+opt" respectively, yield values compatible within the errors, and consistent with a high N/O abundance log( N/O)≈ -0.15 ± 0.17.
Similarly, for C/O we use the expressions from <cit.>, <cit.> (PM17, and <cit.> (I+23) using either only the rest-UV or a combination of the UV and optical lines. As seen from Table <ref> the ionic abundance ratios derived in this manner are compatible within uncertainties. For the total C/O abundance we adopt log( C/O)=-0.75±0.11 as our default value. The C/O ratio is therefore clearly subsolar, and in fact very similar to the average of normal star-forming galaxies at the same O/H (see below).
Finally we also derive the Neon abundance from the and lines and applying an ICF from the oxygen lines, following <cit.>.
We find an abundance ratio of log( Ne/O) = -0.63 ± 0.07, somewhat higher than the average value of log( Ne/O) = -0.78 ± 0.01 determined for normal star-forming galaxies by <cit.> at the same metallicity.
Although the abundances derived here assume low densities they are not altered by density effects at the density derived for , as already discussed above. Most importantly, the critical densities for the , , and lines involved in the (N^2++N^3+)/O^2+ ratio derived from the UV are all very high (n_ crit > 10^9 ), which further shows that this important ionic abundance ratio can be determined accurately.
Taken together, the derived abundances of show that this object has a “metallicity” (O/H) of approximately 1/10 solar <cit.>, an exceptionally high N/O abundance, and a normal C/O abundance, when compared to galaxies of similar metallicity (see Fig. <ref>). The interpretation of these abundances and implications will be discussed below (Sect. <ref>).
§.§ Comparison with other studies and caveats
ISM properties and abundances of have been determined by several other studies, with whom we now briefly compared our results.
<cit.> argue that the doublet can be deblended, from which they infer an electron density of n_e = (1.9±0.2) × 10^3 . From inspection of the doublet they suggest that the density could be higher than n_e > 10^4 .
The density inferred here from the doublet (n_e ≈ 10^4 - 10^5 ) is compatible with their finding. Most importantly for the abundance determinations, all available density estimates indicate that the main emission lines should not be affected by density effects.
From their 3-σ detection of <cit.> inferred T_e=18630 ± 3682 K, in excellent agreement with our determination. Based on the T_e determination they infer =7.664 ± 0.508 from an average relation between T_e and O/H determined empirically by <cit.>.
<cit.> determined =7.72^+0.17_-0.14 using the direct method. Within the quoted uncertainties, our results agree with both of these determinations. A slightly higher O/H abundance (=7.97 ± 0.16), but still compatible with the uncertainties, has been derived by <cit.> using a less accurate R23 strong-line calibration.
Finally, assuming AGN models, <cit.> have obtained a higher metallicity for , but similar N/O, C/O, and Ne/O ratios as derived here.
Note also that the abundance ratios determined here assume a homogeneous medium both in abundance and density. If pockets of high density and enriched gas coexist with lower density gas with say normal abundance ratios, only a relatively small fraction of enriched gas – i.e. relatively low amounts of Nitrogen – might suffice to explain the observed emission line ratios, since the emissivity of the forbidden line depends on the density <cit.>. However, in this case the inferred N/O abundance would also be lower limit of the true N/O ratio in the enriched pocket.
§.§ Other physical properties
§.§.§ Morphology
As shown in the left panel of Figure <ref> shows a complex morphology in the NIRCam bands consistent with three different clumps/structures separated by ≃ 0.24^'', or ≃ 1.12 kpc at z=8.678 (4.68^'' kpc^-1). These clumps, labeled as A, B, and C as indicated in Figure <ref>, are very compact, only resolved in the NIRCam bands at short wavelengths.
To investigate the morphology of in more detail, we model the three galaxy substructures following accurately the methodology applied to the study of stellar clumps in <cit.> and <cit.>. Assuming that clumps have Gaussian profiles, we consider a 15×15 pixel region centered on the galaxy and we fit a model consisting of three 2D Gaussian functions, convolved to the NIRCam instrumental PSF in this field from the grizly library. The best fit to their observed profiles (given by least-squares minimization) returns their fluxes and sizes. We assume that the shape of each substructure is the same in all bands. For this reason, the fit is initially performed in F200W, chosen as the reference filter, and then the shape (size, axis ratio, and position angle) of each clump is kept fixed in the other filters, where only the source flux is fitted. Uncertainties are obtained from Monte Carlo sampling.
The results of the model analysis are presented in Table <ref>. Our findings indicate that the morphologies of the three clumps in are compact, with measured FWHMs of 48±5 mas, 62±15 mas, and 43±4 mas for clumps A, B, and C, respectively. Following <cit.> (see also: , , and ), the inferred FWHM suggest that these clumps are resolved, albeit slightly, as their sizes are larger than the pixel size of the NIRCam images (40 mas). Translating these measurements into half-light radii, we find r_ e = 112 ± 12 pc, 145±35 pc, and 101±9 pc for clumps A, B, and C, respectively.
§.§.§ Spectral Energy Distribution
We now analyze the spectral energy distributions (SEDs) of as a whole (named Total) as well as its sub-components (A, B, and C). We use the SED-fitting code CIGALE <cit.> using the available NIRCam photometry from F115W to F444W, covering the rest-frame wavelength ∼ 1200-4600Å. Stellar population models from <cit.> are used along with the <cit.> Initial Mass Function (IMF) and the Small Magellanic Cloud extinction curve (R_v=2.93, ). The metallicity is fixed to Z=0.004, the closest available value inferred for , and is assumed to be the same for nebular emission and starlight. The dust attenuation (E(B-V)) and ionization paramater (log(U)) are treated as free parameters, ranging from 0.0-0.5 mag and -3.5 to -1.0, respectively. Finally, we explore two different star-formation histories: a constant star-formation model applied to the integrated light of (Total) and instantaneous burst episodes for the three sub-components (A, B, and C). For the former, we include the flux measurements of the Hβ + [O iii] λλ 4960,5008 emission lines in the fitting process.
Starting with the integrated emission of (Total), the best-fit model, shown in black in the right panel of Figure <ref>, finds a continuous star-formation rate SFR=161±23 M_⊙ yr^-1 over 14±7 Myr. The stellar mass is M_⋆^ total/ M_⊙=(2.0±0.6)× 10^9 attenuated by E(B-V)=0.17±0.02, in agreement with the values reported in <cit.>.
For the three individual components A, B, and C we find burst masses of M_⋆^ A/ M_⊙=(5.7±0.5)× 10^8, M_⋆^ B/ M_⊙=(4.6±0.1)× 10^8, and M_⋆^ C/ M_⊙=(8.6±0.2)× 10^8, respectively. Clumps A and B are well-fitted with very young burst models, having ages of 4.0±0.26 Myr and 5.6±0.7 Myr, respectively. On the other hand, clump C is older than the other components, with a burst age of 15.0± 2.9 Myr. Indeed, the color obtained for clump C F356W - F444W = 0.32 ± 0.29 is significantly lower than those measured in clumps A and B, F356W - F444W ≃ 0.75-1.16, suggesting a weak contribution of nebular emission in F444W (e.g., Hβ and [O iii]), thus negligible star formation over the last ≲ 10 Myr.
§.§.§ Stellar mass and SFR surface densities
Based on the stellar masses and half-light radii obtained for the individual clumps (Table <ref>), we obtained high stellar mass surface densities of log(Σ_M)=3.86±0.11, 3.55±0.53, and 4.14±0.14 M_⊙ pc^-2 for clumps A, B, and C, respectively (defined as Σ_M = M_⋆ / (2 π r_ eff^2)). It is worth noting that the inferred values of Σ_M may even be higher if each substructure comprises multiple unresolved stellar systems. Nevertheless, these values are already comparable to the densest systems identified at high redshift by <cit.> or <cit.>, and significantly higher than the average log(Σ_M) ≃ 2 M_⊙ pc^-2 observed in nearby young clusters <cit.>. Similarly, the compactness index, defined as C_5 = (M_⋆/10^5 M_⊙)(r_ eff/pc^-1) is also high in the case of . It ranges from C_5≃ 30-90 depending on the clump, exceeding the values of old globular clusters and young massive clusters by at least one order of magnitude <cit.>, suggesting high cluster formation efficiencies <cit.>.
The SFR surface density is also found to be very high for clumps A and B with log(Σ_ SFR)=3.27±0.11 and 2.81±0.21 kpc^-2, respectively. In contrast, clump C does not show significant star formation over the last 10 Myr, yielding an upper limit of log(Σ_ SFR)<2.27 kpc^-2.
Finally, the derived mass and SFR surface densities in are comparable with those of other prominent N-emitters discussed below, such as GN-z11 (log(Σ_M) ∼ 4.6 M_⊙ pc^-2, ), SMACSJ2031 (log(Σ_M) ∼ 4.0 M_⊙ pc^-2, log(Σ_ SFR) ∼ 1.4 kpc^-2, ), the Sunburst cluster (log(Σ_M) ∼ 4.1 M_⊙ pc^-2, log(Σ_ SFR) ∼ 3.7 kpc^-2, ), or Mrk 996 (log(Σ_M) ∼ 2.8 M_⊙ pc^-2, ). This suggests a potential connection between compactness and a high production efficiency of nitrogen.
§.§ Mass of the enriched material
The total mass of enriched, ionized gas, which is directly observable, can easily be estimated assuming ionization equilibrium and a constant ISM density <cit.>:
M_ ionized = m_p Q_H/α_B n_e = 2.5 × 10^6 (10^3/n_e) (Q_H/10^54) ,
where Q_H is the ionizing photon production rate which can be determined from H recombination lines, n_e the electron density, m_p the proton mass, and α_B the recombination rate coefficient.
For we thus find M_ ionized∼ 1.2 × 10^5 , from the observed luminosity and adopting n_e=10^5 , very similar to M_ ionized∼ 2 × 10^5 inferred for GN-z11 by <cit.>. <cit.> argue that the amount of enriched gas in GN-z11 could be even smaller if the N-emitting gas is found at higher densities, as they suggest.
§ DISCUSSION
§.§ Observed heavy element abundances in comparison to “normal” objects
The main elemental abundance ratios derived for are shown in Fig. <ref>, and compared to measurements in other galaxies and regions. To do so we use in particular the recent CNO abundances determined and compiled by <cit.>, who primarily included data from low-redshift star-forming galaxies observed with HST/COS, and data on individual regions from the works of <cit.>, <cit.>, and <cit.>.
As well known, the majority of galaxies and regions follow a fairly well-defined sequence of N/O versus O/H and C/O versus O/H <cit.>, which can be understood with chemical evolution models <cit.>. In N/O, for example, only few strong outliers with a large nitrogen excess are known at low redshift <cit.>.
In comparison, clearly stands out by having an extremely high Nitrogen abundance, log( N/O) = -0.13 ± 0.11, which is approximately 5.6 times the solar ratio <cit.> and more than a factor 10 higher than the N/O values generally observed at similar metallicities (O/H). This exceptionally high N abundance reflects the very peculiar UV spectrum of , showing unusually strong Nitrogen lines.
In contrast to N/O, with log( C/O)=-0.75 ± 0.11, the C/O abundance is fairly normal for the observed metallicity. The Ne/O abundance, log( Ne/O)=-0.63 ± 0.07 is somewhat higher (by ∼ 0.15 dex) than the average value for normal star-forming galaxies derived by <cit.> at the same metallicity.
Interestingly, these observed abundance ratios of resemble those of globular cluster stars, similarly to what was pointed out by <cit.> and <cit.> for GN-z11. The origin of these peculiar abundances ratios will be discussed below.
§.§ Abundances in other N-emitters
Interestingly, the abundance ratios found in resemble those found by
<cit.> for the z=10.6 galaxy GN-z11 observed recently with JWST by <cit.>, which are shown by boxes in Fig. <ref>.
As shown, the abundances in GN-z11 suffer from large uncertainties, which are in particular due to the fact that the line is shifted beyond the range accessible with NIRSpec and no direct O/H abundance determination is possible for this object from the present data.
Using photoionization modeling, <cit.> have further constrained the abundances in GN-z11, obtaining total gas abundances of =7.84±0.06 and
log( N/O)=-0.38±0.05, which are quite similar to those obtained here for .
Clearly, both and GN-z11 are significantly enriched in Nitrogen, reaching exceptionally high N/O values.
The carbon abundance cannot be well constrained in GN-z11, since the electron temperature remains undetermined in this object. The allowed range, derived by <cit.>, is indicated in Fig. <ref>.
Very few other galaxies or regions with a high N/O abundance and/or clear detections of nebular lines of N in the UV can be found in the literature.
<cit.> list known AGN and galaxies with O vi, N v, or emission lines in the rest-UV. Among the non-AGN in their list one finds the peculiar galaxy named the Lynx arc (at z=3.36), which has been studied by <cit.> and <cit.>, although <cit.> have argued that this object may be an obscured QSO. According to the photoionization models of <cit.>, both the N/O and C/O abundance ratios of this object are elevated, as seen in Fig. <ref>. Although suspected, no direct signs of WR stars have been found in this object <cit.> and the inferred abundances are not explained.
Another object showing nebular emission is the strongly lensed galaxy SMACSJ2031.8-4036 at z=3.5 studied in detail by <cit.> and <cit.>. The available VLT observations (with XShooter and MUSE) cover both the rest-UV and optical domain, allowing the detection of numerous emission lines, and thus electron temperature, density and abundance determinations.
Interestingly, this object shows indications for high density (n_e 10^5 ) from the doublet and lower densities from other diagnostics <cit.>. The metallicity = 7.76 ± 0.03 is very similar to and it shows a normal C/O abundance (log( C/O)=-0.80 ± 0.09), according to <cit.>.
Inspection of their spectra, kindly provided by the authors, shows a clear detection of both and lines, which allows us to determine N/O from the
UV lines and the reported T_e using the same methods described above (see Sect. <ref>). We find a relatively high N abundance of log( N/O)=-0.66 ± 0.1, which we also report in Fig. <ref>.
Finally, we also find a normal Neon abundance of log( Ne/O)=-0.82 from the reported line fluxes.
In the list of <cit.> other non-AGN spectra showing UV lines of Nitrogen show only N v P-Cygni lines, which are most likely due to stellar emission, or are stacked spectra with weak detections, not suitable for our purpose.
Another high-redshift object where emission has recently been detected is the strongly lensed and multiply imaged stellar cluster at z=2.368 in the Sunburst arc <cit.>, an exceptional object studied in depth by various authors <cit.>. From a detailed analysis and photoionization modelling, <cit.> infer in particular a high N/O abundance ratio (log N/O = -0.21^+0.10_-0.11), and normal C/O and Ne/O ratios for a metallicity (O/H) of approximately ∼ 0.22 solar.
The N/O ratio of this object fares thus among the highest values, comparable to , and C/O is also similar, as also shown in Fig. <ref>.
To extend our comparison, we have also examined the low-redshift galaxy Mrk 996, which is a well-known Blue Compact Dwarf (BCD) galaxy with peculiar properties, such as a high electron density, broad emission line components in , and other lines, the presence of Wolf-Rayet stars of WN and WC type, and a high N/O abundance <cit.>.
This galaxy also shows N iii] and N iv] emission lines in the UV <cit.>.
From integral-field observations <cit.> have found a normal N abundance (log( N/O) ≈ -1.43) across the galaxy and a N-enhancement by a factor ∼ 20 (log( N/O) ≈ -0.13) in the broad line component, emitted in the central region. The two measurements are plotted in Fig. <ref>.
The C/O abundance of Mrk 996 can be derived from the and line ratio, which is taken from the
HST/COS observations from the CLASSY survey <cit.>, and adopting the electron temperature T_e=10^4 K from <cit.>. We find a high Carbon abundance of log( C/O)= -0.22, close to solar, for this galaxy. However, for its metallicity <cit.> the C/O abundance ratio is comparable to that of other galaxies and regions, hence not unusual.
Taken together we thus conclude that all of the six N-emitters show an elevated (supersolar) N/O abundance ratio, whereas the C/O abundance is normal in four of them, and only one of them (the Lynx arc) appears enhanced in C/O.
The observed and other properties of these objects are also summarized in Table <ref>.
We will now discuss possible scenarios to explain to observed abundance pattern.
§.§ Possible chemical enrichment scenarios
Galactic chemical evolution models are able to reproduce the observed average trends of the abundance ratios of CNO and H for “normal” galaxies <cit.>, although the evolution of Nitrogen has notoriously been more complicated to explain, since the observations show a behaviour like a primary element at low (subsolar) metallicity <cit.>.
To examine the conditions which may be more appropriate for low metallicity dwarf galaxies and regions, which dominate the current samples
of extra-galactic CNO measurements in galaxies (the samples shown here), various authors have studied the effects of variable or bursty star-formation
histories, outflows and different star-formation efficiencies. Again, such models are able to reproduce the average trends of C/O, N/O and C/N as a function of metallicity and they can also explain the observed scatter in the data, e.g. by the presence of burst phases <cit.>.
Since the observed abundance ratios of and possibly other N-emitters are, however, clearly more extreme than those of the bulk of galaxies studied so far, we need to examine the possible nucleosynthetic sources and the conditions capable to explain them. To do so, we first consider two quantitative scenarios, the first involving enrichment from normal massive stars, and the second nucleosynthesis from super-massive stars. These scenarii were considered in recent studies <cit.>.
§.§.§ Enrichment from massive stars – “WR-scenario”
It is well-known that the stellar winds of massive stars can carry important amounts of newly-created elements such as He and N (from H-burning, the latter resulting at the expense of C and O) or C and O (from He-burning); those elements appear at the stellar surfaces and are ejected by the winds
during the so-called Wolf-Rayet (WR) phases, with N enhanced in the WN
phase and C enhanced in the subsequent WC phase <cit.>. The stellar wind yields depend strongly on the initial mass and metallicity of the stars,
and also on other properties such as stellar rotation and the efficiency of mixing in the stellar interiors, or their evolution in close binary systems <cit.>.
Using the recent stellar yields from <cit.> we have computed the cumulative stellar wind yields of a simple stellar population as a function of time, for a <cit.> IMF, three different metallicities ([Fe/H]=-2, -1 and 0, respectively) and three different initial rotational velocities (V_ Rot=0, 150 and 300 km/s, respectively). The latter value of V_ Rot=300 km/s was adopted in <cit.> to discuss the observations of GN-z11.
Assuming that stars more massive than 20–25 do not explode but collapse and become black holes <cit.>, the stellar ejecta have exclusively a wind composition for several million years. In the first couple of Myr, that composition is the original one of the stellar envelope, then it is dominated by H-burning products and subsequently, by He-burning products.
To compare with observed abundance ratios <cit.> assumed a dilution of the wind ejecta with an equal amount of ISM. Here we assume no such mixing, thus maximizing the effect of the stellar winds on the composition. Physically, this may correspond to the situation where the winds of the previous O-star phase, operating for a few Myr, have opened a cavity in the ISM where the winds of the subsequent WR phase are expanding. Actually, there is mixture with pristine ISM material, since we include the winds released by all stars above 12 and in the considered period of 8 Myr the stars less massive than 20 do not reach the WR phase.
In Fig. <ref> we display the evolution of various quantities of the "WR scenario" for stars of [Fe/H]=-1, a value reasonably close to the metallicity of the extragalactic systems studied here. Results are shown up to 8 Myr after the formation of a stellar population of total mass 10^8 with a normal IMF <cit.>. During that period, stars below 25 have not yet ended their lives (by assumption), so that only the wind ejecta populate the cavity crafted by the winds and the radiation of the stars. The mass of the wind ejecta increases steadily, from ∼10^4 after the first Myr to ∼10^6 at 4 Myr and more slowly after that. In Sec. <ref> we discussed the amounts of ionized gas estimated in and GN-z11, which are compatible with the model results for this earliest period after the starburst (horizontal dashed lines in the top panel).
The evolution of the wind composition differs between the non-rotating and the rotating stars.
The former (solid red curves) have practically no mixing between their convective core and radiative envelope; in consequence, the signatures of H-burning (high N/O and N/C) appear abruptly in the wind, once the mass loss uncovers the former H-burning core. The latter (solid blue curves) undergo rotational mixing, bringing slowly the H-burning products to the surface; as a result, the N/O and N/C ratios increase slowly but steadily, up to the equilibrium value, which is similar to the case of non-rotating stars. The timescale for the appearance of high N abundance is ∼ 3 Myr, in good agreement with the time window inferred by <cit.> for GN-z11. About a Myr later, some amounts of He and He-burning products – mainly C and insignificant O amounts – appear in the wind ejecta of the most massive rotating stars (from 120 to ∼ 70 ) while the less massive ones never reach the WC phase; the combined effect is a strong increase of C/O, a strong decrease of N/C and a small variation of N/O. In contrast, none of the non-rotating stars reaches the WC phase at such low metallicity, and all the CNO ratios remain basically unchanged. After that, the situation is expected to change drastically, as the first SN from M<25 stars explode and eject their core material in the ISM.
As shown in Fig. <ref> in the early evolution of a stellar population, there is a period of several Myr during which the N/O ratio in the stellar winds reaches the high N/O ratios observed in and in the other N-emitters analyzed here.
However, rapidly after reaching the maximum N/O value, the carbon abundance also increases (very strongly in rotating star or less so without rotation), implying C/O and N/C ratios that are incompatible with the observations of , SMACS2031, and the Sunburst cluster over most of the time (see also Fig. <ref>). In the results displayed here, there is thus only a fairly short period of ∼ 0.5 Myr (yellow shaded area in Fig. <ref>) where all three ratios N/O, N/C, and C/O are compatible with the observations of for the case of rotating stars. In view of the timescales involved (several Myr), the probability of such an occurrence is small but certainly non-negligible. We note that this occurs rather early in the evolution of the starburst, but well within the time window found by the analysis of <cit.> for GN-z11 (violet horizontal segments in the 2nd and 3d panels). We also note that other stellar models than those used here could result in more extended periods of high N/O and N/C ratios. This could be the case, for instance, of stars rotating more rapidly than 300 km/s <cit.>, binary stars, or stars calculated with higher mass loss rates, etc. <cit.>.
On the other hand, for the central region of Mrk 996 which shows both N and C enrichment, we find that all the abundance ratios are well reproduced by the models. Furthermore, in this galaxy the WR-scenario is directly supported by the presence of WR stars both of WN and WC types <cit.>. Similarly, N and C enrichment found in the Lynx arc could also be explained by the WR scenario, and earlier studies have argued for the presence of WR stars, from emission line modelling of this peculiar object <cit.>.
Is there any direct evidence for WR stars in the N-emitters discussed here? In short, WR stars have been reported only in the low-z galaxy Mrk 996, as mentioned earlier.
In the spectral range covered by the observations of , the strongest WR features could be and in the rest-UV and the so-called blue WR-bump centered around . None of these features is detected in the current NIRSpec spectra and the same holds for GN-z11 <cit.>. However, the JWST spectra of these very high-z objects, and in particular for , are of insufficient spectral resolution and S/N to rule out, e.g., emission with equivalents widths 7-10 Å (depending on the adopted FWHM of the WR line), and therefore stellar populations comparable to those of Mrk 996, which has EW(1640)≈ 3-4 Å, cannot be ruled out from the present data.
The rest-UV spectrum of SMACS2031 from <cit.> also shows no clear feature of WR stars. is present with an EW(1640)=0.99 ± 0.1 Å, but it is only marginally broader than nebular emission lines.
The very high-S/N spectrum of the Sunburst cluster, discussed by <cit.>, also shows no signature of WR stars. Except for the nebular lines, the Sunburst spectrum resembles in fact strongly the spectrum of the well-known massive star cluster R136 in the LMC, which is known to be very young (∼ 1.5 Myr) and to host very massive stars with masses up to ∼ 200 <cit.>. The Sunburst cluster also appears to be too young to host WR stars.
Finally, <cit.> have suggested the presence of WR in the Lynx arc, in particular to explain the hard observed ionizing spectrum, but no direct signatures are detected in the relatively low S/N spectra available for this object.
In conclusion, except for Mrk 996 where the presence of important populations of WR stars (both of WN and WC types) is established, no direct evidence for WR stars is found in the other N-emitters studied here. However, this does not necessarily exclude the WR-scenario, since WR stars may be present below the detection threshold.
§.§.§ Enrichment from super-massive stars (M 1000 ) – SMS scenario
An alternate scenario, already invoked by
<cit.> to explain the high N-abundance in the compact galaxy GN-z11 at z=10.6 , is that of super-massive stars (SMS), which have previously been proposed to explain
the abundance anomalies of the multiple stellar populations seen in old Galactic and extra-galactic globular clusters (GC) and in extra-galactic massive star clusters with ages down to ∼ 1.7 Gyr
<cit.>.
In essence, this model proposes that gas accretion and collisions of proto-stars in the densest clusters lead to the runaway formation of one or several SMS, with masses M 10^3 that increase with the cluster mass.
During some time before two-body relaxation heats the cluster, this mostly convective SMS undergoes accretion (from proto-stars in the cluster and infalling gas) and it ejects processed matter, whose composition reflects the conditions in its hot H-burning core. Namely, the ejected material is strongly enriched in N, Na, and Al, and strongly depleted in O and C as a result of CNO, NeNa, and MgAl nuclear reactions at high temperature. As initially shown by <cit.>, the whole range of abundance anomalies (C-N, O-N, Na-O, Mg-Al anticorrelations) in GC stars is very well accounted for after dilution of the SMS ejecta with proto-GC gas.
The constant supply of unprocessed material to the SMS “freezes” its evolution close to the zero-age main sequence, preventing strong He-enrichment of the SMS yields, in agreement with GC multiple band photometry <cit.>. This also
solves the so-called “mass budget" problem encountered by all the other scenarios that try to explain the presence and properties of multiple stellar populations in globular clusters <cit.>.
For example, <cit.> find that a SMS forming into a dense cluster hosting 10^7 proto-stars can reach and process respectively ∼ 5% and ∼ 45% of the cluster mass. This is significantly higher than the ∼ 2% of wind mass ejected in the massive star scenario (cf. Fig. <ref>).
In particular, the super-linear scaling predicted between the amount of material nuclearly processed by the SMS and the cluster mass explains the observed increase of the fraction of
second population stars with GC mass <cit.>. This picture is dubbed the “conveyor-belt” SMS model.
The high amount of processed matter also implies that any additional matter ejected by the SMS during its final phase (once the conveyer-belt stops) will have very little impact on the final abundance ratios.
In Figs. <ref> and <ref> the solid lines show, for three different initial metallicities (0.34 Z_⊙, 0.12 Z_⊙, and 0.018 Z_⊙), the predicted chemical abundance ratios resulting from the mixture of ejecta of 10^4 M_⊙ SMS in the conveyer-belt scenario with different amounts of ISM gas with a normal, initial abundance (stellar models from ). The composition of the SMS ejecta reflects the yields from H-burning via the CNO-cycle. It is very strongly enriched in Nitrogen, with N/O >10, i.e. nearly 100 times super-solar, and very strongly depleted in Oxygen and Carbon.
With an increasing fraction of matter from the SMS mixed into the ISM, the predicted N/O and N/C ratios increase strongly. The resulting mixture also shows a decreasing O/H abundance (metallicity) while C/O remains relatively constant.
The observed N/O ratio of GN-z11 and can be explained by mixing approximately equal amounts of SMS ejecta with ISM gas, as already shown by <cit.>. The N/O abundance of all other N emitters considered here could also be explained with the SMS scenario. The same is also true for the C/O and N/C abundance ratios, except for the two objects which show a high C/O ratio, Mrk 996 and the Lynx arc. As already mentioned before, C/O in these galaxies reveals the presence of both H- and He-burning products, which, in the case of Mrk 996, is compatible with its observed WR star population. In short, the comparison of the observed N/O, C/O, and N/C ratios suggests that , SMACS2031, and the Sunburst cluster might be explained by the SMS conveyor-belt scenario, implying that they should contain one or several proto-GC, and Mrk 996 and the Lynx arc by the WR scenario. From the available data and the lack of accurate C/O measurements, the case of GN-z11 remains inconclusive.
<cit.> have computed the composition of the material ejected through winds along the entire evolution of SMS with masses between 10^3 and 10^5 for 0.1 Z_⊙, neglecting the conveyor belt rejuvenation of the star discussed above (they assume that SMS form through gravitational collapse during the merger of gas-rich galaxies at high-z, see ).
In addition, they estimate if and when the SMS become GR unstable as they evolve, as well as the modifications of the composition of the material that can be ejected at the end of the life of the stars in the case they eventually explode due to the CNO cycle and the rp (rapid proton capture) process (for details see ).
Their 10^3 and 10^4 models – not shown here – predict strong N-enrichment on the main sequence, confirming the results of <cit.> and <cit.>. However, these two models do not become GR unstable and make it until carbon-oxygen burns. As a consequence, their winds reach super-solar C and O abundances because of the dredge-up of C and O from the core during central He-burning, and they are strongly enriched in He.
This implies that without undergoing the conveyor-belt episode that is required to solve the mass budget and the photometric constraints for the GC case, the total yields of such models cannot explain the GC abundance anomalies, nor can they explain the N/O and C/O ratios in CEERS-1019
and in GN-z11 as discussed by <cit.>.
On the other hand, <cit.> find that their 5 × 10^4 and 10^5 models at 0.1 Z_⊙ become GR unstable close to or at the end of the main sequence, implying that their winds contain super-solar N and sub-solar C and O before the stars eventually collapse to a black hole or are disrupted by a thermonuclear explosion.
The dashed lines in Figs. <ref> and <ref> show the range of abundances expected when the ejecta of their 10^5 model is diluted to various degrees with ISM of initial composition. In addition to the N-enrichment along the main sequence, this includes their estimate of the additional N that is produced during the expected CNO-powered explosion <cit.>.
This model accounts well for the observed abundance N/O ratios in , GN-z11, and SMACS2031.
And it also produces enough enriched material to be able to pollute sufficient ionized gas, i.e. masses in the observed range (see Sect. 4.7), as shown by <cit.>.
From this, we conclude that SMS over a wide range of masses can simultaneously explain the GC abundance anomalies and the N/O and C/O ratios in CEERS-1019, GN-z11, and SMACS2031, if they eject large quantities of H-processed material early on the main sequence, as predicted by the conveyor-belt SMS scenario <cit.>, or if the SMS sheds large amounts of processed material due to instabilities and an explosion during the CNO-cycle <cit.>.
In Sect. <ref> we will further argue whether the N-emitters are proto-GCs, and discuss possible implications of the SMS scenario, including the possible formation of an intermediate-mass black hole (IMBH).
§.§.§ Other scenarios to explain strong N emission
<cit.> have discussed different processes or sources which could explain the observed N-enhancement in GN-z11, including enrichment from AGB stars, pollution from Pop III star-formation, stellar encounters in dense star clusters, or tidal disruption of stars from encounters with black holes. The main conclusions of their qualitative discussion is that these explanations would need very fine-tuned conditions and that the origin of N-enrichment is currently largely unknown.
The predictions of classical chemical evolution models including AGB stars are shown e.g. in the studies of <cit.>. <cit.> also show predictions of such models in comparison with GN-z11. Indeed, as well known from earlier works, such models cannot produce high N/O abundance ratios at low metallicity
(as observed in the N-emitters discussed here), since these models include also the yields of massive stars and core-collapse supernovae, which produce large amounts of oxygen, and hence no extreme N/O ratios.
The pure WR-wind models of <cit.> are essentially the same as our massive star models (WR-scenario).
<cit.> have recently argued that GN-z11 shows signs of a type 1 AGN, with emission from very high density and a Broad Line Region (BLR). They further argue that the exceptionally high nitrogen abundance “becomes much less problematic” in the AGN scenario, for several reasons. First, they point out that several “nitrogen-loud” AGN have been found, making GN-z11 less peculiar. And second, they mention that only small amounts of enriched gas are needed if the observed gas is at very high densities. Finally, they mention supernovae from supermassive stellar progenitors, rapidly recycled secondary nitrogen production, or bloated atmospheres of giant/supergiant stars as possible sources of the observed enrichment, without providing quantitative estimates.
Clearly, the spectra of and the other N-emitters discussed here are very different from nitrogen-loud AGN, as discussed in Sect. <ref>. Furthermore, except for GN-z11 for which <cit.> show indications of densities n_H 10^10 typical of BLR, the densities inferred here are much lower, typically n ∼ 10^4-5 , and all observed emission line properties are compatible with photoionization from star-formation (Sect. <ref>). The qualitative scenarios sketched by <cit.> for GN-z11 may therefore not be applicable to the other N-emitters discussed here. In any case, more quantitative studies on the detailed chemical abundances of nitrogen-loud AGN and their source of enrichment could be of interest to better understand the common points and differences with other N-emitters.
For the Sunburst cluster, <cit.> proposed a model where the super star cluster is surrounded by low- and high-density photoionized clouds and regions (channels) through which ionizing radiation can escape, and they argue that only the high-density clouds in the vicinity of the star cluster are N-enriched and confined by strong external pressure. They estimate that ∼ 500 of nitrogen is needed – an amount which can be produced by the star cluster with a mass ∼ few × 10^7 – and suggest that it originates from young massive stars, ejected, e.g., in dense LBV winds or non-conservative binary mass transfer. SN ejecta are not favored, since the Sunburst is not enriched in C, and the inferred age (4 Myr) is consistent with this explanation.
The model of <cit.> is essentially the same as our massive star scenario, although they do not use a specific model to predict the chemical yields of the cluster and its temporal evolution, and our massive star scenario does not include ejecta from mass transfer in binary systems.
As already discussed above, such a scenario requires some specific “tuning”, in particular the selection of a fairly specific age at which the composition of ejecta matches the observed abundances. For the Sunburst cluster this seems very plausible; however, it is not clear if this could be generalized to and the other N-emitters.
§.§ Are and other N-emitters proto-GC in formation or related to the formation of intermediate-mass black holes ?
The unusually high N/O abundances derived for GN-z11 and the Sunburst arc and similarities with the abundance pattern of stars in globular clusters have led several authors to suggest a link between these peculiar objects and GC formation <cit.>.
With the finding of a highly supersolar N/O ratio and normal C/O in and similar results for other objects from the literature (in total six N-emitters analyzed here), the question of the nature of the N-emitters must be rediscussed in light of new and additional evidence. We summarize basic observational evidence and our favourite scenarii/explanations in Table <ref>.
First, the observed abundance ratios of N/O and C/O, which are accurately measured for five objects, suggest that two of them (the low-z galaxy Mrk 996 and the Lynx arc) are probably explained by pollution from WR stars, as discussed above.
If correct, it implies that the cluster(s) dominating presumably these objects cannot be progenitors of GCs. This is due to the fact that massive star wind scenario suffers from the so-called mass budget problem of GCs <cit.>, which basically means that the massive stars cannot produce sufficient amounts of enriched material to explain the observed population of “polluted” (second population) stars in GCs without this first population being much more massive than the second one, in contradiction with observations.
In Mrk 996 WR features are detected, and the presence of WR stars is suspected in the Lynx arc. We therefore suggest that they are somewhat peculiar star-forming galaxies (WR galaxies), although we note that <cit.> have also considered a hidden AGN to explain the emission line properties of the Lynx arc.
For , GN-z11, SMACS2031, and the Sunburst cluster, the N/O, C/O, and N/C ratios could be explained by the two scenarii discussed earlier, with the enriched matter originating from normal massive stars or from supermassive stars.
We favour the SMS scenario for several reasons. First, the WR scenario requires a very special and shorter timing than the SMS scenario. Second, these galaxies contain at least one sufficiently massive and compact region (the Sunburst cluster is of course a cluster) with extreme conditions (very high SFR and mass surface density), and unusually high ISM densities. Such rare conditions may be necessary for the formation of supermassive stars through runaway collisions and for the conveyer-belt model, as proposed by <cit.>. This would also naturally explain why N-emitters are rare. We therefore propose that , SMACS2031, and the Sunburst cluster have been enriched by SMS and that they host (or are) proto-GCs in star-forming galaxies. Finally, the finding of such objects at look-back times between 11.2–13.3 Gyr is also compatible with them hosting proto-GCs.
The case of GN-z11 may be somewhat different as it may host an AGN, as suggested by <cit.>. If the high density of the ionized gas (n_e 10^10 ) inferred by these authors is confirmed, it would significantly reduce the amount of ionized gas which needs to be polluted, but it still leaves the source of chemical enrichment unexplained <cit.>. However, this does not exclude pollution from one or several SMS, which might even have seeded the “small” massive black hole (with log(M_ BH/) ∼ 6.2±0.3) or contributed to its growth.
Indeed, the final fate of SMS is difficult to predict since in addition to metallicity and mass, other input parameters of the stellar models (mass loss, convection, overshooting, rotation, etc.) may impact the occurrence of the GR instability, its timing, and whether the collapse would trigger an explosion through the CNO-cycle <cit.>. In any case, the formation of IMBH with masses ∼ 10^4 to 10^5 from SMS seems possible at metallicities comparable to that of GN-z11, as shown e.g. by <cit.>. We therefore propose that N-emitters could also be an indication of black hole seed formation from SMS. And these objects could evolve to N-loud quasars, a rare sub-population of quasars showing strong N lines in the UV <cit.>, and which have been suggested to be objects with high N/O and sub-solar metallicities in a rapid growth phase <cit.>.
We therefore mark GN-z11 as a possible AGN with BH-formation related to SMS in Table <ref>.
Finally, we also consider that the formation of an IMBH
with mass 1000
from an SMS is incompatible with the proto-GC scenario, as the presence of such a BH in old GCs seems to be ruled out observationally <cit.>. This is also reflected in Table <ref>.
Finally, we wish to remind the reader that <cit.> suggested that also hosts a black hole, although our analysis does not show significant evidence for this and suggests that the object is dominated by star-formation (see Sect. <ref>). If harbours an AGN, the situation could be similar to that of GN-z11, just discussed and point to a possible link between SMS and black hole formation.
Also, we note that <cit.> have considered a hidden AGN to explain the emission line properties of the Lynx arc, although the nature of this source remains unclear.
To conclude, we also recall that none of the four other N-emitters discussed here show any AGN indication.
We are therefore probably left with three good candidates for SMS and proto-GCs, , SMACS2031, and the Sunburst cluster.
§.§ Future steps and improvements
Clearly, better data and more N-emitters would be helpful to better understand the origin of the strong N emission lines, to further test the proposed enrichment scenarios and the possible presence of SMS, and thus to understand the nature of these rare objects.
An important test for the massive star scenario would be to detect direct spectral signatures of WR stars. Deeper, very high S/N spectra, in the rest-optical domain would be ideal for this. The massive star scenario also predicts important amounts of helium in the ejecta, which might be measurable from the analysis of nebular He and H emission lines in rest-optical spectra of sufficient quality.
In the SMS scenario, a strong enrichment of aluminum, originating from H-burning from the MgAl chain <cit.>, is predicted (Ramirez-Galeano, in prep.), as observed in GC stars <cit.>. In contrast, massive stars should produce less aluminum <cit.>.
Aluminum has spectral signatures in the rest-UV (Al ii λ1670, Al iii λλ1855,1863), which are often seen in absorption in high-z galaxy spectra <cit.>, and which are in emission in some AGNs <cit.>. These features might be an interesting test of the relation between N-emitters and proto-GCs, and to distinguish between the WR and SMS scenarii.
To examine if the strong N lines could be related to large density variations and found preferentially in pockets of high density, it will be of interest to obtain multiple density measurements probing the widest possible range of density, regions of different ionization, and possibly also spatial variations. Both high S/N and high-resolution spectra are needed for this, and measurements of fine-structure lines of oxygen and nitrogen with ALMA could also provide insights into this question.
Future studies may reveal new N-emitters, improving the statistics and providing more test cases. If strongly enhanced N-emitters are found at significantly lower metallicities (say 7) the SMS scenario might be favored, since WR stars should be less present at low O/H. Also, objects with even higher N/O abundances could exist, if the SMS scenario is correct.
§ CONCLUSION
In this work, we have presented the detailed analysis of at z=8.678 using deep spectroscopy and imaging with NIRSpec and NIRCam obtained from the JWST CEERS program. Low- and medium-resolution NIRSpec spectra covering 1-5μm reveal a wealth of rest-frame UV and optical nebular emission lines of various transitions and ionizing states from H, He, C, N, O, and Ne. In particular, shows remarkably intense Nitrogen emission of N iii and N iv, with N iv] λ1486 emerging as the strongest line within the rest-frame UV spectrum. These emission lines are very rarely seen in galaxy spectra, and – which shows some resemblance with the peculiar object GN-z11 revealed recently by JWST <cit.> – is thus the second “N-emitter” found at z>8. From the analysis of these data, we arrive at the following main results:
* Using the well-detected auroral [O iii] λ4363 line we determined the O/H abundance using the direct method, resulting in = 7.70 ± 0.18. We derived the electron temperature from both rest-frame UV and optical [O iii] lines, yielding consistent values of T_e≈ 18000 K. The density-sensitive lines of N iv] 1483/1487 = 0.50± 0.22 and [O ii] 3727/3729=0.98±0.23 suggest a relatively high electron density of n_e≈ 10^3-5 cm^-3.
These values are consistent with those reported by other studies for this object <cit.>.
* Metal abundances were derived for different ions of C, N, O, and Ne. Notably, we found an exceptionally high N/O abundance of log(N/O)=-0.13±0.11, approximately 5.6 times higher than the solar ratio. Conversely, exhibits relatively normal C/O and Ne/O ratios for its metallicity (O/H), with log(C/O)=-0.75± 0.11 and log(Ne/O)=-0.63±0.07, respectively. This translates to high N/O and N/C, and normal C/O ratios, typically found in globular cluster stars, and which reflect the abundance ratios from H-burning via the CNO-cycle at very high temperature <cit.>.
* We have discussed possible chemical enrichment scenarios to explain these peculiar C, N, and O abundance ratios observed in . Enrichment from massive star winds through the WR phase can explain the observed ratios but requires a very short and specific time window (and the presence of WN stars only); it would also come with a very strong He enrichment. Furthermore, no signatures of WR stars are detected in , although their presence cannot be ruled out from the available data. Alternatively, models of super-massive stars (>1000 M_⊙) mixed with ISM with a normal composition can explain the abundance ratios of . In this scenario, the ejected processed material via SMS will exhibit H-burning products only, strong enriched in N and possibly some depletion in O and C, and a normal He content.
* We have investigated the possibility of an AGN in , a scenario recently suggested by <cit.> due to the detection of a broad component in Hβ. Our own reduction of the NIRSpec spectrum shows a tentative, broad component in Hβ (FWHM≃ 1150 km s^-1) but detected with a fairly low significance (≃ 2.2 σ). Line ratios using rest-UV lines (N v, N iv], C iv, C iii], O iii], and He ii) suggest that the gas is primarily photoionized by star formation, and any contribution from an AGN would likely be residual. The non-detection of the high-ionization lines of N v λ 1240 and He ii λ 1640 further support this scenario.
* shows a complex morphology with three resolved clumps. By analyzing the light distribution of these substructures, we found very compact morphologies with characteristic half-light radii of ≃ 100-150 pc. Multi-wavelength SED fits for each individual clump predict stellar masses of log(M_⋆/M_⊙)≃ 8.66-8.94, resulting in very high stellar mass surface densities log(Σ_M_⋆/(M_⊙ pc^-2) ≃ 3.55-4.14. The star formation rate appears very intense in two clumps (SFR ≃ 80-150 M_⊙ yr^-1), while the remaining clump displays a negligible level of ongoing star formation.
represents thus the second example of a rare population of strong N-emitting galaxies at z>8 with highly super-solar N/O abundances, very compact regions, and a high-density ISM.
To put this object into context and better understand these N-emitters, we have (re-)analyzed other known N-emitting star-forming galaxies from the literature. This includes three lensed objects, two galaxies (SMACS2031 and the Lynx arc), and one star-cluster (the Sunburst cluster) at z ∼ 2.3-3.5, plus a nearby blue compact dwarf galaxy (Mrk 996), all of them without any clear indications of AGN activity.
Similar to , these sources show peculiar abundance ratios with a supersolar N/O ratio along with very dense clustered mass and star formation (log(Σ_M_⋆/(M_⊙ pc^-2)) ≳ 3.5) and high ISM densities (n_e ∼ 10^4-10^5 ). Two galaxies, Mrk 996 and the Lynx arc, show an enhanced C/O ratio compared to normal galaxies at the same metallicity (O/H), indicative of enrichment from WR stars.
We have also presented quantitative predictions for the chemical enrichment in two different scenarios, including enrichment from winds of massive stars (called the WR-scenario) or from ejecta of supermassive stars (SMS) with masses 10^3-10^5 , which have been invoked to explain the abundance anomalies observed in present-day globular clusters <cit.>. The WR scenario explains well the two galaxies with enhanced C/O and is supported by direct evidence of WN and WC stars in Mrk 996. As already found by <cit.> for GN-z11, we found that the SMS scenario reproduced well the observed abundance ratios in , SMACS2031, and the Sunburst cluster.
These observations probably provide the best indirect evidence so far for the possible existence of SMS in galaxies.
Finally, considering the preferred enrichment scenarii and other physical properties, we have also examined which of the N-emitters could host proto-GCs and what their nature is. From our analysis we concluded that , SMACS2031, and the Sunburst cluster host most likely proto-GCs. We also suggested that the peculiar abundances of GN-z11 could be due to SMS, even if this object was confirmed to host an AGN, as proposed by <cit.>. This could also point to the formation of intermediate-mass black holes from SMS and suggest a link between the N-emitters and N-loud quasars.
In short, the newly discovered N-emitter and other N-emitters show tantalizing similarities with stars in GCs and the conditions expected during the formation of GCs. They may also offer a unique window into the formation of SMS, their role during the formation of GCs, and also their possible importance as seeds for the formation of massive black holes. More detailed studies and further discoveries of these rare objects will shed further light on these exciting topics and questions.
We thank Lise Christensen and Johan Richard for sharing spectra from their VLT observations of SMACS2031.
We also thank Mark Gieles, Eros Vanzella, Laura Ramirez Galeano, Anastasios Fragos, Holger Baumgardt, Montse Villar-Martin and other colleagues for stimulating discussions.
CC acknowledges support from the Swiss National Science Foundation (SNF; Project 200020-192039). M.M. acknowledges the support of the Swedish Research Council, Vetenskapsrådet (internationell postdok grant 2019-00502).
Y.I. acknowledges support from the National Academy of Sciences of Ukraine (Project No. 0123U102248) and from the Simons Foundation.
aa
|
http://arxiv.org/abs/2307.06135v1 | 20230712123755 | SayPlan: Grounding Large Language Models using 3D Scene Graphs for Scalable Task Planning | [
"Krishan Rana",
"Jesse Haviland",
"Sourav Garg",
"Jad Abou-Chakra",
"Ian Reid",
"Niko Suenderhauf"
] | cs.RO | [
"cs.RO",
"cs.AI"
] |
Theory of Elastic Microphase Separation
David Zwicker
=======================================
Large language models (LLMs) have demonstrated impressive results in developing generalist planning agents for diverse tasks. However, grounding these plans in expansive, multi-floor, and multi-room environments presents a significant challenge for robotics. We introduce SayPlan, a scalable approach to LLM-based, large-scale task planning for robotics using 3D scene graph (3DSG) representations. To ensure the scalability of our approach, we: (1) exploit the hierarchical nature of 3DSGs to allow LLMs to conduct a semantic search for task-relevant subgraphs from a smaller, collapsed representation of the full graph; (2) reduce the planning horizon for the LLM by integrating a classical path planner and (3) introduce an iterative replanning pipeline that refines the initial plan using feedback from a scene graph simulator, correcting infeasible actions and avoiding planning failures. We evaluate our approach on two large-scale environments spanning up to 3 floors, 36 rooms and 140 objects, and show that our approach is capable of grounding large-scale, long-horizon task plans from abstract, and natural language instruction for a mobile manipulator robot to execute. We provide real robot video demonstrations and code on our project page https://sayplan.github.iosayplan.github.io.
§ INTRODUCTION
“Make me a coffee and place it on my desk" – The successful execution of such a seemingly straightforward command remains a daunting task for today's robots. The associated challenges permeate every aspect of robotics, encompassing navigation, perception, manipulation as well as high-level task planning. Recent advances in Large Language Models (LLMs) <cit.> have led to significant progress in incorporating common sense knowledge for robotics <cit.>. This enables robots to plan complex strategies for a diverse range of tasks that require a substantial amount of background knowledge and semantic comprehension.
For LLMs to be effective planners in robotics, they must be grounded in reality, that is, they must adhere to the constraints presented by the physical environment in which the robot operates, including the available affordances, relevant predicates, and the impact of actions on the current state. Furthermore, in expansive environments, the robot must additionally understand where it is, locate items of interest, as well comprehend the topological arrangement of the environment in order to plan across the necessary regions. To address this, recent works have explored the utilization of vision-based value functions <cit.>, object detectors <cit.>, or Planning Domain Definition Language (PDDL) descriptions of a scene <cit.> to ground the output of the LLM-based planner. However, these efforts are primarily confined to small-scale environments, typically single rooms with pre-encoded information on all the existing assets and objects present. The challenge lies in scaling these models. As the environment's complexity and dimensions expand, and as more rooms and entities enter the scene, pre-encoding all the necessary information within the LLMs context becomes increasingly infeasible.
To this end, we present a scalable approach to ground LLM-based task planners across environments spanning multiple rooms and floors. We achieve this by exploiting the growing body of 3D scene graph (3DSGs) research <cit.>. 3DSGs capture a rich topological and hierarchically-organised semantic graph representation of an environment with the versatility to encode the necessary information required for task planning including object state, predicates, affordances and attributes using natural language – suitable for parsing by an LLM. We can leverage a JSON representation of this graph as input to a pre-trained LLM, however, to ensure the scalability of the plans to expansive scenes, we present three key innovations.
Firstly, we present a mechanism that enables the LLM to conduct a semantic search for a task-relevant subgraph 𝒢' by manipulating the nodes of a `collapsed' graph, which exposes only the top level of the full 3DSG 𝒢, via and API function calls – thus making it feasible to plan over increasingly large-scale environments. In doing so, the LLM maintains focus on the small, informative subgraph, 𝒢' during planning, without exceeding its token limit. Secondly, as the horizon of the task plans across such environments tend to grow with the complexity and range, there is an increasing tendency for LLMs to hallucinate or produce infeasible action sequences <cit.>. We counter this by relaxing the need for the LLM to generate the navigational component of the plan, and instead leverage an existing optimal path planner such as Dijkstra <cit.> to connect high-level nodes generated by the LLM. Finally, to ensure the feasibility of the proposed plan, we introduce an iterative replanning pipeline that verifies and refines the initial plan using feedback from a scene graph simulator in order to correct for any unexecutable actions, e.g., missing to open the fridge before putting something into it – thus avoiding planning failures due to inconsistencies, hallucinations, or violations of the physical constraints and predicates imposed by the environment.
Our approach SayPlan ensures feasible and grounded plan generation for a mobile manipulator robot operating in large-scale environments spanning multiple floors and rooms. We evaluate our framework across a range of 90 tasks organised into four levels of difficulty. These include semantic search tasks such as (“Find me something non-vegetarian.”) to interactive, long-horizon tasks with ambiguous multi-room objectives that require a significant level of common-sense reasoning (“Let's play a prank on Niko”). These tasks are assessed in two expansive environments, including a large office floor spanning 36 rooms and 150 interactable assets and objects, and a three-storey house with 32 rooms and 121 objects. Our experiments validate SayPlan's ability to scale task planning to large-scale environments while conserving a low token footprint. By introducing a semantic search pipeline, we can reduce full large-scale scene representations by up to 82.1% for LLM parsing and our iterative replanning pipeline allows for near-perfect executability rates, suitable for execution on a real mobile manipulator robot.[https://sayplan.github.iosayplan.github.io]
§ RELATED WORK
Task planning in robotics aims to generate a sequence of high-level actions to achieve a goal within an environment. Conventional methods employ domain-specific languages such as PDDL <cit.> and ASP <cit.> together with search techniques <cit.> and complex heuristics <cit.> to arrive at a solution. The requirement to specify tasks and the environment via these languages as well as the need for such complex heuristics limits the versatility of the approach when scaling to larger environments and more complex tasks. Learning-based alternatives <cit.>, hierarchical and reinforcement learning <cit.>, face challenges with data demands and scalability. Our work leverages the in-context learning capabilities of LLMs to generate feasible long-horizon task plans across 3D scene graphs, a scalable approach to representing large-scale scenes. Tasks, in this case, can be naturally expressed using natural language, and search heuristics are derived by the LLM based on its internet scale knowledge, the task's semantics and the information present within the scene graph representation.
Task planning with LLMs, that is, translating natural language prompts into task plans
for robotics, is an emergent trend in the field. Earlier studies have effectively leveraged pre-trained LLMs' in-context learning abilities to generate actionable plans for embodied agents <cit.>. A key challenge for robotics is grounding these plans within the operational environment of the robot. Prior works have explored the use of object detectors <cit.>, PDDL environment representations <cit.> or value functions <cit.> to achieve this grounding, however, they are predominantly constrained to single-room environments, and scale poorly with the number of objects in a scene which limits their ability to plan over multi-room or multi-floor environments. In this work, we explore the use of 3D scene graphs and the ability of LLMs to generate plans over large-scale scenes by exploiting the inherent hierarchical structure of these representations.
Integrating external knowledge in LLMs has been a growing line of research combining language models with external tools to improve the reliability of their outputs. In such cases, external modules are used to provide feedback or extra information to the LLM to guide its output generation. This is achieved either through API calls to external tools <cit.> or as textual feedback from the operating environment <cit.>. More closely related to our work, CLAIRIFY <cit.> iteratively leverage compiler error feedback to re-prompt an LLM to generate syntactically valid code. Building on these ideas, we propose an iterative plan verification process with feedback from a scene graph-based simulator to ensure all generated plans adhere to the constraints and predicates captured by the pre-constructed scene graph. This ensures the direct executability of the plan on a mobile manipulator robot, operating in the corresponding real-world environment.
§ SAYPLAN
§.§ Problem Formulation
We aim to address the challenge of long-range planning for an autonomous agent, such as a mobile manipulator robot, in a large-scale environment based on natural language instructions. This requires the robot to comprehend abstract and ambiguous instructions, understand the scene and generate task plans involving both navigation and manipulation of a mobile robot within an environment. Existing approaches lack the ability to reason over scenes spanning multiple floors and rooms. Our focus is on integrating large-scale scenes into planning agents based on Language Models (LLMs) and solving the scalability challenge. We aim to tackle two key problems: 1) representing large-scale scenes within LLM token limitations, and 2) mitigating LLM hallucinations and erroneous outputs when generating long-horizon plans in large-scale environments.
§.§ Preliminaries
Here, we describe the 3D scene graph representation of an environment and the components of the scene graph API which we leverage throughout our approach.
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Hierarchical Structure of a 3D Scene Graph. This graph consists of 4 layers. Notes that the room nodes are connected to one another via sequences of pose nodes which capture the topological arrangement of a scene.
Scene Representation: The 3D Scene Graph (3DSG) <cit.> have recently emerged as an actionable world representation for robots <cit.>, which hierarchically abstracts the environment at multiple levels through spatial semantics and object relationships while capturing relevant states, affordances and predicates of the entities present in the environment. Formally, a 3DSG is a hierarchical multigraph G = (V, E) in which the set of vertices V comprises V_1∪ V_2∪…∪ V_K, with each V_k signifying the set of vertices at a particular level of the hierarchy k. Edges stemming from a vertex v ∈ V_k may only terminate in V_k-1∪ V_k ∪ V_k+1, i.e. edges connect nodes within the same level, or one level higher or lower.
We assume a pre-constructed 3D scene graph representation of a large-scale environment generated using existing techniques <cit.>. The entire 3D scene graph can be represented as a NetworkX object <cit.> and text-serialised into a JSON data format that can be parsed directly by a pre-trained LLM. An example of a single asset node from the 3D scene graph is represented as:
. The 3D Scene Graph (3DSG) is organized in a hierarchical manner with four primary layers: floors, rooms, assets, and objects as shown in Figure <ref>. The top layer contains floors, each of which branches out to several rooms. These rooms are interconnected through pose nodes to represent the environment's topological structure. Within each room, we find assets (immovable entities) and objects (movable entities). Both asset and object nodes encode particulars such as state, affordances, additional attributes such as colour or weight, and 3D pose. The graph also incorporates a dynamic agent node, denoting a robot's location within the scene.
Scene Graph API: The LLM is given access to an external API which provides it with a set of tools required to manipulate and operate over 3DSGs. It enables the LLM to manipulate scene graphs through and functions, revealing connected nodes in a lower layer, or reversing the process respectively. Furthermore, generated plans can be verified through a task-agnostic which consists of a set of rules which verify if actions performed on the nodes adhere to the physical constraints, predicates and affordances present in the corresponding environment.
§.§ Approach
Given a 3D scene graph representations 𝒢 and a task instruction ℐ defined in natural language, we can view our framework SayPlan as a high-level task planner π(a|ℐ,𝒢), capable of generating long-horizon plans a grounded in the large scale environment within which a mobile manipulator robot operates. The plan can then be fed to a low-level visually grounded motion planner for real-world execution. An overview of the SayPlan pipeline is illustrated in Figure <ref> and the corresponding pseudo-code is given in Algorithm <ref>.
We address the challenges that arise when planning across these large-scale scenes by decomposing the planning pipeline into two key stages: semantic search and iterative replanning. During semantic search, the LLM explores from a collapsed representation of the full scene graph for a suitable subgraph 𝒢' that contains the necessary items required to solve the given task. This relaxes the need for providing the entire scene to the LLM, which would typically contain items that are not required for the given task, consuming unnecessary tokens from the LLM's input context. Once the desired subgraph is identified, the LLM switches to the iterative planning phase. Here, we reduce the planning horizon for an LLM by integrating a classical path planner to fill in optimal navigation paths within the high-level plan generated by the LLM. To ensure the plan adheres to the specific constraints and predicates imposed by the environment, it iteratively goes through a plan verification stage where it is executed within a scene graph environment. In the event of a failed plan, a feedback message is returned and appended to the output plan before being fed back to the LLM for replanning. This process is repeated until an executable plan is achieved. We provide more detail on each component in the following sections.
Semantic Graph Search:
The semantic search phase begins with a collapsed representation of the full 3D scene graph 𝒢, exposing only the highest level of the hierarchy to the LLM eg. room layer as shown in Figure <ref>. Given a natural language task description ℐ, the goal of this phase is to conduct a search, for a sub-graph 𝒢' which contains all the asset and object nodes necessary for solving the task. The search is governed by the LLM's common sense reasoning capabilities and in-context learning from a set of input-output examples <cit.>. We leverage Chain-of-Thought (CoT) <cit.> reasoning to help the LLM decompose complex tasks into intermediate steps to facilitate its ability to decide on the appropriate nodes to or using the available API calls.
At each step, the subgraph 𝒢' in the LLM's previous input is updated and passed again to the LLM until a suitable 𝒢' is identified. The ability to nodes that are not required for solving the task reduces the token footprint over the course of long search sequences (see Fig. <ref>). To avoid expanding already-contracted nodes, we maintain a list of expanded nodes, passed as an additional input to the LLM. This leads to a fully Markovian decision-making process, where the current subgraph 𝒢' and the history of expanded nodes are the only state inputs required for the LLM to make its next decision. This allows it to scale to long search sequences, unlike <cit.> which has to maintain the full history of interactions. Once the LLM agent identifies that the current subgraph has visibility over all the assets and objects required to solve the task, it autonomously switches to the planning phase. An example of the LLM-scene graph interaction during semantic search is provided in Appendix <ref>.
Iterative replanning:
Given the identified subgraph 𝒢', we generate correct and feasible long-horizon task plans, via two key mechanisms. First, we shorten the LLM's planning horizon by delegating pose-level path planning to an optimal path planner, such as Dijkstra. For example, a typical plan output such as is simplified to . The path planner handles finding the optimal route between high-level locations, allowing the LLM to focus on essential manipulation components of the task. Secondly, we utilise the to evaluate if the generated plan complies with the scene graph's predicates, state, and affordances. For instance, a action might fail if the robot is already holding something, if it is not in the correct location or if the fridge was not opened beforehand. Such failures are transformed into textual feedback (e.g., "cannot pick banana"), appended to the LLM's input, and used to generate an updated, executable plan. This iterative process, involving planning, validation, and feedback integration, continues until a feasible plan is obtained. This plan is then passed to a low-level motion planner for robotic execution. An example of the LLM-scene graph interaction during iterative replanning is provided in Appendix <ref>.
Implementation Details: We utilise <cit.> as the underlying LLM agent unless otherwise stated. We follow a similar prompting structure to <cit.> as shown in Appendix <ref>. We define the agent's role, details pertaining to the scene graph environment, the desired output structure and a set of input-output examples which together form the static prompt used for in-context learning. This static prompt is both task- and environment-agnostic and takes up approximately 3900 tokens of the LLMs input. During semantic search, both the and components of the input prompt get updated, while during iterative planning only the component gets updated with information from the .
§ EXPERIMENTAL SETUP
We design our experiments to evaluate the 3D scene graph reasoning capabilities of LLMs with a particular focus on high-level task planning pertaining to a mobile manipulator robot. The plans adhere to a particular embodiment consisting of a 7-degree-of-freedom robot arm with a two-fingered gripper attached to a mobile base. We use two large-scale environments, shown in Figure <ref>, which exhibit multiple rooms and multiple floors which the LLM agent has to plan across. To better ablate and showcase the capabilities of SayPlan, we decouple its semantic search ability from the overall causal planning capabilities using the following two evaluation settings:
Semantic Search: Here, we focus on queries which test the semantic search capabilities of an LLM provided with a collapsed 3D scene graph. This requires the LLM to reason over the room and floor node names and their corresponding attributes in order to aid its search for the relevant assets and objects required to solve the given task instruction. We evaluate against a human baseline to understand how the semantic search capabilities of an LLM compare to a human's thought process. Furthermore, to gain a better understanding of the impact different models have on this graph-based reasoning, we additionally compare against a variant of SayPlan using .
Causal Planning: In this experiment, we evaluate the ability of SayPlan to generate feasible plans to solve a given natural language instruction. The evaluation metrics are divided into two components: 1) Correctness, which primarily validates the overall goal of the plan and its alignment to what a human would do to solve the task and 2) Executability, which evaluates the alignment of the plan to the constraints of the scene graph environment and its ability to be executed by a mobile manipulator robot. We note here that for a plan to be executable, it does not necessarily have to be correct and vice versa. We evaluate SayPlan against two baseline methods that integrate an LLM for task planning:
LLM-As-Planner, which generates a full plan sequence in an open-loop manner; the plan includes the full sequence of both navigation and manipulation actions that the robot must execute to complete a task, and LLM+P, an ablated variant of SayPlan, which only incorporates the path planner to allow for shorter horizon plan sequences, without any iterative replanning.
§ RESULTS
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3cOffice 3cHome
Subtask
1cHuman
1cc]@c@SayPlan
()
1cc]@c@SayPlan
()
1cHuman
1cc]@c@SayPlan
()
1cc]@c@SayPlan
()
(lr0.75em)1-1
(lr0.75em)2-4
(lr0.75em)5-7
Simple Search 100% 6.6% 86.7% 100% 0.0% 86.7%
Complex Search 100% 0.0% 73.3% 100% 0.0% 73.3%
Evaluating the semantic search capabilities of . The table shows the semantic search success rate in finding a suitable subgraph for planning.
We summarise the results for the semantic search evaluation in Table <ref>. SayPlan () consistently failed to reason over the input graph representation, hallucinating nodes to explore or stagnating at exploring the same node multiple times. SayPlan() in contrast achieved 86.7% and 73.3% success in identifying the desired subgraph across both the simple and complex search tasks respectively, demonstrating significantly better graph-based reasoning than .
While as expected the human baseline achieved 100% on all sets of instructions, we are more interested in the qualitative assessment of the common-sense reasoning used during semantic search. More specifically we would like to identify the similarity in the semantic search heuristics utilised by humans and that used by the underlying LLM based on the given task instruction.
§.§ Semantic Search
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1c c]@c@Full Graph
(Token Count) c]@c@Collapsed Graph
(Token Count) Compression Ratio
Office 4962 888 82.1%
Home 4602 1827 60.4%
3D Scene Graph Token Count Number of tokens required for the full graph vs. collapsed graph.
We present the full sequence of explored nodes for both SayPlan () and the human baseline in Appendix <ref>. As shown in the tables, SayPlan () demonstrates remarkably similar performance to a human's commonsense reasoning for most tasks, exploring a similar sequence of nodes given a particular instruction. For example when asked to "find a ripe banana", the LLM first explores the kitchen followed by the next most likley location, the cafeteria. In the case where no semantics are present in the instruction such as "find me object K31X", we note that the LLM agent is capable of conducting a breadth-first-like search across all the unexplored nodes.
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Scene Graph Token Progression During Semantic Search. This graph illustrates the scalability of our approach to large-scale 3D scene graphs.
An odd failure case in the simple search instructions involved negation, where the agent consistently failed when presented with questions such as "Find me an office that does not have a cabinet" or "Find me a bathroom with no toilet". Other failure cases noted across the complex search instructions included the LLM's failure to conduct simple distance-based and count-based reasoning over graph nodes. While trivial to a human, this does require the LLM agent to reason over multiple nodes simultaneously, where it tended to hallucinate or miscount connected nodes.
Scalability Analysis: We additionally analyse the scalability of SayPlan during semantic search. Table <ref> illustrates the impact of exploiting the hierarchical nature of 3D scene graphs and allowing the LLM to explore the graph from a collapsed initial state. This allows for a reduction of 82.1% in the input tokens required to represent the Office environment and a 60.4% reduction for the Home environment. In Figure <ref>, we illustrate how endowing the LLM with the ability to contract explored nodes which it deems unsuitable for solving the task allows it to maintain near-constant input memory from a token perspective across the entire semantic search process. Note that the initial number of tokens already present represents the input prompt tokens as given in Appendix <ref>.
C[1]>p#1
§.§ Causal Planning
The results of causal planning across simple and long-horizon instructions are summarised in Table <ref> (left). We compared SayPlan's performance against two baselines: LLM-As-Planner and LLM+P. All three methods displayed consistent correctness in simple planning tasks at 93%, given that this metric is more a function of the underlying LLMs reasoning capabilities. However, it is interesting to note that in the long-horizon tasks, both the path planner and iterative replanning play a role in improving this correctness metric by reducing the planning horizon and allowing the LLM to reflect on its previous output.
The results illustrate that the key to ensuring the task plan's executability was iterative replanning. Both LLM-As-Planner and LLM+P exhibited poor executability, whereas SayPlan achieved near-perfect executability as a result of iterative replanning, which ensured that the generated plans were grounded to adhere to the constraints and predicated imposed by the environment. Detailed task plans and errors encountered are provided in Appendix <ref>. We summarise these errors in Table <ref> (right) which shows that plans generated with LLM+P and LLM-As-Planner entailed various types of errors limiting their executability. LLM+P mitigated navigational path planning errors as a result of the classical path planner however still suffered from errors pertaining to the manipulation of the environment - missing actions or incorrect actions which violate environment predicates. SayPlan mitigated these errors via iterative replanning, however in 6.67% of tasks, it failed to correct for some hallucinated nodes. While we believe these errors could be eventually corrected via iterative replanning, we limited the number of replanning steps to 5 throughout all experiments. We provide an illustration of the real-world execution of a generated plan using SayPlan on a mobile manipulator robot coupled with a vision-guided motion planner in Appendix <ref>.
§ LIMITATIONS
SayPlan naturally inherits the limitations and biases inherent in current large language models, which are dependent on their training data, and can adversely affect the correctness of the generated plans causing misinterpretation of instructions; examples of such failures are illustrated in Appendix <ref>. More specifically, SayPlan is limited by the graph-based reasoning capabilities of the underlying LLM which fails at simple distance-based reasoning, node count-based reasoning and node negation. Future work could explore fine-tuning these models for these specific tasks or alternatively incorporate existing and more complex graph reasoning tools <cit.> to facilitate decision-making. Secondly, SayPlan's current framework is constrained by the need for a pre-built 3D scene graph and assumes that objects remain static post-map generation, significantly restricting its adaptability to dynamic real-world environments. Future work could explore how online scene graph SLAM systems <cit.> could be integrated within the SayPlan framework to account for this. Lastly, a potential limitation of the current system lies in the scene graph simulator and its ability to capture the various planning failures within the environment. While this works well in the cases presented in this paper, for more complex tasks involving a diverse set of predicates and affordances, the incorporation of relevant feedback messages for each instance may become infeasible and forms an important avenue for future work in this area.
§ CONCLUSION
SayPlan is a natural language-driven planning framework for robotics that integrates hierarchical 3D scene graphs and LLMs to plan across large-scale environments spanning multiple floors and rooms. We ensure the scalability of our approach by exploiting the hierarchical nature of 3D scene graphs and the semantic reasoning capabilities of LLMs to enable the agent to explore the scene graph from the highest level within the hierarchy. In this way, we are no longer required to present all lower-level entities to the LLM, resulting in a significant reduction in the tokens required to capture larger environments. Once explored, the LLM generates task plans for a mobile manipulator robot, and we ensure the plan is feasible and grounded to the state of the environment via iterative replanning using a scene graph simulator. The framework produces the highest number of correct and executable plans that a mobile robot can follow compared to existing baseline techniques. Finally, we translate a select number of the generated plans to a real-world mobile manipulator agent, capable of completing natural language-defined tasks spanning long-range, long-horizon across a wide range of rooms.
The authors would like to thank Ben Burgess-Limerick for assistance with the robot hardware setup, Nishant Rana for creating the illustrations and Norman Di Palo and Michael Milford for insightful discussions and feedback towards this manuscript. The authors also acknowledge the ongoing support of the QUT Centre for Robotics. This work was partially supported by the Australian Research Council (Project DP220102398) and by an Amazon Research Award to Niko Sünderhauf.
§ ENVIRONMENTS
We evaluate SayPlan across a set of two large-scale environments spanning multiple rooms and floors as shown in Figure <ref>. We provide details of each of these environments below:
Office: A large-scale office floor, spanning 36 rooms and 150 assets and objects which the agent can interact with. This scene graph represents a real-world office floor within which a mobile manipulator robot is present. This allows us to embody the plans generated using SayPlan and evaluate their feasibility in the corresponding environment. A full and collapsed 3D scene graph representation of the office floor is provided in Appendix <ref> and <ref> respectively.
Home: An existing 3D scene graph from the Stanford 3D Scene Graph dataset <cit.> which consists of a family home environment () spanning 32 rooms across 3 floors and contains 121 assets and objects which the agent can interact with. A 3D visual of this environment can be viewed at the 3D Scene Graph project website.[https://3dscenegraph.stanford.edu/Klickitat.html3dscenegraph.stanford.edu/Klickitat]
§ TASKS
We evaluate SayPlan across 4 instruction sets which are classified to evaluate different aspects of its 3D scene graph reasoning and planning capabilities:
Simple Search: Focused on evaluating the semantic search capabilities of the LLM based on queries which directly reference information in the scene graph as well as the basic graph-based reasoning capabilities of the LMM.
Complex Search: Abstract semantic search queries which require complex reasoning. The information required to solve these search tasks is not readily available in the graph and has to be inferred by the underlying LLM.
Simple Planning: Task planning queries which require the agent to perform
graph search, causal reasoning and environment interaction in order to solve the task. Typically requires shorter horizon plans over single rooms.
Long Horizon Planning: Long Horizon planning queries require
multiple interactive steps. These queries evaluate SayPlan's ability to reason over temporally extended instructions to investigate how well it scales to such regimes. Typically requires long horizon plans spanning multiple rooms.
The full list of instructions used and the corresponding aspect the query evaluates are given in the following tables.
§.§ Simple Search
§.§.§ Office Environment
§.§.§ Home Environment
§.§ Complex Search
§.§.§ Office Environment
§.§.§ Home Environment
§.§ Simple Planning
§.§ Long Horizon Planning
§ FULL 3D SCENE GRAPH: OFFICE ENVIRONMENT
§ CONTRACTED 3D SCENE GRAPH: OFFICE ENVIRONMENT
§ SEMANTIC SEARCH EVALUATION RESULTS
- Full listings of the generated semantic search sequences for the evaluation instruction sets are provided on the following page -
§ CAUSAL PLANNING EVALUATION RESULTS
- Full listings of the generated planning sequences for the evaluation instruction sets are provided on the following page -
§ REAL WORLD EXECUTION OF A GENERATED LONG HORIZON PLAN.
§ INPUT PROMPT STRUCTURE
Input prompt passed to the LLM for SayPlan. Note that the components highlighted in violet represent static components of the prompt that remain fixed throughout both the semantic search and iterative replanning phases of SayPlan.
white
Agent Role: You are an excellent graph planning agent. Given a graph representation of an environment, you can explore the graph by expanding nodes to find the items of interest. You can then use this graph to generate a step-by-step task plan that the agent can follow to solve a given instruction.
Environment Functions:
goto(<pose>): Move the agent to any room node or pose node.
access(<asset>): Provide access to the set of affordances associated with an asset node and its connected objects.
pickup(<object>): Pick up an accessible object from the accessed node.
release(<object>): Release grasped object at an asset node.
turn_on/off(<object>): Toggle object at agent’s node, if accessible and has affordance.
open/close(<asset>): Open/close asset at agent’s node, affecting object accessibility.
done(): Call when the task is completed.
Environment State:
ontop_of(<asset>): Object is located on <asset>
inside_of(<asset>): Object is located inside <asset>
inside_hand: Object is currently being grasped by the robot/agent
closed: Asset can be opened
open: Asset can be closed or kept open
on: Asset is currently on
off: Asset is currently off
accessible: The object is not accessible if it is inside an asset and the asset state is "closed".
Environment API:
expand_node(<node>): Reveal assets/objects connected to a room/floor node.
contract_node(<node>): Hide assets/objects, reducing graph size for memory constraints.
verify_plan(): Verify generated plan in the scene graph environment.
Output Response Format:
{chain_of_thought: break your problem down into a series of intermediate reasoning steps to help you determine your next command,
reasoning: justify why the next action is important
mode: "exploring" OR "planning"
command: {"command_name": Environment API call
"node_id": node to perform an operation on
"plan": task plan if in planning mode}}
Example: <see Appendix <ref> and <ref>>
Instruction: Natural language description of the task
3D Scene Graph: Text-serialised JSON description of a 3D scene graph
Memory: History of previously expanded nodes
Feedback: External textual feedback from scene graph simulator
Ensure the response can be parsed by Python json.loads.
§ EXAMPLE INPUT-OUTPUT INTERACTION BETWEEN LLM AND SCENE GRAPH SIMULATOR DURING SEMANTIC SEARCH
white
Instruction: make a coffee for Tom and place it in his room
3D Scene Graph: {nodes: {room: [{id: bobs_room}, {id: toms_room}, {id: jacks_room}, {id: kitchen}, {id: livingroom}], pose: [{id: pose1}, {id: pose2}, {id: pose3}, {id: pose4}, {id: pose5}], agent: [{location: bobs_room, id: agent}]}, links: [bobs_room↔pose1, bobs_room↔agent, toms_room↔pose1, toms_room↔pose2, toms_room↔pose5, jacks_room↔pose2, jacks_room↔pose3, kitchen↔pose3, kitchen↔pose4, kitchen↔pose5, livingroom↔pose4]}
Memory: []
SayPlan [Mode->Search]: chain-of-thought: explore graph for coffee mug, coffee machine and somewhere to place the coffee -> perform common sense exploration first -> explore toms room -> explore kitchen -> if items not found then explore other nodes
reasoning: i will expand toms room
command: expand(toms_room)
Scene Graph Simulator: <Updated 3D scene graph>
Memory: [toms_room]
SayPlan [Mode->Search]: chain-of-thought: i have found a wardrobe in tom's room -> leave this node expanded -> the coffee mug is not in his room -> still have not found the coffee machine -> kitchen might have coffee machine and coffee mug -> explore this node next
reasoning: i will expand the kitchen next
command: expand(kitchen)
Scene Graph Simulator: <Updated 3D scene graph>
Memory: [toms_room, kitchen]
SayPlan [Mode->Search]: chain-of-thought: i've found the coffee machine in the kitchen -> the coffee mug however is not in the kitchen -> i will explore other unexpanded nodes now for the coffee mug -> jacks room, bobs room and the living room are unexplored yet -> i will try jack"s room next
reasoning: i will expand jacks room next
command: expand(jacks_room)
Scene Graph Simulator: <Updated 3D scene graph>
Memory: [toms_room, kitchen, jacks_room]
SayPlan [Mode->Search]: chain-of-thought: jacks room does not have anything relevant towards solving this task -> I should contract this node to save token memory -> i will explore other unexplored nodes after doing this
reasoning: contract jack's room to save memory
command: contract(jacks_room)
Scene Graph Simulator: <Updated 3D scene graph>
Memory: [toms_room, kitchen, jacks_room]
SayPlan [Mode->Search]: chain-of-thought: i have found a wardrobe in tom's room and coffee machine in the kitchen -> the coffee mug is still required for the task -> explore a new room next
reasoning: i will expand bob's room as it is unexplored
command: expand(bobs_room)
Scene Graph Simulator: <Updated 3D scene graph>
Memory: [toms_room, kitchen, jacks_room, bobs_room]
SayPlan [Mode->Search]: chain-of-thought: i have found the coffee mug, the coffee machine and tom's wardrobe on the graph -> search complete -> I will switch to planning mode-> collect coffee mug -> generate plan for making coffee -> place coffee mug on Tom's wardrobe
reasoning: I will generate a task plan using the identified subgraph
§ EXAMPLE INPUT-OUTPUT INTERACTION BETWEEN LLM AND SCENE GRAPH ENVIRONMENT DURING ITERATIVE REPLANNING
white
Instruction: make a coffee for Tom and place it in his room
3D Scene Graph: {nodes: {room: [{id: bobs_room}, {id: toms_room}, {id: jacks_room}, {id: kitchen}, {id: livingroom}], pose: [{id: pose1}, {id: pose2}, {id: pose3}, {id: pose4}, {id: pose5}], agent: [{location: bobs_room, id: agent}], asset: [{room: toms_room, state: free, affordances: [release], id: bed2}, {room: toms_room, state: closed, affordances: [open, close, release], id: wardrobe2}, {room: kitchen, state: closed, affordances: [open, close, release], id: fridge}, {room: kitchen, affordances: [turn_on, turn_off], state: off, id: coffee_machine}, {room: bobs_room, state: free, affordances: [release], id: bed1}, {room: bobs_room, state: closed, affordances: [open, close, release], id: wardrobe1}], object: [{affordances: [pickup], state: inside_of(wardrobe1), attributes: "blue", id: coffee_mug}]}, links: [bobs_room↔pose1, bobs_room↔agent, bobs_room↔bed1, bobs_room↔wardrobe1, toms_room↔pose1, toms_room↔pose2, toms_room↔pose5, toms_room↔bed2, toms_room↔wardrobe2, jacks_room↔pose2, jacks_room↔pose3, kitchen↔pose3, kitchen↔pose4, kitchen↔pose5, kitchen↔fridge, kitchen↔coffee_machine, livingroom↔pose4, wardrobe1↔coffee_mug]}
Memory: [toms_room, kitchen, jacks_room, bobs_room]
SayPlan [Mode->Planning]: chain-of-thought: "i have found the coffee mug, the coffee machine and tom's wardrobe on the graph -> search complete -> I will switch to planning mode-> collect coffee mug -> generate plan for making coffee -> place coffee mug on Tom's wardrobe"
reasoning: I will generate a task plan using the identified subgraph
plan: [goto(bobs_room) > access(wardrobe1) > pickup(coffee_mug) > goto(kitchen) > access(coffee_machine) > release(coffee_mug) > turn_on(coffee_machine) > turn_off(coffee_machine) > pickup(coffee_mug) > goto(toms_room) > access(wardrobe2) > release(coffee_mug) > done]
Path Planner:plan: [goto(pose3) > goto(pose4) > goto(pose1) > goto(bobs_room) > access(wardrobe1) > pickup(coffee_mug) > goto(pose2) > goto(pose5) > goto(kitchen) > access(coffee_machine) > release(coffee_mug) > turn_on(coffee_machine) > turn_off(coffee_machine) > pickup(coffee_mug) > goto(pose5) > goto(pose2) > goto(toms_room) > access(wardrobe2) > release(coffee_mug) > done]
Scene Graph Simulator (Feedback): "coffee mug is not accessible"
Memory: [toms_room, kitchen, jacks_room, bobs_room]
SayPlan [Mode->Planning]: chain-of-thought: "the coffee mug was not accessible -> the wardrobe it is in is currently closed -> I update the plan to open the wardrobe first -> generate plan for making coffee -> place coffee mug on Tom's wardrobe"
reasoning: I will generate a task plan using the identified subgraph and the feedback from the simulator
plan: [goto(bobs_room) > access(wardrobe1) > open(wardrobe1) > pickup(coffee_mug) > goto(kitchen) > access(coffee_machine) > release(coffee_mug) > turn_on(coffee_machine) > turn_off(coffee_machine) > pickup(coffee_mug) > goto(toms_room) > access(wardrobe2) > release(coffee_mug) > done]
Path Planner:plan:
[goto(pose3) > goto(pose4) > goto(pose1) > goto(bobs_room) > access(wardrobe1) > open(wardrobe1) > pickup(coffee_mug) > goto(pose2) > goto(pose5) > goto(kitchen) > access(coffee_machine) > release(coffee_mug) > turn_on(coffee_machine) > turn_off(coffee_machine) > pickup(coffee_mug) > goto(pose5) > goto(pose2) > goto(toms_room) > access(wardrobe2) > release(coffee_mug) > done]
Scene Graph Simulator: Plan Verified
|
http://arxiv.org/abs/2307.04050v1 | 20230708212820 | Optimization-based Learning for Dynamic Load Planning in Trucking Service Networks | [
"Ritesh Ojha",
"Wenbo Chen",
"Hanyu Zhang",
"Reem Khir",
"Alan Erera",
"Pascal Van Hentenryck"
] | cs.AI | [
"cs.AI",
"cs.LG",
"cs.SY",
"eess.SY"
] |
A Robust and Efficient Optimization Model for Electric Vehicle Charging Stations in Developing Countries under Electricity Uncertainty
[
=========================================================================================================================================
*Co-first authors
The load planning problem is a critical challenge in service network
design for parcel carriers: it decides how many trailers (or loads),
perhaps of different types, to assign for dispatch over time between
pairs of terminals. Another key challenge is to determine a flow plan, which specifies how parcel volumes are assigned to planned loads. This paper considers the Dynamic Load Planning Problem (DLPP) that considers both flow and load planning challenges jointly in order to adjust loads
and flows as the demand forecast changes over time before the
day of operations. The paper aims at developing a decision-support
tool to inform planners making these decisions at terminals across the
network. The paper formulates the DLPP as a MIP and shows that it
admits a large number of symmetries in a network where each commodity
can be routed through primary and alternate paths. As a result, an
optimization solver may return fundamentally different solutions to
closely related problems (i.e., DLPPs with slightly different inputs),
confusing planners and reducing trust in optimization. To remedy this
limitation, the paper proposes a Goal-Directed Optimization (GDO) that
eliminates those symmetries by generating optimal solutions staying
close to a reference plan. The paper also proposes an optimization
proxy to address the computational challenges of the optimization
models. The proxy combines a machine learning model and a
feasibility restoration model and finds solutions that satisfy real-time constraints imposed by planners-in-the-loop. An extensive computational study on industrial instances shows that the optimization proxy is around 10 times faster than the commercial solver in obtaining the same quality solutions and orders of magnitude faster for generating solutions that are consistent with each other. The proposed approach also demonstrates the benefits of the DLPP for load consolidation, and the significant savings obtained from combining machine learning and optimization.
§ INTRODUCTION
The e-commerce market continues to show robust growth and leading
analysts project that today's $3.3 trillion market could grow further
to $5.4 trillion annually by 2026 (<cit.>). Much of e-commerce
relies on home delivery of small packages or parcels and other boxed
freight. Key freight carriers like UPS and FedEx continually seek to
redesign and operate profitable logistic networks that meet e-commerce
customer service expectations. Beyond physical network design
including the location and sizing of various freight processing
terminals, these companies face challenging service network design
problems. A critical service network design challenge for package
carriers are the so-called load planning problems (for
background, see <cit.>). Here, load planning refers to
decisions related to the number of trailers or container loads, perhaps of different types, to plan for dispatch over time between
pairs of terminals. Such planned loads are the transportation capacity
of the network. Flow planning decisions represent another key challenge, where the flow plan specifies how to allocate parcel volumes to planned loads to feasibly and cost-effectively serve network demand. As each package moves from its origin to destination, it is transported by a sequence of planned loads where it is unloaded and sorted at a transfer (hub) terminal between each loaded dispatch. Together, the flow and load plan decisions define a service network that moves package
volume from origins to destinations in order to meet customer service
expectations. The research described in this paper is conducted
directly with a leading global parcel carrier that operates a massive
network moving large volumes of packages each day. Figure <ref> illustrates the load planning operations at an example
terminal. It highlights the planner-in-the-loop environment in which load planning takes place; an important consideration underlying this
research.
Packages at a terminal with the same destination and service class are
referred to as a commodity. A flow plan defines flow
rules for each commodity in the service network; these flow rules
specify how a commodity is routed through the network over time. Since parcel carriers operate massive terminal networks with large numbers of transfer locations, a flow plan may include alternate flow rules that specify loading paths for commodities in addition to the default path specified by the primary flow rules. Both the primary (default) and alternate paths specify how a commodity moves through the network, and these planned paths are service feasible, i.e., they ensure that commodities arrive on time given their service guarantees.
This paper considers the Dynamic Load Planning Problem (DLPP)
faced by the load planner at a terminal as depicted in Figure
<ref> during a short time period (one or two weeks) leading up to the day of operations. The goal of the planner, and thus of the DLPP, is to decide (1) how many loads should be planned for outbound dispatch to other terminals at various times during the day of operations and (2) how to allocate commodity volumes across planned loads respecting the capacity constraints and the primary and alternate flow rules. These two decisions define what is called a load plan in this paper. The objective of the DLPP is to obtain a load plan that
minimizes the number of loads, consolidating the commodities as best
as possible. In practice, the DLPP is solved by planners, who adjust
existing load plans manually to reflect changes in commodity volumes
arriving at the terminal. This process is typically myopic
and creates inefficiencies across the network.
The goal of this research is to develop a decision support tool to
assist planners in solving the DLPP, suggesting load plans that remove
existing inefficiencies. Moreover, for terminals that do not have a
planner, the tool can fully automate the DLPP, bridging the gap
between network design and operations. To develop such a tool, this paper first investigates optimization models for the DLPP. In its general form, the DLPP is strongly NP-hard and its MIP formulation is challenging for state-of-the-art solvers given the size of the instances encountered in practice. Moreover, the natural MIP model exhibits significant symmetries which is highly undesirable for the planner-in-the-loop environment of the industrial partner. Indeed, planners will be extremely confused if small changes in commodities result in completely different load plans. To address this challenge, this paper presents a Goal-Directed Optimization (GDO) that solves a first model to find the optimal solution to the DLPP and uses a second model to find a plan that is as close as possible to a reference plan. GDO is shown to produce consistent plans, i.e., plans that are close for inputs that only differ slightly. Unfortunately, the GDO approach is too time-consuming to be used in planner-in-the-loop environments. To address this final difficulty, this research proposes the use of
optimization proxies that combine a Machine-Learning (ML) model and a
feasibility restoration procedure to obtain near-optimal solutions in
a few seconds, even for the largest terminals. The ML model uses
supervised learning to mimic the GDO approach and predicts the optimal set of planned loads. The feasibility restoration procedure then solves a small MIP model to determine the final allocation of commodity volumes to planned loads, adding extra capacity as needed to ensure feasibility. The proposed approach is practical since it produces high-quality plans that are consistent with each other, where small changes in inputs leads to very similar load plans by virtue of the ML training that mimics the GDO optimization.
The main contributions of the paper can be summarized as follows:
* The paper formalizes the DLPP and develops a natural MIP
formulation to solve it.
* The paper proposes a Goal-Directed Optimization approach to
remedy the limitations of the MIP formulation; it uses a 2-stage
approach to eliminate symmetries and provide optimal load plans that
are close to a reference plan.
* The paper proposes an optimization proxy to address the computational
difficulties of the GDO approach; the optimization proxy uses a
machine learning model to predict the loads and a feasibility
restoration procedure to adjust the predictions to satisfy the problem
constraints and determine the commodity flows. Once trained, the optimization proxy provides high-quality solutions in a few seconds.
* The paper presents extensive computational results on industrial
instances, including some of the largest terminals in the network;
the results demonstrate the significant benefits of optimization and
the ability of the optimization proxy to find high-quality and
consistent solutions in real time. More precisely, the paper shows
that the optimization proxy outperforms a greedy heuristic and the
MIP model solved by a commercial solver both in terms of the
objective function value and consistency metrics. The
optimization proxy is around 10 times faster than the commercial
solver in obtaining solutions with the same objective function value
and orders of magnitude faster in terms of generating solutions that
are consistent with a reference plan. Empirical experiments show
the value of breaking symmetries by GDO, which helps the proxy to
produce high-quality and consistent load plans.
* From a business and sustainability perspective, the experiments
demonstrate the value of having alternate flow paths for the
commodities, in addition to the primary flow paths. The proposed load
plans allocate approximately 17% commodity volume to the alternate
flow paths and reduce the required load capacity by 12%-15%.
The rest of this paper is organized as follows. Section
<ref> summarizes related work. Sections
<ref> and <ref> introduces the DLPP
and its modeling. Sections <ref> and <ref>
present the GDO approach and the optimization proxy. Section
<ref> describes a heuristic that mimic human
planners and serve as a baseline. Section <ref>
describes the computational results. Section <ref>
discusses the benefits of the DLPP formulation, optimization, and
machine learning, quantifying the cost and sustainability benefits and
the important factors driving them.
§ RELATED WORK
Service Network Design.
There is abundant research on network design for the Less-than-truckload (LTL) trucking industry (see
<cit.>). Interested readers
can consult erera2013creating for a detailed description
of LTL operations. <cit.> present a detailed
description of the mathematical models and heuristics for the problems
arising in trucking service network design. The authors describe the
tactical flow and load planning problem which is solved weeks
in advance for “typical” commodity volume (e.g., average daily
origin-destination commodity volume) for a network of terminals. The
goal of the flow and load planning problem is to determine
effective primary flow paths for the commodity volume and the total
trailer capacity required on each flow path in a network of terminals.
Most of these network design problems are formulated over time-space
networks using integer programming models. The flow and load planning
problem with both primary and alternate flow paths for industry-scale
instances can be modeled as large-scale integer programming models
which, unfortunately, cannot be solved directly by commercial
solvers. Therefore, previous work in this area focused mainly on
finding a single cost-effective primary flow path for the
commodities. Exact approaches to solve these problems have been
proposed by <cit.>, <cit.>,
and <cit.>. However, these approaches can only
solve instances with a few thousand packages. For industry-scale
instances, researchers have resorted to various heuristics including
variants of local search heuristic algorithms
(<cit.>,
<cit.>) and greedy algorithms (<cit.>).
Flow and Load Planning with Alternate Paths.
Tactical flow and load planning is typically based on average daily
estimates of origin-destination commodity volume. However, commodity
volume differ substantially from day to day and from week to week
(<cit.>). Hence, planners at a terminal locally
modify the load plans on a daily basis, using the latest estimates of
commodity volume until the day of operations. More specifically, the
planners take advantage of both primary and alternate flow paths to
improve trailer consolidation at their respective
terminals. It is worth highlighting that the primary flow paths come from flow and load planning. Once primary options are available alternate flow paths, that are time feasible, are identified. To the best of our knowledge no paper carefully studies the problem of allocating volume across alternate flow paths in operations. Alternate flow paths are useful to reduce trailers when commodity volume can be split across paths. This is especially useful because of demand uncertainty. <cit.> present a study on the value of
having these alternate flow paths to hedge against demand
uncertainty. They show that it is sufficient to have just one
alternate to contain the impact of most of the fluctuations
in demand; the authors refer to such a load plan as a 2-alt load plan. Subsequently, the authors in <cit.> study the operational decisions that LTL carriers need to make to effectively operate a 2-alt load plan when demand changes dynamically on a day-to-day basis. However, the proposed approach cannot be solved for practical sized instances. This paper proposes a ML-based solution approach for the allocation of volume across both primary and multiple alternate flow paths; the proposed approach is shown to be effective for large scale instances experienced in practice.
Dynamic Load Planning.
Network-wide simultaneous optimization of load planning adjustments is
a daunting challenge due to the scale of the network, number of commodities and the number of transfer hubs for the commodities.
Existing research in the literature may be applicable to the problem of selecting a single primary flow path
(non-splittable) for each commodity at each terminal for each sorting
period in order to minimize the cost of the resulting load plan. Splitting commodity volume across alternate flow paths is likely to improve trailer utilization as it introduces more flexibility in the load planning process. This research considers the DLPP problem at a terminal in which the commodity volume can be split (among primary and alternate flow paths) to promote better trailer utilization, lower transportation cost, and increased sustainability. The flexibility to adjust plans enables terminal planners to better manage daily operations while maintaining service guarantees. This problem is mentioned as an
interesting and useful future research direction by <cit.>.
One paper in the literature, <cit.>, does introduce the problem of re-routing freight volume on alternate flow paths to improve on-time performance of load plans on the day-of-operations; this becomes necessary when the actual volume deviates from the forecasted volume on the day-of-operations. In this work, commodity volume is assigned to exactly one flow path (it is not splittable) such that the total (fixed) trailer capacity is respected and the objective is to minimize the total lateness of shipments. The authors develop MIP models for this problem and propose heuristic algorithms to solve them. Note that a key difference between this approach and the approach proposed in the current paper is that we allow volume to be split across multiple flow paths on the day-of-operations. Furthermore, we also adjust the load plan to identify opportunities to reduce outbound capacity (and improve utilization) as demand forecasts are updated.
The DLPP is also similar to the variable-sized bin packing problem
described by <cit.> where the objective is to
minimize the total space used to pack a set of items into bins
(available in different sizes), such that each item is packed into
exactly one bin. In the DLPP, the packages are the items and trailers
are bins but the key difference is that the DLPP allows for the
splitting of the package volume into compatible trailers in order to
further reduce the transportation cost by promoting better
consolidation or packing.
Machine Learning for Optimization.
In recent years, there has been a notable surge of interest among
researchers in the development of ML surrogates for solving MIPs. This
emerging field has attracted attention due to the potential of ML
techniques to provide efficient approximations for computationally
intensive calculations involved in solving MIPs. We refer the reader
to (<cit.>,<cit.>) for a comprehensive
overview on the topic. The techniques can fall into one of the two
categories. The first category includes methods based on
reinforcement learning
(<cit.>),
where the ML model is trained by interacting with simulation
environments. The second category comprises supervised learning
(<cit.>),
where the ML model imitates the optimization model and replaces
expensive calculations with a quick approximation. This research
focuses on the latter category since the proposed optimization model
could be used as the expert for supervised learning. Optimization
proxies, which combine learning with feasibility restoration, has
emerged from supervised learning. Recent work in this area includes
(<cit.>).
§ PROBLEM DESCRIPTION AND MODELING
Parcel carriers operate massive terminal networks with hundreds of
facilities to move large volumes of parcels each day. Each day
at a terminal is divided into time windows (typically three to four
hours in length), called sort periods or sorts,
during which parcels are sorted. A typical operational day includes
“day”, “twilight”, “night” and “sunrise” sorts that are
non-overlapping in time. All parcels sorted at a terminal
during a given sort with the same service class (e.g., one-day service
or two-day service) and the same destination are referred to as a
commodity. Suppose then that each commodity has a primary flow path and one or more alternate flow paths that each specify a sequence terminals and sorts that parcels will traverse en route from origin to destination. For a specific commodity at a specific terminal at a specific sort, each flow path will determine the next terminal and sort to which packages will be loaded. Typically, shipments are loaded on trailers moving along the primary flow path for the commodity; however, when there are better consolidation opportunities, commodity volume can be split over primary and alternate flow paths, or completely allocated to alternate flow paths. The rest of this section describes the main concepts underlying the DLPP. Section <ref> describes some key terminology and presents examples to illustrate the operations at terminals. Section <ref> describes the DLPP that includes splitting of commodity volume across
primary and alternate flow paths.
§.§ Definitions
Let 𝒢 = (𝒩,𝒮) denote a time-space network. Each node n ∈𝒩 represents a terminal location at a particular time period and is defined by a tuple, i.e. n=(terminal, sort, day). Each arc s ∈𝒮 represents a directed dispatch of loads from one timed node to another. Henceforth in the paper, we refer to each such an arc as a sort pair. Figure <ref> illustrates an example time-space network for terminal A during a single twilight sort period. In this example, three sort pairs are outbound from terminal A on day 1, namely, (A,Twilight,1)→(X,Twilight,2),
(A,Twilight,1)→(Y,Twilight,2), and
(A,Twilight,1)→(Z,Twilight,3). Figure <ref> illustrates another example of terminal B that operates multiple sort periods, i.e., the day, twilight, night sorts on a given day, and seven sort pairs (b_1,b_2,b_3,b_4,b_5,b_6,b_7) outbound from terminal B.
A key objective in load planning is to determine the number of
trailers (possibly of different types) to operate on each sort pair to containerize the total commodity volume allocated to the sort pair. During a sort, each loading door at a terminal builds/loads trailers for a specific sort pair destination. In a single sort facility, as shown in Figure <ref>, if there is commodity volume allocated on each of the three sort pairs, then at least three trailers (one on each sort pair) should be opened at the loading doors corresponding to the sort pair destinations.
In practice, commodities outbound from an origin terminal that arrive over consecutive sorts and that are heading to the same time-space destination can be consolidated together. For that, the concept of load pairs is introduced, where a load pair represents a set of consecutive sort pairs that share the same destination node. Combining sort pairs into load pairs allows better consolidation and trailer utilization, since trailers can be held partially loaded from one sort to the next prior to dispatch to the destination. Figure <ref> illustrates an example of a load pair that is composed of three different sort pairs.
We now relate primary and alternate flow paths to sort pairs. If we consider volume for commodity k ∈𝒦 at some time-space location n, its primary flow path specifies the next (terminal, sort, day) to which it should be loaded. Thus, the primary path identifies a unique outbound sort pair for k at n. Similarly, each alternate flow path identifies a (possibly different) outbound sort pair for k. Recall that primary and alternate flow paths for each k at n are specified in advance, and we assume that loading outbound on any of these options will lead to volume arriving on-time to its destination.
We will define compatible sort pairs for k at n to be the primary path sort pair (the primary sort pair) and any alternate path sort pair (an alternate sort pair). Furthermore, any sort pairs that are in load pairs with compatible sort pairs with an earlier origin sort are also compatible. When volume is assigned to such earlier sort pairs, the decision is to assign volume to trailers that are opened first for loading in those earlier sorts and held for dispatch. Figure <ref> illustrates four compatible sort pairs (outbound from terminal B) for a commodity k sorted in the twilight sort at terminal B.
§.§ Dynamic Load Planning Problem (DLPP)
Parcel carriers typically build a load plan in two phases: (1) the tactical flow and load planning phase specifies an initial plan and provides an input to the scheduling team; and (2) the load plan adjustment allows adjustments to the initial plans up to the day-of-operation. The scheduling and load dispatching teams then execute the
adjusted load plan. Weekly plans that determine the number of loads or
trailers to operate on each sort pair are fixed approximately two
weeks in advance of the operating week. However, due to demand
uncertainty, the volume forecast for commodities may change, and
adjustments to the load plan may be necessary to accommodate actual
volumes. These adjustments may lead to cost decreases when unnecessary
load capacity is removed from the plan.
Consider the following optimization problem during the two weeks
leading into the day-of-operation. Each terminal in the network has
a set of forecasted inbound commodities during some time period (for
example, a single operating day and multiple sorting periods). Each
such commodity arrives during a specific sorting period and has a
destination terminal and service class (specifying a due date at the
destination). Given this information, the fixed flow plan specifies a
primary flow path (next terminal and arriving sorting period) for each
commodity, and possibly also one or more alternate flow paths. Recall
that, if the commodity is assigned to any of these flow paths, then it
will reach its final destination on time according to plan. The
adjustment optimization problem is to assign each commodity to its
primary and/or one of its alternate flow paths while simultaneously
determining how many loads of different types are required for each
proposed flow paths. Note that existing flow and load planning
literature typically assumes that all commodities, arriving
at a terminal during a specific sorting period should be assigned to
the primary flow path. Here, the challenge is different, and is
instead to determine specifically how to split each commodity volume
among its possible compatible flow paths or sort pairs to drive high
load utilization levels and low costs while still meeting service
promises.
Consider the example shown in Figure <ref> with three
commodities (4 units destined to terminal C, 3 units destined to
terminal E, 3 units destined to F) sorted in the twilight sort of day 1 at
terminal B. In this example, we denote each commodity by its
destination terminal name. The commodity destined for terminal F has
three compatible sort pairs: (B,Twilight,1)→ (C,Sunrise,2)
is the primary sort pair, and (B,Twilight,1)→ (E,Sunrise,2)
and (B,Twilight,1)→ (D,Twilight,2) are the alternate sort
pairs. Splitting commodity volume destined to terminal F between the two
alternate sort pairs to C and D yields better consolidation (and
lower transportation cost) as the solution requires one less trailer
on the two arcs: (B,Twilight,1)→ (D,Twilight,2) and
(D,Twilight,2)→ (F,Day,3).
For a
given terminal, define S to be the set of outbound sort pairs and
let K be the set of commodities sorted at the terminal. Each
commodity k ∈ K has a cubic volume of q^k, and a set of
compatible sort pairs S^k. For every outbound sort pair s, there
is a set V_s of trailer types, that can be used to containerize the
total commodity volume allocated to the sort pair. Each sort pair can
have different set of allowed trailer types, i.e., V_s_1 can be
different from V_s_2 for two different sort pairs s_1,s_2 ∈
S. Each trailer type v ∈ V_s has a cubic capacity Q_v and has a
per-unit transportation cost c_v. A solution of the DLPP
determines the number of trailers of each type assigned to each sort
pair, as well as the volume of each commodity allocated to each
trailer. A solution must ensure that all the volume is assigned to
trailers and that the capacities of the trailers are not violated. The
goal of the DLPP is to find a solution that minimizes the costs of the
trailers. Appendix <ref> provides the complexity
results. The DLPP is strongly NP-hard. It becomes weakly NP-hard when
each commodity is compatible with exactly one or with all sort
pairs and there are multiple trailer types. It becomes polynomial when each commodity is compatible with exactly one or with all sort
pairs and there is only one type of trailer.
§.§ A Mixed-Integer Programming Formulation
An optimization model for the DLPP can be defined as follows in Model <ref>:
x,yMinimize ∑_s ∈ S∑_v ∈ V_s c_v y_s,v
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_k ∈ K:s ∈ S^k x^k_s,v≤ Q_v y_s,v, ∀ s ∈ S, v ∈ V_s,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
y_s,v∈ℤ_≥ 0 ∀ s ∈ S, v ∈ V.
It uses a non-negative continuous decision variable x^k_s,v to
represent the volume of commodity k allocated to trailer type v
operating on a sort pair s, and an integer decision variable
y_s,v to determine the number of trailers of type v installed on
sort pair s. The objective (<ref>) minimizes the total cost of creating
loads. In the experiments, c_v = Q_v ∀ v ∈ V, i.e., the
model minimizes the total trailer capacity required to containerize
the total commodity volume in the problem instances. Constraints
(<ref>) ensure that the total volume of each commodity
is assigned to its compatible sort pairs. Constraints
(<ref>) ensure that the total volume on a sort pair
respects the installed trailer capacity on it. Constraints
(<ref>)-(<ref>) define the domain and range of
variables.
§ GOAL-DIRECTED OPTIMIZATION
The optimization model of the DLPP has a large number of
symmetries. Figure <ref> depicts a simple instance with multiple optimal solutions that are operationally different from one another, yet they are equivalent from Model <ref> perspective as they require the same number of trailers of the same type. This is because in Model in <ref>, commodities are indifferent to the
sort pairs they are assigned to, as the volume allocation decisions
(x-variables) do not incur any cost.
Such symmetries are undesirable for many reasons. Paramount among them
are the realities in the field: the model is intended to be used and
validated by planners. If small variations of inputs produce
fundamentally different solutions, planners are unlikely to trust the
model. Indeed, since the model is used multiple times a day, it is
important to ensure that the successive optimal solutions are as
consistent as possible with each other. Fortunately, in practice, a
reference plan is always available and the DLPP should ideally
produce optimal solutions that are as close as possible to the
reference plan.
This section explores how to refine the model presented earlier to
satisfy this requirement, and presents a Goal-Directed Optimization
(GDO) approach to the DLPP. It uses a reference plan
to eliminate symmetries and ensure that the solution is compatible
with the planner-in-the-loop reality in the field. The use of a
reference plan eliminates many symmetries but not all. To break more
symmetries, the GDO approach also adds a flow diversion cost
that captures the cost of using alternate paths instead of the primary
path. For instance, in the example depicted in Figure <ref>,
only the solution shown in Figure <ref> is optimal following our assumptions. The flow
diversion cost is chosen to be proportional to the distance between
the next alternate terminal and the destination of the commodity, as
there is incentive to move commodities as close as possible to their
destination. For example, suppose a commodity k is in
Atlanta and is destined for Chicago. Let the primary
next terminal be Louisville (with flow diversion cost 0),
alternate 1 be Nashville, and alternate 2 be
Memphis. As Nashville is closer to Chicago than
Memphis, the flow diversion cost of allocating volume to
alternate 1 is lower than that of alternate 2. As a result, the
GDO approach has at its disposal a reference plan γ, where
γ_s,v denotes the number of trailers of type v planned to
operate on sort pair s. It also leverages the flow diversion
cost d^k_s that denotes the cost of allocating a per-unit volume
of commodity k ∈ K to a compatible sort pair s ∈ S^k.
The GDO approach first solves Model <ref>
to obtain the optimal objective value Z^*. It then solves a second
MIP Model to bias the trailer decisions so that they are as close as
possible to the reference plan and minimize diversion costs.
The second-stage model is defined as follows:
x,yMinimize ∑_s ∈ S∑_v ∈ V_s| y_s,v - γ_s,v| + ϵ∑_k ∈ K∑_s ∈ S_k∑_v ∈ V_sd^k_s x^k_s,v
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_k ∈ K:s ∈ S^k x^k_s,v≤ Q_v (y_s,v), ∀ s ∈ S, v ∈ V_s,
∑_s ∈ S∑_v ∈ V_s c_v y_s,v≤ Z^*,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
y_s,v∈ℤ_≥ 0 ∀ s ∈ S, v ∈ V.
The objective function (<ref>) minimizes the weighted sum of the Hamming
distance of the trailer decisions from the reference plan γ and
the flow diversion costs. The weight ϵ for the flow diversion cost is sufficiently small such that the cost does not dominate over the Hamming distance term in the objective function. The purpose of the flow diversion cost in (<ref>) is to break the symmetry between solutions with the same Hamming distance; it biases the solution to have more volume allocated to primary sort pairs than alternate sort pairs. Constraints (<ref>),
(<ref>), (<ref>) and (<ref>) are
the same as in Model <ref>. Constraint
(<ref>) ensures that the optimal solution does not use more
trailer capacity than Z^*. Note that the objective function is
non-linear due to the Hamming distance term. It can be linearized by
replacing | y_s,v - γ_s,v| with new variables
w_s,v≥ 0 (s ∈ S, v ∈ V_s) and imposing the following
constraints
y_s,v - γ_s,v≤ w_s,v ∀ s ∈ S, v ∈ V_s,
γ_s,v - y_s,v≤ w_s,v ∀ s ∈ S, v ∈ V_s,
Figure <ref> illustrates the sensitivity of the
trailer decisions (y-variables) subject to increases in the total
commodity volume (∑_k ∈ Kq^k) (x-axis) for the two models:
Model <ref> (red plot) and the GDO approach (blue
plot). As the total commodity volume increases, Model
<ref> exhibits solutions where the trailer decisions
fluctuate dramatically between 1 and 6 trailers for sort pair 1,
and between 1 and 5 trailers for sort pair 2. However, when
using GDO, the trailer decisions in GDO are more consistent and vary
between 1 and 2 trailers on sort pair 1, and is constant at 2 trailers on sort pair
2.
§ LEARNING-BASED OPTIMIZATION PROXIES
The GDO approach produces consistent solutions to the DLPP, but it is
too slow to be used with planners in the loop. This section proposes a
Machine Learning (ML) approach to the DLPP. Its goal is to move some
of the optimization burden offline and produce high-quality solutions
in real time. More precisely, the approach uses the concept of optimization proxies to produce high-quality solutions to an
optimization problem by learning its input/output mapping (see, for
instance,
(<cit.>)
for an overview of this concept and its applications).
The overall methodology underlying optimization proxies is depicted in Figure
<ref>. It consists of two stages,
* an offline stage where an ML model learns the input/output
mapping of the optimization problem;
* an online stage which is used in real time: it receives
an instance, applies the ML model to predict a (possibly infeasible) solution
and uses a repair procedure to deliver a feasible solution.
For the DLPP, the ML model learns the mapping between the (input)
commodity volumes and the (output) trailer decisions; in other words, given the
commodity volumes, the ML model predicts trailer decisions for every
sort pair. The trained ML model may sometimes underestimate
the number of trailers on some sort pairs when executed in real time. To circumvent this issue, the feasibility
restoration step projects the predicted trailer decisions back into the
feasible region; in addition, the feasibility restoration also
computes the volume allocation on the sort pairs. A key element in the
ML training is data augmentation that complements historical
data by generating realistic instances through input perturbations. The ML model formulation is introduced and discussed in more details in what follows.
§.§ The ML Model Formulation
This section defines a machine learning model f, parameterized by
θ, that maps the input parameters, i.e., the commodity volume,
to the optimal trailer decisions:
(<ref>)-(<ref>).
f_θ: ℝ_≥ 0^|K|⟶ℤ_≥ 0^|S| × |V|
𝐩⟼𝐲
The ML inputs are assumed to be taken from a distribution 𝒫
that captures the actual instances.
Given a dataset of input parameters {𝐩_i}_i ∈ N∼𝒫, where N is the set of instances, parametrization θ^* can be obtained by minimizing
the empirical risk shown in (<ref>), where
(<ref>) denotes the optimization problem
solved by Model <ref>, and l denotes the loss function
that measures the L1-distance of the predicted
(f_θ(𝐩)) and optimal (y^*) trailer decisions.
θMinimize 1/N∑_i∈ N l(f_θ(𝐩_i), 𝐲_i^*)
subject to (𝐱_i^*, 𝐲_i^*) = 𝐱, 𝐲∈𝒞(𝐩_i) c(𝐱, 𝐲) ,
It is important to highlight that an ML model could be used to predict
commodity volume allocation on the sort pairs (x-variables) instead
of the trailer decisions (y-variables). This may seem to be a good
approach since, after predicting volume allocation, one can easily
recover the trailer decisions and hence a feasible solution, by
setting y_s,v=⌈∑_k ∈ K:s ∈ S^k
x^k_s,v/Q_v⌉ ∀ s ∈ S, v ∈ V_s.
However, this approach has some shortcomings. First, the output
dimension is significantly larger than the input dimension which makes
it very difficult to develop an effective ML model even for the smallest
instances. Second, recovering trailer decisions is very sensitive to
the predicted volume allocation decisions. Consider an example where
100 cubic volume is allocated to a sort pair which requires two
trailers, each with capacity 50 cubic volume, in the optimal solution.
If the ML model predicts the volume on the sort pair to be 100.5,
then the total number of trailers required is ⌈100.5/50⌉ = 3 which generates a poor solution in terms of the objective function value of Model <ref>.
Experimental results confirmed that it is beneficial to learn the
mapping from input parameters to the trailer decisions rather than the
volume decisions. The trailer decisions 𝐲∈ℤ_≥ 0^|S| × |V| are more aggregated than the
volume allocation decisions ℝ_≥ 0^|K| × |S|
× |V|. The benefits comes from the significant reductions in
output dimensionality and variability. In addition, as presented in
section <ref>, once the trailer
decisions are known, restoring the feasibility of the solution is
relatively easy as the feasibility restoration MIP has a small number
of binary decision variables and therefore, it is easy to solve.
The ML model used in this paper is a deep neural network as illustrated in Figure
<ref>. It uses a Multi-Layer Perceptron (MLP),
where each dense layer is followed with a batch normalization (<cit.>),
a dropout (<cit.>), and a ReLU (Rectified Linear Unit) function.
It maps the input parameter 𝐩
to the flattened trailer decision 𝐲. The
last ReLU guarantees that the output of the neural network is
non-negative. The compatible trailer decisions 𝐲
are then generated by reshaping the flattened decision
𝐲 and masking it with the compatible
trailer mask 𝐦, where m_s, v = 1 indicates that
equipment type v ∈ V is compatible with sort pair s ∈ S. In the training
phase, the loss function is computed by measuring the distance of
predicted compatible trailer decision 𝐲 with the
optimal trailer decisions. Specifically, this work used smooth l_1 loss.
The loss is used to update the parameters of the MLP using stochastic gradient descent (<cit.>)
with back propagation (<cit.>). At inference time (i.e., in real time), the
compatible trailer decisions are rounded to an integer value.
§.§ MIP-based Feasibility Restoration
The proposed ML model predicts the number of trailers
y_s,v for each sort pair s ∈ S and equipment type v
∈ V_s. Let the total trailer capacity installed on each sort pair
s ∈ S be Λ_s = ∑_v ∈ V_sQ_v
(y_s,v). The system of equations
∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v≤Λ_s, ∀ s ∈ S,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
is then used to determine the volume of every commodity k ∈ K
allocated to its compatible sort pairs. However, it is possible that
some of the sort pairs do not have sufficient trailer capacity because
the ML model may underestimate the capacity. In that case,
(<ref>) is infeasible. The following linear program
zMinimize ∑_s ∈ S z_s
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v - z_s ≤Λ_s, ∀ s ∈ S,
x^k_s,v,z_s ≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
can be used determine the sort pairs with trailer capacity violations.
Its objective function (<ref>) minimizes the capacity
violations on the sort pairs. Constraints (<ref>)
ensure that total volume of every commodity is assigned to compatible
sort pairs. Constraints (<ref>) determine the sort
pair capacity violations. Constraints (<ref>) define the
domain and range of variables. When Model <ref>
has an optimal objective value equal to 0, it has recovered a
feasible solution to Model <ref>. Otherwise,
additional trailer capacity is required on sort pairs with capacity
violations.
This paper proposes a two-stage MIP-based feasibility restoration
process. In the first stage, Model <ref> is
solved to obtain an optimal solution z^*. Let the set of sort pairs
with trailer capacity violation be S = {s ∈ S: z^*_s >
0}. The feasibility restoration then identifies the cheapest
equipment v to serve the excess volume on sort pair s ∈S. The extra trailer capacity is given by ξ_s =
( ⌈z_s/Q_v⌉ * Q_v ) and the option to add the extra capacity to sort pair s ∈S is
added using a binary decision variable. The second stage
solves the following MIP model:
uMinimize ∑_s ∈S u_s ξ_s
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v≤Λ_s + u_s ξ_s, ∀ s ∈S,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v≤Λ_s, ∀ s ∈ S\S,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
u_s ∈{0,1} ∀ s ∈S,
The objective function in (<ref>) minimizes the total
trailer capacity added on each sort pair. Constraints
(<ref>) ensure that the commodity volume is
assigned to the compatible sort pairs. Constraints
(<ref>) and (<ref>) ensure
that commodity volume allocated to each sort pair respects the trailer
capacity. Constraints (<ref>) and
(<ref>) define domain and range of variables. The
number of binary variables in this model is at most the number of sort
pairs in the instance, i.e. | S |. The main difference
between Model <ref> and Model
<ref> is that Model <ref> uses
binary variables u_s instead of continuous variable z_s, to denote
the option of adding extra trailer capacity ξ_s on sort pairs s
∈S. When u_s = 1, extra trailer capacity is added to
sort pair s.
After solving Model (<ref>), adding ξ_s
capacity on every sort pair s ∈S yields a feasible
solution to Model (<ref>). However, the goal is to
use Model (<ref>) to obtain a better feasible
solution. Consider an example with a set of commodities all of which
can be allocated to any of the two sort pairs s_1 and s_2 and
trailer with capacity 2 units. Suppose the optimal solution of Model
<ref> is z_s_1=z_s_2=1. In this case, a
feasible solution to Model <ref> can be recovered by
adding two trailers, one on each sort pair. However, Model
(<ref>) (which has two binary variables) yields a
solution with only one trailer on any one of the two sort pairs. Algorithm 1 provides a summary of the feasibility restoration procedure.
§.§ Value of Symmetry-Breaking Data Generation for Learning
The optimization proxies are trained using the solutions provided by
the GDO which uses the same reference plan for all instances of a given terminal. As
a result, the proxies are consistent by design and do not rely on a
reference plan. The GDO approach is not only critical for
environments with planners in the loop, but it also has an additional
benefit: it makes the learning problem easier. This section provides
theoretical insights about why the data generation using GDO results
in better function approximation than data generation from Model
(<ref>) alone.
Observe that the solution trajectory associated with different
instances can often be effectively approximated by piecewise linear
functions, as depicted in Figure <ref>. This
approximation becomes exact in the case of linear programs and mixed
integer programs when the input reflects incremental changes in the
objective coefficients or right-hand sides of the constraints. This
paper utilizes ReLU-based neural networks to approximate the solutions
of optimization problems. These neural networks are capable of
capturing piecewise linear functions, which makes them well-suited for
this purpose. However, the ability of representing a target piecewise
linear function accurately depends on the model capacity. As the
complexity of the function grows with more pieces, a larger model is
required to obtain a high-quality approximations.
(Model Capacity) (<cit.>)
Let f: ℝ^d →ℝ be a piecewise linear function with p pieces.
If f is represented by a ReLU network with depth k+1, then it must have size at least 1/2kp^1/k-1. Conversely, any piecewise linear function f that is represented by a ReLU network of depth k+1 and size at most s, can have at most (2s/k)^k pieces.
Due to the symmetry in optimal solutions of Model
(<ref>), as shown in
Figure <ref>, the solution trajectory varies
dramatically. Theorem <ref> states that the approximation
of a more volatile solution trajectory (i.e., a piecewise linear
function with more pieces) requires a deep neural network with greater
capacity, which makes the learning task more challenging. In other
words, given a fixed-size ReLU network, higher variability of the
solution trajectory typically results in higher approximation
errors. These errors are bounded by the following theorem.
(Approximation Error) (<cit.>)
Suppose a piecewise linear function f_p', with p' pieces each of width h_k for k ∈ [p'], is used to approximate a piecewise linear f_p with p pieces, where p' ≤ p. Then the approximation error
f_p - f_p'_q ≤1/2h^2_max∑_1≤ k ≤ p|L_k+1 - L_k|,
holds where L_k is the slope of f_p on piece k and h_max is the maximum width of all pieces.
Theorem <ref> relates the approximation error of
a piecewise linear function with the total variation of its slopes.
It implies that the data generated using GDO (which exhibits lower
sensitivity than the data from Model (<ref>))
should facilitate learning and result in lower approximation errors.
§ GREEDY HEURISTIC (GH)
This section proposes a greedy heuristic to construct feasible
solutions for Model (<ref>) and benchmark the quality
of the solution obtained from optimization proxies. This heuristic
iteratively solves linear programs (LP) until all the y-variables
are integers, i.e., they satisfy the integrality tolerance
(10^-5). In each iteration, the algorithm identifies fractional
variables with minimum (⌈y_s,v⌉-y_s,v) value, updates the lower bound of variable y_s,v
to ⌈y_s,v⌉, and re-solves the LP as
shown in Algorithm <ref>. The main idea is that
for a given sort pair s ∈ S and trailer type v ∈ V_s, if
y_s,v has a fractional value very close to an integer ⌈y_s,v⌉, then, this indicates that there is
enough commodity volume to have at least ⌈y_s,v⌉ trailers on the sort pair. GH
greedily adjusts the lower bound of a y-variable in each
iteration till all y-variables can be labelled as integers, in which
case a feasible solution to Model (<ref>) has been
found.
§ COMPUTATIONAL STUDY
This section reports a series of experiments conducted on real-life
instances provided by our industry partner. Section <ref>
presents statistics for the problem instances. Section
<ref> discusses the experimental setup for the
optimization models and proxies. Section
<ref> evaluates the computational
performance of the optimization proxies against the greedy heuristic
(GH) and the optimization models (Model (<ref>) and GDO). Section <ref> evaluates the benefits
of GDO for learning.
§.§ Instances
The experiments are based on industrial instances for three different
terminals in the service network of our industry partner: medium (M),
large (L), and extra-large(XL). Each category
has a reference plan for a
terminal on a particular day as provided by our industry partner. Table <ref> reports the
statistics of the instances: #Arcs denoting the total
number of unique outgoing sort pairs or arcs from the terminal,
#Commodities denoting the number of commodities that are sorted at the terminal and loaded into
outbound trailers (rounded to nearest
multiple of 1,000), and #Loads denoting the number of planned loads in the reference load plan for the
corresponding terminals (rounded to the nearest multiple of 50). Note that, in addition to the planned loads,
small package companies typically operate empty trailers on
the outbound sort pairs for trailer repositioning. This study only
considers trailers that are filled with commodity volume and do not
include empty trailer capacity.
It is worth highlighting that the XL
instance operates more volume and capacity than the M and L instances
combined. Table <ref> reports some statistics for Model
(<ref>) for the three instances: #Integer-Vars and
#Continuous-Vars denoting the number of integer and continuous
decision variables, respectively, and #Constraints denoting the total number
of constraints.
§.§ Experimental Setup
Parameters for GDO
The cost of assigning commodity k to a sort pair s ∈ S^k
(denoted by d^k_s) is defined as
d^k_s =
0, if s is primary flow path for commodity k
(α^k_s + 10*β^k ) otherwise
β_k =
1, if commodity k belongs to one-day service class
2, if commodity k belongs to two-day service class
3, if commodity k belongs to three-day service class
4, otherwise
ϵ = 1/(max_k ∈ K, s ∈ S^k(α^k_s + 10*β^k) )∑_k ∈ K q^k
where α^k_s denotes the distance between the alternate next
terminal and the destination of commodity k ∈ K for sort s, and parameter β_k depends on the commodity service level.
Recall that a commodity k ∈ K is defined as all packages with the
same destination and service class. The term α^k_s ensures that
two commodities with different destinations have different flow
diversion cost. However, two commodities with different service class can
have the same destination. β^k ensures that such commodities have different flow diversion cost for the same
destination. The weight for the flow diversion cost is defined in (<ref>).
Data Generation for ML Model
The dataset is generated by perturbing the input parameters of
real-life instances provided by the industry partner with up to
20,000 commodities. Denote by 𝐩^ref the volume of
different commodities in a given reference plan. The DLPP instances are
generated by perturbing this reference commodity volume. Namely, for
instance i, 𝐩^(i) = γ^(i)×η^(i)×𝐩^ref, where γ^(i)∈ℝ denotes a
global scaling factor and η∈ℝ^|K| is the commodity
level multiplicative white noise. γ is sampled from a uniform
distribution U[80%, 120%], and for every commodity η is
sampled from a normal distribution with mean equals to 1 and standard deviation
of 0.05. For every category, 10,000 instances are generated, and a commercial solver is used to solve the GDO model for each instance. The dataset of
10,000 instances for each category is then split as follows: 80%
for the training set, 10% for the validation set, and 10% for
the test set.
Performance Metrics
The performance metrics in this study are designed to compare the
total trailer cost and the consistency of the
solutions generated by the optimization proxies against the total
trailer cost from Model (<ref>) and then the consistency
of solution from Model (<ref>) of the GDO approach. Given
an instance 𝐩 with optimal trailer decision 𝐲^*
and a feasible trailer decision 𝐲̂, the optimality gap
is defined as
Gap = (Ẑ - Z^*)/|Z^*|,
where Z^* is the optimal trailer cost of Model
(<ref>), and Ẑ is the trailer cost computed
from 𝐲̂. Recall that the total trailer cost does not
increase in Model (<ref>) of the GDO approach due to constraint
(<ref>). If Model (<ref>) cannot be solved
to optimality in 30 minutes, then the best lower bound obtained from
the solver run is used to compute the optimality gap instead of Z^*.
This paper proposes two metrics to quantify the consistency. The
first one is a normalized distance (Δ) between the optimized load plan and the load plan
𝐲, using shifted
geometric means as given by
Δ_s,v =
|y_s,v - y_s,v| if y_s,v = 0
|y_s,v - y_s,v|/y_s,v, otherwise ∀ s ∈ S, v ∈ V_s
Δ = exp(1/|S||V|∑_s∈ S∑_v ∈ V_slog (Δ_s,v + 0.01) ) - 1%.
From a planner perspective, this metric captures the deviation of the
optimized load plan with respect to the reference load plan. As mentioned
in Section <ref>, load plans that are as
close as possible to the reference plan are highly desirable.
The second metric is the total variation of the set of trailer decisions
across a set of N instances (for each terminal). For simplicity,
instances are ordered such that ∑_k ∈ K q^k_i+1≥∑_k
∈ K q^k_i ∀ i ∈{1,2,⋯,N-1}. The goal is to
analyze the variation in trailer decisions on sort pairs when the total
commodity volume is incrementally increased from ∑_k ∈ K
q^k_1 to ∑_k ∈ K q^k_N. Let {𝐲_i}^N_i=1
denote the set of trailer decisions of N instances. The total
variation is defined as:
TV({𝐲_i}^N_i=1) = ∑_i=1^N-1𝐲_i+1 - 𝐲_i_p,
where p=2.
This metric captures the sensitivity of the models, i.e., the impact of changes in total commodity volume on the trailer decisions of different sort pairs. Lower total variation implies that the trailer decisions are less sensitive to changes in total commodity volume. Planners are more amenable to such solutions because fewer (but effective) load plan modifications reduce the solution evaluation effort and is also easier to execute in practice.
The computational efficiency of different models is measured by the
training time of optimization proxies including the data-generation
time and the inference time. Unless specified otherwise, the average
metrics on the test dataset are reported in shifted geometric means:
μ_s(x_1, …, x_n) = exp(1/n∑_i log (x_i + s) ) - s,
where the shift is set as 0.01 for the optimality gap and normalized
distance, 1 second for the inference/solving time, and 1 cube for
the distance between the optimized load plan to the reference load plan.
Implementation Details
All optimization problems are formulated using the Gurobi Python interface,
and solved with Gurobi 9.5 (<cit.>) with 8 CPU threads and
default parameter settings, except for MIPFocus which is set
to a value of 3. All deep learning models are implemented using
PyTorch (<cit.>) and trained using the Adam
optimizer (<cit.>). The ML models are multiple layer
perceptron and are hyperparameter-tuned using a grid search with
learning rate in {10^-1, 10^-2}, number of layers in {3, 4,
5}, and hidden dimension in {128, 256}. For each system, the best
model is selected on the validation set and the performances on the
test set are reported. Experiments are conducted on dual Intel Xeon
[email protected] machines running Linux, on the PACE Phoenix cluster
(<cit.>). The training of ML models is performed on Tesla
V100-PCIE GPUs with 16GBs HBM2 RAM.
§.§ Computational Performance of the Optimization Proxies
This section presents numerical experiments used to assess the
performance of the proposed optimization proxies (Proxies) against the
optimization models (GDO) and the greedy heuristic (GH).
Optimality Gap
Table <ref> presents the optimality gaps of various
approaches, including the results of Model (<ref>)
under various time constraints. In the table, the columns under “Gap
of Model (<ref>)” denote the optimality gaps of the
model under various time limits. Similarly, columns Gap for GH
and Proxies denote optimality gaps for GH and the
optimization proxies. In addition, columns Time(s) denote the solving
times for GH and Proxies.
Recall that Model (<ref>) produces solutions that
exhibit considerable variability when the total commodity volume is
perturbed as detailed in Table <ref> and <ref>. As such, it is unlikely to be practical in
scenarios with planners in the loop. Hence, the table compares the
optimization proxies and the heuristics GH with an “idealized”
benchmark. With this caveat in place, observe the performance of the optimization
proxies under tight time constraints. Proxies generate solutions with
low optimality gaps and may be up to 10 to 50 times faster than GH,
and around 10 times faster than Model (<ref>) solved
with Gurobi. Second, although Model (<ref>)
efficiently produces solutions with low optimality gaps, closing the
optimizality gap proves to be a significant challenge due to the poor
LP relaxation. The performance of GH is also impeded by the
inefficiencies of the LP relaxation, as it solves the LP relaxations
over many iterations; it takes the GH around 30 iterations for
terminal M, 200 iterations for terminal L, and more than 1000
iterations for terminal XL to generate a feasible solution.
Consistency
Tables <ref> and <ref> report
the consistency of solutions obtained from different models in terms
of the normalized distance to the reference load plan and the total
variation of the generated solutions. As GDO requires running Model
(<ref>) and Model (<ref>) sequentially,
these experiments set the same time limits for the two stages. For
example, if a time limit of 30 seconds is set, GDO runs Model
(<ref>) for 30 seconds and subsequently runs Model
(<ref>) using the best upper bound obtained from Model
(<ref>) for another 30 seconds.
The high-level result is that proxies are ideally suited to
produce consistent plans. Table <ref> shows
that the proxies accurately predict, in a few seconds, the results
produced by GDO after an hour. Furthermore, Table <ref> shows that proxies produce solutions that have at
least an order of magnitude smaller total variations in trailer
decisions than both GDO and GH. Proxies produce load plans that
exhibit great stability with changing total commodity volume.
The fact that proxies improve the consistency of the GDO plans is
especially interesting: it means that the optimization proxies, by
virtue of the learning step, avoid oscillations present in the GDO
approach. Of course, it does so at a small loss in objective value
(if, for instance, the GDO model is allowed a minute to run instead of
the 2.5 seconds of the optimizations). But the consistency benefits
are substantial as shown in Table <ref>. The
proxies also provide dramatic improvements over the GH heuristic. Note
also that GDO itself brings significant improvements over Model
(<ref>).
In Table <ref>, observe that the normalized
distance for solution from GDO for the large (L) instance first increases from
0.45% to 11.40%, and then follows the expected
decreasing trend with increase in computational time limit. Recall
that GDO first minimizes the total trailer capacity required in Model
(<ref>) and then solves Model (<ref>) to
minimize the Hamming distance of the solution (and the flow diversion
cost) from the reference load plan. As shows in Table <ref>
the feasible solution obtained from Model <ref> is of
poor quality and is closer to the reference load plan in terms of the
number of trailers. Hence, the normalized distance value is small. As
the computational time limit increases, the feasible solution obtained
from Model (<ref>) exhibits a reduced total trailer
capacity compared to the reference load plan. Hence, the normalized
distance increases as the model tries to find more cost effective solutions. With further increases in computational time, the
normalized distances decrease as the solver finds a better solution
with a smaller Hamming distance using Model <ref>.
It is also interesting to observe that the total trailer capacity
predicted by the ML model, i.e., the capacity provided by all the
trailers predicted to be needed by the ML model, is very close to a
feasible solution. Only a few trailers must be added to recover a
feasible solution. Figure <ref> shows the
distribution of the predicted trailer capacity as a percentage of the
total trailer capacity in the feasible solution generated by the
proxies for each type of terminal. The results show that more than
98% of the trailer capacity is predicted correctly and less than
2% comes from feasibility restoration step generated by algorithm <ref>. More
accurate predictions might even result in a feasibility restoration
model that has fewer decision variables, hence, requiring less
computational time to produce a feasible solution. Appendix <ref> shows that one of the key benefits of the optimization proxies is that it replaces a model with large number of integer decision variables with a prediction model, and requires to solve a relatively simpler feasibility restoration model with small number of binary variables.
§.§ Value of Symmetry-breaking Data Generation
As discussed in section <ref>, the optimal (or near-optimal)
trailer decisions of Model (<ref>) are very sensitive
to changes in total commodity volume due to the presence of symmetries
in the model and the randomized nature of MIP solvers. The solutions
to Model (<ref>) are reported in red in the
plots of Figure <ref>, which illustrates this
behavior. This is not desirable in environments with
planners-in-the-loop, where similar solutions are expected for similar
instances. The GDO approach is much more consistent and its solutions
are shown in orange in the plots of Figure <ref>. The ML component of the optimization proxy uses GDO as
the expert to imitate and learn solution patterns from. As
shown in blue in the plots of Figure <ref>,
the ML model is effective in producing solutions that are close to the
solutions generated by GDO. It should be highlighted that the GDO
approach has two benefits. First, it generates consistent solutions that
are amenable to planner-in-the-loop environments. Second, it makes the
learning problem much more tractable. Designing an ML model for
(<ref>) is really challenging due to the high
sensitivities in small changes: typically an ML model for learning
Model (<ref>) would return an average value.
§ BENEFITS OF DYNAMIC LOAD PLANNING, OPTIMIZATION, AND LEARNING
This section discusses the benefits of the dynamic load planning approach,
optimization, and learning. The load planning methodology studied in
this paper is based on the concepts of primary and alternate flow
paths. With the availability of optimization models, it is possible to
evaluate the benefits of this approach for load consolidation, at
least from a local perspective.
The results in the paper also make it possible to evaluate the
benefits of optimization compared to human planners. During
operations, planners typically assign commodities to the primary flow
paths. If there is no capacity available on the primary flow path,
then planners allocate the remaining volume on the first alternate
flow path and, if there is no capacity on the first alternate, they
turn to the second alternate flow path, and so on. Observe that this
is a greedy strategy of loading commodity volume on trailers, and
hence, it is myopic in nature. A comparison between such a greedy approach
and the optimization models help assess the value of optimization. Of
course, the optimization models are too slow to be used with planners
in the loop. The optimization proxies proposed in the paper are the
technology enabler for translating the benefits of optimization into
practice.
The first results in this section aim at quantifying the value of a
network with alternate flow paths relative to a network with primary
flow options only. Figure <ref> presents some characteristic
of the networks studied in this paper: it shows the distribution of
the number of commodities with a specific number of alternate flow
paths for each instance. It highlights that the network has some
significant flexibility for many of its commodities.
Figure <ref> presents the benefits of the load
planning methodology. It compares the variation in trailer cubic
capacity required to containerize the total commodity volume
(blue curve) and the total volume allocated to alternate flow
paths (green curve) across four different load plans: Primary
Only, Reference Plan, 1-Alt and All-Alt for
the three instance categories. In the Primary Only plan, each
commodity can be assigned only to its primary flow path. The
Reference Plan, referred to as the P-Plan, is the reference load plan from our industry
partner. Note that in the reference plan each commodity can use any number
of compatible alternate flow paths. In the 1-Alt plan, each
commodity can be assigned to either its primary path or the
cheapest alternate path. In the All-Alt plan, each commodity
can be assigned to all the available paths, i.e. splitting is allowed. Observe that the curves are
on different scales: the left scale for the blue curve and the right
scale for the green curve. The P-Plan is produced by the
planners using the greedy approach proposed earlier.
Figure <ref> demonstrates a consistent trend in
cubic capacity required in the four different load plans: the capacity
monotonically decreases and the decreases are significant. Allowing
spittability of commodity volume across primary and alternate flow
paths improves trailer consolidation. These benefits are already
apparent in the P-Plan of the planners, despite the fact that
this is a greedy approach. The optimization model with a single
alternate flow path, i.e., the 1-Alt plan, brings another
step change in consolidation, highlighting the benefits of
optimization. This benefit stems for the fact that a large number of
commodities have at least one alternate flow path in all instances
(see Figure <ref>). Note also that the 1-Alt load
plan requires significantly smaller total trailer capacity than the
P-Plan, although the P-Plan has the flexibility of
using any number of alternate flow paths. The All-Alt plan
brings further benefits but they are rather incremental. Part of the
reasons comes from the fact that a relatively small fraction of the
commodities have more than one alternate flow path. It would be
interesting to study a network with more flexibility as this may bring
further load consolidation benefits.
There is an interesting phenomenon that appears in the medium-sized instance M: the volume
assigned on the alternate flow paths decreased when moving from
1-Alt to an All-Alt plan. This comes from the fact
that this instance has many commodities, with a smaller volume, that
have new alternate flow paths options available in the
All-Alt setting. As a result, commodities with larger volume
are allocated to their primary flow path (as the flow diversion cost
is proportional to the total commodity volume assigned to alternate
flow paths) and the commodities with smaller volume can be allocated
to the alternate flow path that is the primary for the commodities
with larger volume (and not the cheapest alternate path of the
1-Alt setting). Hence, the total volume assigned to the
alternate flow paths reduces, although the total number of commodities
that use alternate flow paths increases.
Figure <ref> compares the percentage of the total
commodity volume that is assigned to the alternate flow paths in the
P-Plan and the All-Alt plan. It is undesirable to
allocate a major proportion of the volume to the alternate flow paths
because the downstream buildings may not be better equipped to handle
or process the large inbound volume. Observe that, on average across
all the instances, the All-Alt plan (resp. P-Plan)
allocates around 17% (resp. 9%) commodity volume on the
alternate flow paths. The All-Alt plan reduces the total
trailer capacity by roughly 12%-15% relative to the
P-Plan. For the XL instance, there is a significant gap
between the P-Plan and All-Alt plan statistics
because most of the commodities in the P-Plan are allocated
to the primary flow paths. This is why the total commodity volume
allocated to the alternate flow paths in the P-Plan and
the Primary Only have a small difference; see Figure
<ref> for XL category.
These results show that optimization proxies can bring substantial
benefits in practice. They provide, in real time, significant
improvements over the existing planning process. Moreover, by virtue
of their training mimicking the GDO optimization, that makes sure that
plans evolve smoothly during the planning process: small changes in
inputs will not result in large changes in the proposed solutions.
These results are eminently practical. One of the challenges
in the operational setting is the need for additional trailers when
the total commodity volume increases. Planners can acquire these
trailers either through empty trailer repositioning or by engaging in
short-term trailer leasing with local companies. Conversely, if the
commodity volume decreases, planners are left with a plan that has low
trailer utilization. The optimization proxies address this issue
directly. Planners can also use the proposed optimization proxies to obtain
recommendations for load plan adjustment in the event of a disruption
(due to uncertainty in commodity volume), even for the largest
terminal, within a matter of seconds. Furthermore, the recommendations
from the optimization proxies are consistent with existing load plans,
which makes it easy for the planners to evaluate and implement the
suggestions. Finally, new terminals in the service network often do
not have dedicated planners to develop load plans and extra capacity
is built in the system to handle the commodity volume in the
worst-case scenario. Optimization proxies can be used as a decision
support tool at such terminals.
§ CONCLUSIONS AND FUTURE WORK
This paper studies the Dynamic Load Planning Problem (DLPP) that
considers both load and flow planning challenges jointly in order to adjust loads and flows
as the demand forecast keeps changing over time before the day of
operations. The paper is motivated by the need of a
decision-support tool to advice planners making these decisions at
terminals across the network. The paper formulates the problem as a
MIP and shows that it admits many symmetries. As a result, the
optimization solver may return fundamentally different solutions to
closely related problems (i.e., DLPPs with slightly different inputs),
confusing planners and reducing trust in optimization. To remedy this
limitation, the paper proposes a Goal-Directed Optimization (GDO) that
eliminates those symmetries by generating optimal solutions staying
close to a reference plan. The paper also proposes an optimization
proxy, combining learning and optimization, to provide
high-quality and consistent load plans. An extensive computational
study on industrial instances shows that the optimization proxy is
around 10 times faster than the commercial solver in obtaining the
same quality solutions and orders of magnitude faster for generating
solutions that are consistent with each other. The proposed approach
also highlights the benefits of the DLPP for load consolidation, and
the significant savings from the combination of machine learning and
optimization.
This research is the first stage of a multi-stage project with our
industry partner (a large parcel carrier) for solving load planning
problems. Future research will extend the proposed approach to clusters of
terminals, taking into account their capacities for processing
commodities. The resulting problem thus requires to determine both
inbound and outbound planning decisions at each terminal, which
significantly complicates the optimization and learning models.
§ ACKNOWLEDGEMENT
This research was partly supported by the NSF AI Institute for Advances in Optimization (Award 2112533).
§ APPENDIX
§.§ Complexity Results
Model <ref> is difficult to solve because in addition to determining the right combination of trailer types to contain volume on each arc, we need to determine the right splits of commodity volume on the given set of compatible arcs. We will analyze the complexity of Model <ref> using the special cases described below.
Case 1: There is only one trailer type available at the terminal, i.e., | V_s | = 1 ∀ s ∈ S. Each commodity k ∈ K is compatible with exactly one sort pair s_k, i.e., S^k = {s_k} ∀ k ∈ K
Case 2: There is only one trailer type available at the terminal, i.e., | V_s | = 1 ∀ s ∈ S. Each commodity k ∈ K is compatible with all sort pairs, i.e., S^k = S ∀ k ∈ K
Cases 1 and 2 are polynomial time solvable
In Case 1, the volume of each commodity k is assigned to its only compatible sort pair, s_k, i.e. x^k_s_k = q^k. Then, the optimal solution has y_s = ⌈∑_k ∈ K: s ∈ S^k x^k_s/Q⌉ = ⌈∑_k ∈ K: s ∈ S^k q^k/Q⌉ ∀ s ∈ S.
In Case 2, the optimal solution is to assign the volume of all commodities on any sort pair s ∈ S and set x^k_s = q^k ∀ k ∈ K, y_s = ⌈∑_k ∈ K q^k/Q⌉, y_s' = 0 ∀ s' ∈ S, s' ≠ s.
Case 3: Same as Case 1, but with more than one trailer type available at the terminal
Case 4: Same as Case 2, but with more than one trailer type available at the terminal
Cases 3 and 4 are weakly NP-Hard
In the optimal solution in Case 3 the volume of each commodity k is assigned to its only compatible sort pair s_k. Thus, it remains to decide the optimal combination of trailer types required to containerize the volume on every sort pair. This is the minimum knapsack problem (see <cit.> for the problem definition) for each sort pair (that has more than one trailer type) as shown in <ref> which is known to be weakly NP-Hard.
For every s ∈ S: yMinimize ∑_v ∈ V_s c_v y_s,v
subject to ∑_k ∈ K: s ∈ S^k q^k≤∑_v ∈ V_sQ_v (y_s,v),
y_s,v∈ℤ_≥ 0 ∀ v ∈ V_s
Similarly, for Case 4 there exists an optimal solution in which the volume of all commodities is assigned to one sort pair s^* ∈ S, i.e. x^k_s^* = q^k ∀ k ∈ K and it remains to solve a minimum knapsack problem for the sort pair s^* due to which Case 4 is weakly NP-Hard.
Case 5: Each commodity k ∈ K is compatible with a subset of sort pairs, i.e., S^k ⊂ S, and has unit volume, q^k = 1. There is only one trailer type with per-unit cost c_s=1 ∀ s ∈ S and capacity Q=max_s ∈ S{∑_k ∈ K1_s ∈ S^k}; hence, y_s ∈{0,1}∀ s ∈ S, as installing one unit of trailer is enough to containerize the total volume that can be assigned to the sort pair. Note that we ignore the index v for trailer because each sort pair has exactly one and the same trailer type.
In the optimal solution of Case 5, each commodity is assigned to exactly one compatible sort pair (i.e. there is no splitting of volume)
We will present a proof by contradiction. WLOG, suppose there exists an optimal solution in which the volume of a commodity k̂ is split between two sort pairs and the volume of all other commodities k ∈ K \{k̂} is assigned to exactly one sort pair s_k. Thus, we have x^k_s_k = q^k ∀ k ∈ K \{k̂} and x^k̂_s_1 + x^k̂_s_2 = q^k. Consider a solution x^k_s =x^k_s ∀ k ∈ K \{k̂} and x^k̂_s_1 = x^k̂_s_1 + ϵ,x^k̂_s_2 = x^k̂_s_2 - ϵ, where ϵ > 0 is a small real number. Note that x^k̂_s_1 + x^k̂_s_2 = q. Consider another solution x̅^k_s = x^k_s ∀ k ∈ K \{k̂} and x̅^k̂_s_1 = x^k̂_s_1 -ϵ,x̅^k̂_s_2 = x^k̂_s_2 + ϵ. Note that both solutions x and x̅ satisfy constraints (<ref>) and are feasible to constraints (<ref>) because we choose Q = max_s ∈ S{∑_k ∈ K1_s ∈ S^k}. The solution x can be written as a convex combination of the solution x and x̅ (x^k_s = 1/2x̅^k_s + 1/2x^k_s ∀ k ∈ K, s ∈ S^k) which contradicts the optimality of the solution.
Case 5 is strongly NP-Hard
We will show that this special case can be solved as a set cover problem which is known to be strongly NP-Hard (<cit.>). An instance of a set cover is given by a ground set U = {x_1, x_2, ⋯, x_n} and a collection of m subsets E_i ⊆ U ∀ i ∈{1,2,⋯,m} of the ground set U. The optimization problem is to find the smallest number of subsets i ∈{1,2,⋯,m} such that ⋃_i ∈{1,2,⋯,m} E_i = U.
From claim <ref> we know that each commodity is assigned to exactly one compatible sort pair in the optimal solution. Let commodity k ∈ K denote element x_k ∈ U, | K | = n and set of sort pairs S = {1,2,⋯,m}. Define K_i = {k ∈ K : x_k ∈ E_i} as the set of commodities or elements that can be covered by selecting sort pairs i ∈{1,2,⋯,m}. Now note that finding the smallest number of sort pairs s ∈ S such that all commodities in K are covered is equivalent to finding the smallest number of subsets i ∈{1,2,⋯,m} to cover all elements in U.
§.§ Additional Experimental Results
Table <ref> compares the the number of integer decision variables in Model <ref> and the average number of binary decision variables in Model <ref> across multiple test instances. The number of integer decision variables remain the same for each instance category because it depends in the number of arcs or sort pairs and trailer types; only the commodity volume changes across different test instances. However, the size of the feasibility restoration model <ref> depends on the predictions of the ML model. Recall that the ML model predicts the value of the integer decision variables of Model <ref>. Hence, if the predictions are accurate, then fewer sort pairs would have capacity violations. Consequently, there would be fewer binary decision variables in Model <ref>; the number of binary decision variables in Model <ref> is equal to the number of sort pairs with capacity violation. As the ML predictions can vary for different test instance with the same set of sort pairs due to different commodity volume, the number of binary variables in Model <ref> can be different for different test instances. This is why the Table <ref> reports fractional values for the average number of binary variables. It is worth highlighting that one of the key benefits of the optimization proxies is that it replaces a model with large number of integer decision variables with a prediction model, and requires to solve a relatively simpler model with small number of binary variables.
|
http://arxiv.org/abs/2307.05581v1 | 20230710142653 | Exploring the Dynamics of the Specialty Insurance Market Using a Novel Discrete Event Simulation Framework: a Lloyd's of London Case Study | [
"Sedar Olmez",
"Akhil Ahmed",
"Keith Kam",
"Zhe Feng",
"Alan Tua"
] | q-fin.GN | [
"q-fin.GN",
"cs.CE",
"cs.MA"
] |
Reliable Devices Yield Stable Quantum Computations
The manuscript is authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan: https://www.energy.gov/doe-public-access-plan.
Samudra Dasgupta^1, 2^*, and Travis S. Humble^1,2^†
^1Quantum Science Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
^2Bredesen Center, University of Tennessee, Knoxville, Tennessee, USA
^*[email protected], ORCID: 0000-0002-7831-745X
^†[email protected], ORCID: 0000-0002-9449-0498
February 2023
====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
This research presents a novel Discrete Event Simulation (DES) of the Lloyd's of London specialty insurance market, exploring complex market dynamics that have not been previously studied quantitatively. The proof-of-concept model allows for the simulation of various scenarios that capture important market phenomena such as the underwriting cycle, the impact of risk syndication, and the importance of appropriate exposure management. Despite minimal calibration, our model has shown that it is a valuable tool for understanding and analysing the Lloyd's of London specialty insurance market, particularly in terms of identifying areas for further investigation for regulators and participants of the market alike. The results generate the expected behaviours that, syndicates (insurers) are less likely to go insolvent if they adopt sophisticated exposure management practices, catastrophe events lead to more defined patterns of cyclicality and cause syndicates to substantially increase their premiums offered. Lastly, syndication enhances the accuracy of actuarial price estimates and narrows the divergence among syndicates. Overall, this research offers a new perspective on the Lloyd's of London market and demonstrates the potential of individual-based modelling (IBM) for understanding complex financial systems.
§ ACKNOWLEDGEMENTS
A special thanks to the Alan Turing Institute's Internship Network (TIN) in supporting the project. Accenture for their partnership with academia. Ki and Brit syndicates for bridging the relationship between academia and industry, providing the resources to undertake the research project. A special thanks to Reuben Thomas-Davis for developing and providing access to the HADES framework.
§ INTRODUCTION
The specialty insurance market is a large-scale complex system with many uncertainties, complex business relationships and non linear dynamics and interactions amongst participants. To quantify the behaviour of this complex system, researchers have turned to various traditional approaches ranging from time-series methods <cit.> to differential equation based mathematical models <cit.>. This top-down, aggregate approach of modelling complex systems may fail to capture the interactions which occur at a micro-scale which ultimately lead to large-scale emergent phenomena. Indeed, many researchers attest to this challenge within the specialty insurance modelling literature. For instance, the “Underwriting Cycle", an important phenomena where the market undergoes periods of high profitability, less competition and low profitability, high competition among insurers (in the insurance literature known as the hardening and softening of the market), is seldom accurately captured using the vast number of traditional aforementioned methods. Boyer et al. demonstrates the challenges in modelling this phenomena in their paper <cit.> where they show time-series methods fail to model this important stylised fact of the specialty insurance market. All of the above points reflect the need for an alternative approach, this is where Individual-Based Modelling (IBM) approaches can help.
Individual-Based Models, which encompass modelling frameworks such as Discrete Event Simulations (DES) to Agent-Based Models (ABMs), can alleviate the above challenges posed by traditional methods utilised in the study of insurance markets. For example, these models allow complex systems to be built from the bottom-up, focusing on interactions at the micro-scale between autonomous “agents" which can represent anything from people, organisations to more granular entities such as molecules. Since the late 90's, IBM applications have grown in popularity, starting from computational social science to ecological studies, biology and environmental sciences. Given its influence in various research fields, the IBM method continues to advance across disciplines, to more relevant areas of research including the modelling of economic markets such as housing, insurance and energy <cit.>.
The IBM methodology, can enable researchers to observe the emergent phenomena at the aggregate level which arise from the micro level interactions of autonomous heterogeneous agents. Each type of agent can embed behaviours and actions reflective of real-world concepts. In fact, these powerful IBM methodologies have reached industry practitioners within the wider retail insurance market, evidencing the appeal of such approaches in modelling market dynamics. The Institute and Faculty of Actuaries (IFA), the professional body and regulator of actuaries in the UK, have gone so far as to start an “Agent-Based Modelling Working Party" where workshops have taken place, and research such as <cit.> presented. Moreover, the Bank of England has published working papers utilising the approach <cit.>. On the other hand, academics from the Bayes Business School <cit.> and The Institute for New Economic Thinking <cit.> have all applied ABMs to the insurance market with significant success.
Given the spotlight on the modelling of insurance markets and a lack thereof in specialty insurance markets, this paper will utilise the findings from the existing ABM insurance literature. We combine these findings with the expert knowledge gained through workshops and interviews with underwriters, actuaries, capital modellers, portfolio analysts, brokers and algorithm engineers at Ki and Brit syndicates (insurers) within Lloyd's of London (the oldest specialty insurance marketplace). These findings allow us to create a novel, discrete event simulation (DES) to study the emergent properties and drivers of the specialty insurance market at various spatio-temporal resolutions. The model proposed in this research article is novel in several aspects:
* The model utilises a DES framework open-sourced by Ki insurance (HADES). HADES consists of two main components:
* Processes - these components are responsible for performing some actions and subsequently emitting events as a response to input events, i.e., Broker process responding to Day, Month or Year events.
* Events - a piece of information that can be altered given its interactions with processes.
* The model incorporates functionality for different event types, such as catastrophe losses (typically low-frequency, high-severity events) and attritional losses (high-frequency, low-severity events). While previous research endeavours have only sought a single type of loss.
* Interviews and workshops conducted with experts within Ki and Brit insurance have shaped the conceptualisation of the agents represented in the model.
* The model simulates the dynamics of the Lloyd's of London specialty insurance market by incorporating unique features such as lead and follow insurers.
This research article will conduct four experiments with varying complexity to demonstrate the emergent properties of the specialty insurance market <cit.>. The first experiment, explores the interactions between syndicates and brokers with simplistic pricing methods to demonstrate a profitable market with low volatility. The second experiment incorporates catastrophe events that increase volatility in the market which can lead to insolvencies and pronounced cyclicality. Thirdly, the decision-making of syndicates is advanced with the introduction of improved exposure management, where syndicates become better equipped in dealing with catastrophe losses. Finally, we introduce the syndication of risk in the marketplace, via lead-follow mechanics, which significantly reduces volatility and tightly couples syndicates' loss experiences.
Some initial findings from the model have demonstrated stylised facts observed in the specialty insurance market, Lloyd's of London, thus validating the model. For example, when insurers price, based on past loss history with uniform risks and attritional losses only, we find that premium prices tend towards the fair analytical price but with substantial variance. When catastrophe events occur which affect multiple risks simultaneously, this leads to large industry losses, which cause step drops in syndicate capitals and increases in premium prices immediately afterwards. When syndicates are able to use exposure management processes to manage tail risks, this leads to smaller and better placed syndicate portfolios. Lastly, when lead and follow quote negotiation and line sizes are applied to premiums and claims, this improves the actuarial price estimates and reduces the spread between syndicates. Syndicate performance becomes more correlated as a result.
The following sections of the article are Literature Review <ref> where pre-existing insurance based literature is disseminated and strengths/weaknesses highlighted. Model Description <ref> describes every process, event and underlying functionality of the model. Results & Discussion <ref> introduces the specialty insurance market in the real-world, then delves into the aforementioned four experiment scenarios and discusses the observed quantitative and qualitative results. Lastly, the Conclusion <ref> highlights the findings from the model, how it contributes to specialty insurance research and future avenues to be explored.
§ LITERATURE REVIEW
The specialty insurance market differs from the conventional retail insurance market. The main difference is that the risks are intrinsically complex, offering cover for perils such as kidnap and ransom, cybersecurity breaches, terrorist attacks and political violence, commercial property damage and personal accident. These risks typically require specialist/expert assessment by insurers/syndicates <cit.>. Simplifying the process extensively, a client reaches out to a broker with a risk, the broker brings the risk to the Lloyd's of London marketplace. Underwriters that operate within the Lloyd's market are reached out to, the underwriters utilise their subject matter expertise, actuarial pricing strategies and portfolio management approaches to decide if they should offer a lead, follow line or no line at all. If an underwriter offers a lead line, they underwrite a portion of the risk and tender a policy as a lead insurer, usually the lead covers a bigger portion of the risk and thus is paid a bigger share of the premiums, subsequently taking on more risk. Every risk is insured by a single lead insurer who usually shapes the policy and multiple follow insurers who agree to the terms and offer a line size. Once the risk has been covered proportionally across many syndicates, premiums are distributed depending on the line size agreed by all parties. Conversely, if a claim comes through, then everyone pays out proportionally, see <cit.>.
The hardening and softening of the insurance market has been an area of academic enquiry over the years. The specialty insurance market can be a very costly, dynamic and highly volatile market to operate within, one characteristic of the market is the underwriting cycle. In short, when capacity is high, rates decrease and profits decrease, conversely, losses occur, insurers go insolvent or withdraw and capacity decreases, with subsequent upwards pressure on premium levels to generate acceptable return on investment. This is a typical soft to hard market transition, respectively <cit.>, which attracts new capacity. The cycle then repeats.
The majority of the literature, has focused on this underwriting cycle phenomena, and have utilised agent-based, mathematical and time-series models to investigate the emergence and drivers of these cycles <cit.>. Surprisingly, there seems to be contradictory views with regard to the causes of market cycles, for example, <cit.> argues that institutional lags are the main reason behind the liability insurance market cycle in the mid-1980s in North America when analysing insurance company level data. However, when analysing market-level data, they claim the opposite. Most in-depth literature reviews <cit.> agree that research into underwriting cycles intensified in the mid-1980s due to the “liability crisis". During this time, an interesting discovery made by <cit.> was that different lines of insurance have quasi-cyclical behaviour but others can't be discerned from white noise fluctuations, i.e., random walk. Due to these discoveries, Venezian et al. proposed auto-regressive (AR) models which replicated these cyclical behaviours. Following the findings from <cit.>, most researchers primarily utilised AR models to investigate cyclicality in the insurance market. Furthermore, to ensure these models lead to actionable insights empirically, <cit.> proposes that data used for analysing market cycles has traditionally been at the aggregate, industry level at a yearly temporal resolution, to remedy this, they argue for comprehensive datasets captured at a more granular spatio-temporal resolution.
There are several hypothesised causes of underwriting cycles discerned by researchers, these are:
* The flow of capital into and out of the market in response to market conditions such as major catastrophe events (capital shock) <cit.>, while some disagree <cit.>.
* General economic conditions, i.e., rise and fall of central bank interest rates <cit.>, contrary views <cit.>.
* Unanticipated inflation <cit.>, while <cit.> disagrees.
* Institutional delays and lags, i.e., data collection, regulation and renewal periods <cit.> while <cit.> also makes a counter argument.
* Forecasting errors, imperfect knowledge of the market <cit.>.
One of the prevailing reasons for why the analysis of market cycles has been inadequate is due to the cycle periods being longer than periods over which data is sampled <cit.>. Furthermore, traditional time-series approaches work well in analysing deterministic systems, however, given the volatile nature of financial systems, this approach can be limited <cit.>. Moreover, if a modelled system can be affected by external factors then these factors should explicitly be included in the time-series model <cit.>. For this and many other aforementioned reasons, this research article is well-timed and necessary.
Given the interest in modelling the insurance market, researchers have adopted individual-based modelling approaches, to capture new insights and overcome some of the earlier described issues <cit.>. Some researchers have utilised individual-based models to discover stylised facts <cit.> such as the emergence of market crashes and bubbles. <cit.> argue that traditional methods such as AR models, fall short in quantifying the existence of a cycle in time-series data and questions the ability for these methods to forecast accurately. The article <cit.> suggests that linear models are insufficient and other approaches used to detect periodicity should be considered. Individual-based modelling approaches are more appealing as they can model the individual behaviours and social interactions at the micro level and generate complex aggregate behaviours such as cycles at the macro level <cit.>. Some researchers agree that the insurance market is complex, irrational and heterogeneous, evidenced by the interactions of individuals such as actuaries, underwriters, claims adjusters, brokers and organisations such as syndicates, regulators, managing agents and capital providers <cit.>. The model developed by <cit.> tests the hypothesis that agents with simple decision-making rules interacting at the individual-level can produce aggregate complex behaviours depicting empirically valid market dynamics. <cit.> also tests the theory of plural rationality <cit.>, where they can parameterise the insurers mark-up calculation where a policy is either priced aggressively or priced using the actuarial fair price (market value). They found that as the pricing aggression increased, market volatility also increased due to large price fluctuations.
Research conducted in <cit.> proposes an agent-based simulation of the specialty insurance market. The premise of the research is to investigate the regulatory changes under Solvency II that came into force in 2016. The model proposed, agents and processes represented as re-insurers, insurers, customers, shareholders and cat bonds that interact with each other and share information. The article finds that when the number of risk models increases for an insurer, i.e., an insurer utilises several catastrophe models (these are processes used to evaluate and manage natural and man-made catastrophe risk from perils, for example, hurricanes, wildfires) more risks are insured and conversely fewer risk models lead to lower profitability and higher risk of defaulting. The research article provides many interesting results with regards to catastrophe modelling approaches, to advanced exposure management features, whereby each insurer follows real-world regulations such as Solvency II and the “Solvency Capital Requirement". This requirement is described as Insurers are required to have 99.5% confidence they could cope with the worst expected losses over a year. That is, they should be able to survive any year-long interval of catastrophes with a recurrence frequency equal to or less than 200 years. However, in the model of <cit.> some important aspects of the specialty insurance market have been neglected. These include attritional losses <cit.> as well as the syndication of risk, a unique feature of Lloyd's of London <cit.>, and finally the temporal granularity is fixed. Our proposed solution addresses the limitations of previous work. We believe our model is the first discrete event simulation of the specialty insurance market inspired by real-world actors within Lloyd's of London.
A key insight and novelty of our article is in the use of a novel DES framework as opposed to the traditional ABM approach adopted by the aforementioned articles <cit.>. Unlike the classical ABM approach, the DES approach captures the time-irregular and asynchronous nature of events in the specialty insurance market whereby events (such as insurance claims) occur at irregular time intervals from as little to as many events occurring simultaneously. Handling such unpredictable, irregular and asynchronous events in a classic ABM approach can be challenging, but is seamlessly handled using the DES approach. The benefits of DES have thoroughly been discussed in the following articles <cit.>. Our model architecture is highly modular compared to the other approaches <cit.>, as evidenced in the workflow diagram <ref>, this allows the user to “plug-in" with ease different pricing models (actuarial pricing, underwriter markup, exposure management), relationship networks (the relationships between syndicates and brokers), loss generators (attritional, catastrophic or both) and many output metrics such as yearly, monthly syndicate capitals, premiums offered and accepted, industry statistics (market health indicators), syndicate performance (syndicate health indicators), syndicate insolvency, catastrophe event (the risks affected and extent of losses), claims per risk and syndicates assigned to risk. These configuration processes and metrics will be described in detail in the Model Description <ref> and Results & Discussion <ref> sections. Moreover, data analysis notebooks describing the stylised facts and unique features of the model will be provided as supplementary materials, some of these include, catastrophe losses and its impact on market health, advanced syndicates with exposure management modules and how they deal with losses, syndicate relationships with multiple brokers and its impact on risk coverage, capital venting strategies and what this means for the market.
There are many reasons why insurers and/or researchers would utilise our proposed model to simulate the specialty insurance market dynamics. Practitioners may want a deeper understanding of the market dynamics, such as underwriting cycles, which drive the stability and profitability of the market. Moreover, the model provides both a quantitative and qualitative picture of the impact and drivers of these dynamics. Lastly, the model could be used to develop heuristics which enable better estimation of the current and future market conditions.
§ MODEL DESCRIPTION
The purpose of the Lloyd's of London model is to investigate the emergent, complex characteristics which arise from individual-level interactions of the specialty insurance market. In this pursuit, the model conceptualises the behaviours of syndicates and brokers with which complex processes are undertaken as shown in Figure <ref>. These individual-level interactions from the bottom-up lead to the emergence of stylised facts at the aggregate level which correspond to empirical market trends. In this section, we describe the main agents/processes which the model encapsulates, the events they respond to and emit (i.e. the actions they undertake) and how these features all fit together to produce the DES. We start this section by discussing the notion of time in the model and its importance.
[!htbp]
< g r a p h i c s >
Workflow diagram of the specialty insurance market discrete event simulation. The line colours convey unique relationships between processes and events, i.e., yellow: new risk to quote requested and quote component generated, green: lead quote generation and selection, teal: follow quote generation and selection, red: losses to claims, blue: industry level statistics update, purple: syndicate level capital update.
§.§ Time
Discrete Event Simulations do not necessarily require the notion of time compared to other methods such as ABMs. Instead, events trigger processes which the process in turn responds to by emitting additional events and/or performing actions, i.e., updating an internal state variable. In theory, there is no need for the concept of time. However, in reality, there is a notion of time associated with events. For simulations of complex financial systems, such as the specialty insurance market, the challenge is that events occur asynchronously while the timescale is irregular and varying. Capturing these features of the specialty insurance market is both paramount (in order to ensure the model reflects empirical trends) and difficult to achieve with a typical ABM as mentioned in the following <cit.>. Conversely, the flexibility of the DES framework, allows events to have an associated timestep, which only triggers once the timestep in question occurs, e.g., t = 3. In our DES, a regular timer is incorporated with a standard timesteps of days, months and years. This allows us to capture events which occur in the simulation at different levels of granularity. Moreover, we are able to investigate different emergent phenomena independent of the timescale across which they occur.
The regular timer in our model generates the events shown in Table <ref>. We also detail the processes which respond to the events and the description of the events.
§.§ Broker process
The primary purpose of the Broker process is to bring new risks to the market. This occurs in response to a Day event as noted in Table <ref>. Upon the triggering of a Day event, the broker generates n_r risks according to a Poisson distribution where the λ variable for the distribution is given by the risks per day (RPD) input parameter as described in Table <ref>. The Broker process emits a number of events when the risk is generated in order to broadcast the risk to the insurers/syndicates and to set deadlines for the quotes from the insurers to be finalised.
The broker responds to and generates the events shown in Table <ref>
§.§ Broker-syndicate network process
The broker-syndicate network process is responsible for requesting quotes for new risks entering the market from the registered syndicates. This is facilitated by a network topology which for a given risk, selects a number of syndicates to which a quote is requested. Therefore, a risk does not necessarily have to be broadcasted to all syndicates depending on the parameter choices. The options for the topologies are the following:
* Circular topology: inspired by the model presented in <cit.>.
* Network topology: inspired by the interviews and workshops with stakeholders at Ki & Brit Syndicates.
* Random topology: is a base case feature.
Prevalent in all the topologies is a lead and follow top_k, parameter as mentioned in Table <ref>. This parameter selects the best k syndicates based on the chosen topology methods. For instance, in the circular topology <cit.>, the distance between the brokers and syndicates is used as a measure of the “ease of doing business" where a small distance implies a low cost of doing business conversely a large distance implies a larger cost. The syndicates are then ordered based on the top_k parameter which selects the k lowest cost syndicates. The network topology represents a connected network/graph between the brokers and syndicates where the edge weightings represent the ease of doing business between the brokers and syndicates. The larger the edge weighting, the more likely for brokers and syndicates to do business and vice-versa. Once again, the syndicates are ordered based on the strongest relationship with the top_k parameter filtering this down to the strongest k relationships. Lastly, in the random topology (the adopted network for the experiments conducted below), as the name suggests, the syndicates are randomly ordered with the top_k parameter selecting the first k syndicates in the list.
Given the above, the broker-syndicate network responds to and generates the events shown in Table <ref>.
§.§ Central risk repository process
The central risk repository process is responsible for tracking all of the risks, quotes and underwritten policies in the market. This includes all of the quotes offered for a risk, the policy information if the risk has been underwritten such as who the leader and follower syndicates are. Finally, the central risk repository is responsible for applying any losses, whether attritional or catastrophic, to the underlying syndicates on the policy in question.
Given the above, the central risk repository responds to and generates the events shown in Table <ref>.
§.§ Syndicate process
The syndicate process represents one of the most detailed agents within the model, as it is responsible for a number of important functions. Primarily, the syndicate is responsible for pricing any risks which come to market and decide which line size to give. This is all done in the context of a capital management framework whereby syndicates must ensure they are appropriately capitalised even in the case of tail loss events occurring, in order to avoid insolvency. Lastly, the syndicate must also provide any dividend back to capital providers in the case of profitable performance. As the syndicate is responsible for a number of functions, this section has been split into a number of sub-sections according to the sub-processes which compose the main syndicate process.
Before moving on to the sub-processes, we note that the main syndicate process is responsible for coordinating and organising the sub-processes which compose it. In this regard, the syndicate process responds to and generates the events shown in Table <ref>.
§.§.§ Actuarial sub-process
Pricing of risks in the insurance market is a complex process, risks are heterogeneous, extremely variable and fundamentally stochastic and ambiguous <cit.>. Our model captures all of these features but makes the simplifying assumption that all risks are homogeneous and that they belong to a catastrophe exposed class such as property insurance. The price of a risk will depend on several factors, primarily, the nature of the risk itself (i.e. the frequency and severity with which the risk can occur) and the market's view of the risk (i.e. the supply and demand of services in the market) <cit.>.
These two factors are different from one another. The first, is related to the quantification of the risk itself without considering any external market influences. This process is often carried out by an actuary <cit.> The actuary will assess the risk on experience based metrics (historical data on the risk or similar risks) and/or exposure based metrics (quantification of the risk in absence of data but based on a risk profile). Fundamentally, the actuary attempts to come up with a “fair price" for the risk based on the expectation of losses. That is to say, a price which would cover the expected losses of the risk over its lifetime. The second factor, considers the price of the risk based on the market's view i.e. the supply and demand of insurance services. This process is typically carried out by an underwriter. The underwriter, will take the fair price guideline from the actuary and, in very simplistic terms, scale this price up or down based on the supply and demand in the market. For instance, although the actuary may propose a much higher fair price, the market trends may force the underwriter to reduce this price in order to be competitive.
In this section, we will focus on the actuarial sub-process, inspired by the work of <cit.>. In the next section, we will discuss the underwriting sub-process.
Based on the above, the actuarial sub-process price is given by two main components. The first component is the insurer's expected claim cost:
P_t = zX̅_t + (1-z)λ'_tμ'_t
where P_t is the insurers expected claim cost at timestep t, X̅_t is the insurers past weighted average claims, λ'_t is the industry-wide average claim frequency, μ'_t is the industry-wide expected claim cost and finally z is the internal experience weight input parameter Table <ref> which decides whether a syndicate weighs their own loss or the industry loss experience as more important. As per <cit.>, the insurers expected claim cost, X̅_t, is calculated as a simple exponentially weighted moving average where the weight is an input parameter to the model Table <ref> called the loss experience recency weight.
The final actuarial price, which we denote as P_at, is the sum of the insurers expected claim cost, P_t, and a “risk loading term", α F_t, where F_t, is the standard deviation of the insurer's claims while α is a input parameter to the model Table <ref>, called the volatility weight:
P_at = P_t + α F_t
As can be seen, the actuarial price in Equation <ref>, captures the main idea of syndicates pricing a risk to cover their expected claim losses as well as to allow for some volatility in the losses.
The actuarial sub-process responds to and generates the events shown in Table <ref>.
§.§.§ Underwriting sub-process
As explained in the previous section, the objective of the underwriting sub-process is to “scale" the actuarial price in order to match market supply and demand. We again, employ the equations used by <cit.>, where they apply neoclassic price theory for the price-elasticity of demand. The details of the derivation can be found in <cit.>. The underwriter scaling is given by:
P_t = P_ate^m_t
where P_t is the final price offered by the syndicate after the underwriters scaling, P_at is the actuarial price from Equation <ref> and m_t is the underwriter log markup which attempts to model the price-elasticity of demand in the market.
The underwriter markup, m_t, is also calculated as a simple exponentially weighted moving average where the weight is an input parameter to the model Table <ref> called the underwriter markup recency weighting. Details on how the underwriter markup is calculated can be found in <cit.>.
The underwriting sub-process responds to and generates the events shown in Table <ref>.
§.§.§ Value at Risk (VaR) exposure management sub-process
Exposure management is a critical function for all insurance firms, to such that appropriate levels of exposure management are regulated by law for insurers. Exposure management allows insurers to have a quantitative understanding of the tail risk associated with the policies that they underwrite. By doing so, insurers can quantify the impact of worst case scenarios on their portfolio. This informs insurer's underwriting strategy. For instance, given the state in Figure <ref>, an over-exposed strategy would be for lead syndicate B to underwrite all risks within region 1. Therefore, exposure management can alert syndicates if they are over-exposed to a given catastrophe peril region.
The Value at Risk (VaR) of an insurer's portfolio is a common measure used to quantify the level of risk taken by an insurer. The VaR with some exceedance probability, which we denote as α Table <ref>, identifies the amount of syndicate capital which is at risk in case of any tail events occurring e.g. exceedingly large catastrophe events. The exposure management sub-process is therefore responsible for ensuring that the syndicate capital remains above this threshold value. Given its importance, our model employs the VaR exposure management detailed in <cit.>.
The VaR Exposure Management sub-process responds to and generates the events shown in Table <ref>.
§.§.§ Premium exposure management sub-process
In the previous section, one approach to exposure management was explored via the VaR measure. However, as extensively discussed by <cit.>, VaR exposure management is a complicated process, often relying on computationally expensive monte carlo simulations and in many cases, difficult-to-measure variables. Often insurers seek approximations or proxy approaches which capture the essence of the VaR exposure management. In this section, we detail such a methodology which we refer to as Premium Exposure Management.
The premium an insurer collects for underwriting a risk can be thought of as a proxy for exposure. In particular, when the total premium written is compared to the total capital available, this gives a proxy measure of how exposed an insurer is to the potential risk of insolvency. For example, if an insurer has underwritten a large number of risks, and therefore is collecting a large premium, but their total capital is comparatively smaller. This indicates that the insurer might have insufficient capital available to cover the full range of potential losses. On the other hand, if premiums written were large but the capital available was also large then this would indicate that the insurer has suffered minimal losses and that their current underwriting strategy was profitable. This is the essence of the premium exposure management sub-process.
The Premium Exposure Management sub-process responds to and generates the events shown in Table <ref>.
§.§.§ Line size sub-module
A unique feature of the Lloyd's of London specialty insurance market is the syndication of risk i.e. there will typically be several syndicates on a policy which includes one lead syndicate and several follow syndicates. For this reason, the various syndicates will underwrite only a fraction of the risk, this is termed the line size. In our model, the lead syndicates offer a default line size and set the price of the policy based on the actuarial and underwriting sub-processes. The candidate follow syndicates, also price the risk based on their actuarial and underwriting sub-processes which they then compare to the actual price from the lead syndicate. This allows them to assess the “pricing strength" which in turn is used to decide the line size to offer. The pricing strength is defined as the ratio of the follower's proposed price to the lead price. That is, if the pricing strength is above one, this implies the price of the risk is good and a larger line size is offered and vice versa.
§.§.§ Dividend sub-module
As explored in <cit.>, our DES model also includes, the ability for syndicates to pay a dividend to capital holders on a yearly basis provided they have made a profit. The reason for why this is a class object and not a DES process, is because the only process that relies on its outputs is the syndicate process.
The dividend sub-module only becomes active when a Year event triggers the syndicate process. When this occurs, the syndicate uses the dividend sub-module to check whether a profit has been made and if so calculates the dividend as follows:
D = γ Pr_t
where D is the dividend paid, Pr_t is the profit made by the syndicate, and γ, profit fraction, is an input parameter to the model Table <ref>, which represents the fraction of the profit to pay out as a dividend.
§.§ Attritional loss generator process
Attritional losses (as opposed to catastrophe losses) are defined as those losses which are generally uncorrelated with each other in both space and time, have high frequency, low severity and are fairly predictable <cit.>. On the other hand, catastrophe losses, which will be discussed in the next sub-section, tend to be spatially correlated (affecting a number of policies concentrated by a given peril region, class or industry), low frequency, high severity and difficult to predict.
In our model, we develop the attritional loss generator process inspired by the work of <cit.>. The attritional loss generator process pre-generates a number of AttritionalLossOccurred events when the risk is first brought to the market. The number of claim events is given by the Poisson distribution with the λ value set as the yearly claim frequency, an input parameter as defined in Table <ref>. The severity of the loss is defined by the gamma distribution. The shape parameter of the distribution is given by 1/COV^2, where COV is the gamma distribution's coefficient of variation as defined in Table <ref>. The scale parameter of the distribution is given by μ COV^2 where μ is the mean of the gamma distribution. These AttritionalLossOccurred events are then placed within the event queue within the expiration date of the risk.
Given the above, the attritional loss generator responds to and generates the events shown in Table <ref>.
§.§ Catastrophe loss generator process
In our model, we develop the catastrophe loss generator process inspired by <cit.>. The catastrophe loss generator attempts to capture the phenomena that catastrophe losses are both correlated spatially (e.g. a number of policies concentrated by a given peril region) and temporally. For this reason, all risks in the model have an associated peril region, as observed in Figure <ref>. Unlike the attritional loss generator, which generates attritional events and losses on a risk by risk basis, the catastrophe loss generator generates catastrophe events and losses on a peril region basis. The total loss affecting a given peril region is then cascaded down to the affected risks within the peril region, and subsequently the lead and follow insurers.
When the simulation starts, the catastrophe loss generator pre-generates a number of catastrophe events over the length of the entire simulation. This is done via the Poisson distribution with the λ value set as the product of the mean number of catastrophe events per year Table <ref> and the number of years in the simulation. A peril region is randomly assigned to the CatastropheLossOccurred event. The total loss affecting the peril region is given by a truncated Pareto distribution with the minimum value set as the minimum catastrophe damage Table <ref>. The CatastropheLossOccurred events are then added within the event queue up until the end of the simulation.
Given the above, the catastrophe loss generator responds to and generates the events shown in Table <ref>.
§.§ Industry statistics process
The industry statistics process, as the name suggests, simply keeps a track of all the relevant industry statistics and metrics. This is necessary, as many of the syndicate sub-processes rely on market-wide metrics. Note, that this process does not attempt to mimic any real market agents/processes. It is simply just an aggregator of data.
The industry statistics process responds to and generates the events shown in Table <ref>.
§ RESULTS & DISCUSSION
The Lloyd's of London specialty insurance market is a complex system. Many in academia and the insurance industries have tried to identify the underlying processes that lead to the emergence of phenomena at the aggregate level. One particular phenomena that has been an area of focus, is the underwriting cycle (as described in Section <ref>). As a proof-of-concept we propose several experiments that aim to simulate conditions in the market with varying complexity to reproduce existing industry phenomena. Whereby practitioners and researchers can investigate the underlying features of the market that lead to these trends, better preparing those involved in the market to deal with exogenous shocks.
This section first presents empirical market dynamics, then delves into the four experiments, starting with a base case, introduction of catastrophe events, syndicates adapting to these events and finally the introduction of lead and follow dynamics (described in the previous section <ref>).
In Figure <ref> the market experiences catastrophe events which exacerbates periods of hardening and softening of the market. This phenomena is conveyed as the underwriting cycle, which as a benchmark could be a phenomena we reproduce using our DES model. In the following sub-section, we describe each model scenario (experiment) and subsequently present the results while discussing the findings in relation to Lloyd's of London.
§.§ Model scenarios
To ensure the model is able to tell a story from simplistic behaviours to more advanced features and outcomes, we start with a base case experiment, this and subsequent experiments are described below (see Section <ref> for descriptions of each model feature):
* Base case actuarial pricing, all syndicates use actuarial pricing models, premium exposure management and attritional losses occur (Scenario 1).
* Catastrophe event, all syndicates use actuarial pricing, premium exposure management, attritional and catastrophe losses occur (Scenario 2).
* Syndicates adapting to catastrophe events, all syndicates use actuarial pricing, VaR exposure management, attritional and catastrophe losses occur (Scenario 3).
* Leaders and Followers, syndicates can either be leaders or followers and use actuarial pricing, premium exposure management and only attritional losses occur (Scenario 4).
§.§ Base case actuarial pricing (scenario 1)
This experiment utilises the most basic components of the model, in order to demonstrate the ability for core features to represent important market phenomena. In this experiment, we include five syndicates and twenty-five brokers. This maintains the 1/5 ratio of syndicates to brokers as is the case in the Lloyd's of London market pocket guide <cit.>. As we are only utilising the actuarial pricing model, we expect the premium prices to converge around the fair price, which given the model parameters Table <ref> is $300,000. Secondly, we expect the loss ratio of syndicates to fluctuate between periods of profitability and losses, i.e., early signs of cyclicality. Note, due to computational complexity, the model was only run for a limited number of times.
In sub-figure <ref>, the syndicate response over time, is indicative of capital fluctuations in reality, as observed in <cit.> where the authors discuss capital profiles over time. For instance, some syndicates, i.e., syndicate 0, are bankrupted as a result of their underwriting strategy, while others ride the boom and bust period better. As we hypothesised, the premiums offered have converged to the actuarial fair price of approximately $300,000 sub-figure <ref>, the reason for this (refer to the sub-section <ref>) is because the syndicate's actuarial pricing, offers a price which attempts to cover its prior losses. Given the current model setup, the mean of the prior losses is $300,000.
For context, the loss ratio when ≥ 1 indicates an unprofitable syndicate as their losses are greater than their income, i.e., premiums. Conversely, < 1 leads to profitable syndicates. As mentioned, periods of profitability and losses is a crucial trait of all models that simulate the insurance market as described in <cit.>. Clearly, our model outputs also demonstrate this behaviour as observed in Figure <ref>.
§.§ World shock events, where catastrophes meet the insurance market (scenario 2)
In this experiment, we repeat the input parameters earlier, however, now we introduce the catastrophe loss generator. When catastrophe events occur, this should lead to major losses and the severity should vary across syndicates depending on their underwriting strategies. We expect to see some syndicates go insolvent, while others may not. Furthermore, insurance insiders and academics claim that the primary cause of cyclicality in the market, is due to unexpected catastrophe events <cit.>. Our model allows us to test this hypothesis, which we attempt in this section. These experiments should show exaggerated cyclical trends compared to the previous experiment.
The most intriguing result from this experiment, is shown in Figure <ref>, which demonstrates not only the presence of cyclicality in the premiums offered, but provides an explanation for why this phenomena occurs. Put simply, the cyclicality occurs, in the initial phase, when premiums begin to converge to a fair price, however, catastrophe events which result in large losses, forces syndicates to price premiums higher resulting in an increase in the prices offered. Eventually, once the effect of the catastrophe wears off, the syndicates once again try to converge towards the fair price and the cycle repeats itself.
As mentioned previously, cyclicality is pronounced and in some cases, due to large catastrophe losses, loss ratios go above one Figure <ref>.
§.§ Adapting to market shocks, the utility of VaR exposure management (scenario 3)
The purpose of this experiment, is to specifically showcase the effects of advanced exposure management methods. Due to brevity, we only showcase results indicative of this objective. To this end, syndicates have utilised simplistic premium exposure management. Now, we introduce the VaR exposure management, which allows syndicates to manage their exposure in a more sophisticated manner representing behaviours of real-world syndicates more closely.
VaR exposure management <cit.> should enable syndicates to adopt underwriting strategies which avoid over-subscribing to a given peril-region. A good exposure management strategy, is to distribute risk underwritten uniformly across all peril-regions <cit.>. For this reason, we present the uniform deviation (measures how much the real distribution of risks in the peril regions varies from a perfectly uniform peril-region distribution, where 0 is perfectly uniform). For comparison, we include no exposure management and premium exposure management outputs. As can be seen from Figure <ref>: From left to right, we observe that as the exposure management becoming more sophisticated and stringent, it is clear to see that the uniform deviation moves closer to zero.
§.§ Leaders and followers among syndicates (scenario 4)
The Lloyd's of London specialty insurance market as described in Section <ref> differs from other insurance markets. The most prominent difference is the syndication of risk, i.e., the ability for multiple syndicates to underwrite a risk either as leaders or followers. Lloyd's of London claims that the syndication of complex risks across multiple syndicates, allows these unconventional risks and the potential losses to be distributed among several insurers as opposed to one. This should result in less volatility, and lower likelihoods of insolvency as described in the documentation published by Lloyd's <cit.> and <cit.>. However, as far as the authors are aware, these claims have not been quantitatively tested in academic research, until now where our DES model allows us to investigate this particular claim.
As observed in Figure <ref> compared to Figure <ref> the volatility of the premium offered is significantly lower, with the premiums tightly converging towards the fair price. Due to the syndication of risks, this allows more syndicates to participate significantly in the market. This means that any losses, are shared among all syndicates, implying their loss experience is similar to each other. As a result, they all offer similar prices. This behaviour can also be observed in Figure <ref> where the loss ratios of the syndicates are tightly coupled/correlated, indicating a similar loss experience.
In this scenario, we observe that no insolvencies occurred, compared to the same scenario minus the lead and follow dynamics in sub-section <ref>. These findings, provide strong quantitative justification for the market structure imposed by the regulator Lloyd's of London.
In summary, the proof-of-concept DES can reproduce quantitatively and qualitatively market phenomena, such as the underwriting cycle, merits of lead - follow market structures, and the importance of exposure management.
§ CONCLUSION
This research article proposes a novel DES of the Lloyd's of London specialty insurance market. The model captures granular interactions of significant actors within the marketplace, i.e., syndicates and brokers. The model has demonstrated significant results with regards to market dynamics observed in the real-world, this includes signs of cyclical behaviours congruent of the underwriting cycle, which becomes more severe with catastrophe events. Given the unique model architecture, users can swap components such as pricing models, broker-syndicate relationship networks and other features with ease that may not have been possible in past attempts such as <cit.>. Furthermore, the article has proposed a conceptualisation of the lead and follow syndicate dynamics which is unique to Lloyd's of London, the results from these specific experiments have reinforced the innovative market structure, employed by Lloyd's which reduces volatility in their market.
The proposed DES has many strengths as discussed previously, for example, from a technical perspective, we adopt first-hand knowledge from experts in Lloyd's of London syndicates Ki and Brit and propose a novel, modular DES with changeable components. From a market dynamics and results perspective, we incorporate the lead and follow mechanics unique to Lloyd's of London, integrate both attritional and catastrophe loss events, and quantitatively study many important market phenomena. However, several areas of improvement can also be highlighted, for example, given the nature of the complexity which drives the market, it is difficult to abstract and quantify all aspects of the market appropriately. An example is quantifying the broker-syndicate relationship which is the cornerstone of the market. Capturing these individual-level human relationships is difficult with any modelling framework, however, individual-based modelling has allowed us to acquire new insights regarding this relationship. Furthermore, reinsurance is an important function of the insurance market. Reinsurance has a big effect on the capacity a syndicate can underwrite as they manage the tail risk of a portfolio. Modelling the availability and cost of reinsurance policies is therefore a significant driver of market dynamics. Conventional calibration and validation has not been pursued in this iteration of the proof-of-concept model, i.e., global sensitivity analysis. However, the behaviours of the model have been verified by Ki and Brit. In future there are plans to incorporate proprietary Ki and Brit data in order to calibrate the model so that these validated results can be disseminated.
Given the wealth of engineered features of the model, many future avenues can be explored. The underwriting markup (pricing model) <cit.> was not utilised in the experiments conducted in this article, however, comparing the different pricing models, i.e., actuarial and underwriting may lead to new insights with regards to market competition. Additionally, as discussed and utilised by <cit.>, another major component of a syndicate's activities is to pay out dividends in case of profitable performance. Given that the dividend feature is available in the model as described in Section <ref> this can easily be explored as a path for future work. We hope this model provides researchers and specialty insurance practitioners with new insights that enable future R&D projects and provide the means to reduce market volatility and enhance insurance businesses.
§ SUPPLEMENTARY MATERIALS
* The Hades framework is made open-source by Ki Insurance and can be found at the following link <https://pypi.org/project/hades-framework/>
alpha
|
http://arxiv.org/abs/2307.10200v1 | 20230709023156 | Disentangling Societal Inequality from Model Biases: Gender Inequality in Divorce Court Proceedings | [
"Sujan Dutta",
"Parth Srivastava",
"Vaishnavi Solunke",
"Swaprava Nath",
"Ashiqur R. KhudaBukhsh"
] | cs.CY | [
"cs.CY",
"cs.AI",
"cs.CL",
"cs.LG"
] |
Age of FGK Dwarfs Observed with LAMOST and GALAH: Considering the Oxygen Enhancement
Jinghua Zhang
Received August 12, 2023; accepted August 12, 2023
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Divorce is the legal dissolution of a marriage by a court. Since this is usually an unpleasant outcome of a marital union, each party may have reasons to call the decision to quit which is generally documented in detail in the court proceedings. Via a substantial corpus of 17,306 court proceedings, this paper investigates gender inequality through the lens of divorce court proceedings. While emerging data sources (e.g., public court records) on sensitive societal issues hold promise in aiding social science research, biases present in cutting-edge natural language processing (NLP) methods may interfere with or affect such studies. We thus require a thorough analysis of potential gaps and limitations present in extant NLP resources. In this paper, on the methodological side, we demonstrate that existing NLP resources required several non-trivial modifications to quantify societal inequalities. On the substantive side, we find that while a large number of court cases perhaps suggest changing norms in India where women are increasingly challenging patriarchy, AI-powered analyses of these court proceedings indicate striking gender inequality with women often subjected to domestic violence.
§ INTRODUCTION
The 2011 decennial census in India gave its citizens the following choices to select their marital status – never married, separated, divorced, widowed, married. Based on the census data, a study reported some startling facts <cit.>: 1.36 million of the Indian population is divorced which accounts for 0.24% of the married population, and 0.11% of the total population. More women were separated or divorced than men, and the number of separation was almost three times as high as the number of divorce.
Divorce, a historically taboo topic in India for ages <cit.>, seldom features in mainstream Indian discourse <cit.>. Recent indications of changing social acceptance of divorcees notwithstanding <cit.>, divorce in India still carries a considerable social stigma <cit.>.
How do we quantify gender inequality in Indian divorce? Surveys about divorce often have limited participation and a small sample size <cit.>, perhaps due to the social stigma attached. A vulnerable community – Indian women under conjugal distress – had limited visibility to social scientists. Via a substantial corpus of 17,306 divorce court proceedings, this paper conducts the first-ever computational analysis of gender inequality in Indian divorce based on public court records.
Even though written in English, legal texts are often domain-specific <cit.>. The considerable variation of legal jargon across countries and courts makes domain-specific analysis important. In that vein, Indian legal NLP is an emerging field <cit.>. Most NLP research on legal texts thus far has focused on building robust tools to analyze legal text. Recent research, however, on in-group bias <cit.> and sexual harassment <cit.>, and <Ref> and <Ref> suggest that automated methods to glean social insights from large-scale, legal texts merit investigation. Barring few recent lines of work <cit.>, there is surprisingly little literature on large-scale linguistic analysis of gender bias in India, let alone on legal text zeroing in on divorce.
While emerging data sources (e.g., public court records available on the web) offer opportunities for social scientists to study important and sensitive social issues that previously had limited survey data, applying cutting-edge NLP methods to newer domains requires careful evaluation of the critical question: How much of the (perceived) gender inequality as quantified by the methods truly reflects the corpus and how much of it is due to the inherent biases of the employed NLP methods?
In this paper, we show that the subtleties present in legal text present unique challenges. Unless we consider them and make non-trivial changes to existing methods, we may end up drawing inaccurate social conclusions.
We further show that sophisticated NLP methods built on top of large language models (LLMs) need scrutiny when applied to social inference tasks involving genders. We, in fact, conduct a much broader bias audit of these systems. Our audit reveals well-known LLMs often exhibit gender bias even on simple subject-verb-object sentence completion tasks. Through a corpus-specific text entailment analysis, we demonstrate that downstream applications such as natural language inference (NLI) systems also exhibit sensitivity to gender. We finally, present a novel inconsistency sampling method to mitigate this bias and present our social findings.
To summarize, our contributions are the following:
Social: We create a substantial corpus of 17,306 divorce court proceedings and conduct the first-ever analysis of gender inequality through the lens of divorce proceedings. While a large number of court cases perhaps suggest changing norms in India where women are increasingly challenging patriarchy <cit.>, our analyses reveal widespread domestic violence, dowry demands, and torture of the bride.
Methodological: We address extant gaps and limitations in multiple NLP frameworks. We propose non-trivial modifications to the framework <cit.> to make it suitable for legal text.
We demonstrate a novel application of text entailment <cit.> in quantifying gender inequality. We investigate several potential sources for model bias in NLP resources that can interfere with quantifying gender inequality. We present a novel inconsistency sampling method exploiting counterfactuals to mitigate this bias.
§ DATASET
§.§ Collection
We scrape all the publicly available court proceedings with the word between January 1, 2012 to December 31, 2021 from <https://indiankanoon.org/> (hereafter ), an Indian law search engine launched in 2008 and the largest free online repository of the court proceedings of different courts and tribunals of India. Prior computational law research <cit.> and gender focused social science studies <cit.> have used as source of data.
We download 86,911 case proceedings containing the word from using its advanced search feature. Filtering based on the keyword is a high-recall approach to obtain relevant cases with precedence in computational social science research <cit.>. However, the presence of the keyword may not always indicate a divorce court proceeding; for instance, the keyword can be used to describe the marital status of any of the litigants. It can also be used in an altogether different context (e.g., divorced from reality). We use the following heuristic to further refine our dataset. We also look for other words (e.g., , , ) and phrases (e.g., ), and check if such occurrences repeat for a minimum threshold (set to 5). On a random sample of 100 cases after we apply this cleaning method, a manual inspection reveals that 94 are divorce cases. Hence, our keyword-based filtering is reasonably precise. This pruning step retains 25,635 cases.
§.§ Data Pre-processing
To quantify gender inequality in court proceedings, we must disambiguate the legal parties – the plaintiff and the defendant – and accurately identify of the husband and the wife, who plays which role. Indian legal documents use a wide range of legal terms to denote the plaintiff (e.g., appellant, applicant, complainant, petitioner) and the defendant (e.g., respondent, nonapplicant, opponent). We observe different courts have different formats (sometimes, multiple formats) to summarize the proceedings. The documents also specify which party in marriage represents which role in several different ways (e.g., respondent/wife, respondent-wife, respondent aggrieved wife). We write a regular-expression-based pipeline and consolidate such information to identify the gender of the plaintiff and the defendant across all the states.
The names and salutations (e.g., , , , ) of the plaintiff and defendant also provide gender information. Subcultural naming conventions played a key role in assigning gender to the litigants in some of the cases. For instance, , meaning princess, is a Punjabi last name only for females <cit.>. Or , meaning sister, is solely used in many female names in Gujarat <cit.>. Dependence information of the litigants also provides gender information (e.g., , , ).[We did not find a single mention of in our dataset.]
Of the 25,635 cases, we could unambiguously assign gender to 17,306 cases. For each case, we replace each mention of the litigants as or accordingly. For example, a proceeding snippet “The plaintiff/wife has filed for a divorce. The plaintiff was married to the defendant for three years.”, will be modified to “The wife has filed for a divorce. The wife was married to the husband for three years.” This data set, _divorce, consists of 30,615,754 (30 million) tokens.
§ BRIEF OVERVIEW OF INDIAN LEGAL SYSTEM
Indian Judicial System is largely based on the English Common Law system (where, the law is developed by judges through their decisions, orders, and judgments). The nation has 28 states and 8 union territories (UT), and a total of 25 high courts (some high courts have jurisdiction of more than a state or UT). The federal structure has a supreme court coupled with the high courts that roughly handle the cases in a state or UT. The legal cases of divorce are usually handled by the family or district courts. However, some unresolved cases or sometimes fresh cases are also heard by the high courts. Since the court proceedings are public records and are digitally made available freely by , we found this dataset to be quite appropriate for a large-scale study on gender equality in court proceedings.
§ DOWRY IN DIVORCE PROCEEDINGS
The dowry system involves a transaction of financial assets between the bride's family and the bridegroom's family with the latter being the recipient of the financial assets. While legally prohibited in India since 1961 <cit.>, this practice has continued well after its legal prohibition and has a strong link to social crises such as female feticide <cit.>, domestic abuse and violence <cit.>, and dowry deaths <cit.>. In order to protect the bride from marital cruelty and domestic violence, Indian Penal Code introduced Section 498 in 1983 <cit.>.
Figure <ref> reflects relative proportions of divorce cases containing the text tokens and . For each state, we report the fraction of divorce cases that contain at least one mention of these two tokens. A higher intensity color indicates a larger proportion of such cases. We observe that overall, 24.38% of all cases and 21.86% of all cases mention and , respectively.
Jacob and Chattopadhyay, <cit.> reported that divorce in India does not follow any one-size-fits-all pattern across different states; there exists sufficient interstate variation even for the rate of divorce. We notice a considerable variation in mentions of dowry and section 498-A across different states indicating variance in reported cases of dowry or domestic violence. Among the states and the union territories, the top three entries in terms of dowry mentions are Telangana, Delhi, and Bihar while the top three entries in terms of Section 498-A mentions are Bihar, Telangana, and Andhra Pradesh. Bihar and Telangana have social science literature documenting dowry and domestic violence <cit.>. Apart from the overlap in the top three entries, the statewise dowry and 498-A mentions are moderately correlated (correlation coefficient: 0.67).
We next conduct a qualitative analysis of (alleged) dowry demands [This analysis follows the statements made by the plaintiffs]. On a random sample of 100 court proceedings where the (alleged) dowry demand is explicitly recorded, we observe that the estimated demanded amount is 393,100 ± 544,876. We observe demanded amounts as low as 5,000 to as high as 3,000,000 which explains the staggeringly high variance in our estimation. This also indicates the broad economic spectrum present in India and how far and wide the system of dowry (allegedly) persists. We further observe that cash is not always the solely demanded financial asset. Gold is the second-most commonly demanded asset. Out of the 100 cases, 34 cases report gold demands (71.2 ± 84.6 gm). When we adjust the valuation of demanded gold replacing it with the historical average gold price in India across 2012 and 2021 [Obtained from <https://www.bankbazaar.com/gold-rate/gold-rate-trend-in-india.html>], the estimated (alleged) demanded dowry is 474,798 ± 567,219.
§ METHODS OVERVIEW
We use two NLP methods to quantify gender inequality: (1) Word Embedding Association Test; and (2) a text entailment framework. A brief description follows.
§.§ Word Embedding Based Methods
The first metric is called ord mbedding ssociation est () introduced by <cit.>. To calculate the metric, the words are embedded and the vectors a and b are obtained for the words a and b respectively. The cosine similarity of these words are denoted by cos(a,b). The metric considers two sets of target words given by and , and two sets of attribute words Å and . Then, the score is defined as (, , Å, ) = (_x ∈σ(x, Å, ) - _y ∈σ(y, Å, ))/_w ∈∪σ(w, Å, ),
where, σ(w, Å, ) = _a ∈Åcos(w,a) - _b ∈cos(w,b).
Intuitively, σ(w, Å, ) measures the association of w with the attribute sets, and the score measures the differential association of the two sets of target words with the attribute sets. A positive score implies that the target words in is more associated with the attribute words in Å than and the words in is more associated with than Å.
§.§ Text Entailment Based Methods
Quantifying gender inequality relying on the distributed representation of words presents a diffused, bird's-eye view of the larger trends. Also, these methods are known to be data-hungry <cit.>. Data availability often becomes a limiting factor to conducting contrastive studies at different spatio-temporal granularity. In what follows, we present a novel application of text entailment, a natural language inference (NLI) task <cit.> that bypasses the data size requirement and equips us with a finer lens through which we can compare and contrast gender inequality with respect to individual verbs.
An NLI system take a premise 𝒫 and a hypothesis ℋ as input and outputs entailment, contradiction, or semantic irrelevance. For instance, the hypothesis some men are playing a sport is entailed by the premise a soccer game with multiple males playing <cit.>. As one can see, textual entailment is more relaxed than pure logical entailment and it can be viewed as a human reading 𝒫 would infer most likely ℋ is true. This framework has gained traction in several recent social inference tasks that include estimating media stance on policing <cit.>, aggregating social media opinion on election fairness <cit.>, and detecting COVID-19 misinformation <cit.>.
Formally, let NLI(𝒫,ℋ) takes a premise 𝒫 and a hypothesis ℋ as inputs and outputs o ∈{entailment, contradiction, neutral}. Following <cit.>, we define entailment ratio (denoted by ent(𝒟, ℋ)) for given corpus 𝒟 and a hypothesis ℋ, as the fraction of the individual sentences present in 𝒟 that entails ℋ:
ent(𝒟, ℋ) = ∑_𝒫∈𝒟I(NLI(𝒫, ℋ) = entailment)/|𝒟|,
where I is the indicator function. A larger value of ent(𝒟, ℋ) indicates greater support for ℋ in the corpus.
Consider we are interested in learning how often the husband and the wife are accused of torture (physical or emotional) in our corpus. We analyze this research question in the following way. We first construct a sub-corpus 𝒟_torture from the divorce court proceedings consisting of sentences that (1) mention or at least once; and (2) mention as a verb at least once. We next construct two hypotheses – ℋ_,torture and ℋ_,torture – using a and a as victims and perpetrators interchangeably. ℋ_,torture is A woman tortures a man and ℋ_,torture is A man tortures a woman. We next compute the entailment gap defined as
gap(𝒟_torture,torture) =
ent(𝒟_torture,ℋ_,torture) - ent(𝒟_torture,ℋ_,torture)
Effectively, this means we compute the fraction of sentences that entail A woman tortures a man in 𝒟_torture and subtract it from the fraction of sentences that entail A man tortures a woman in 𝒟_torture. An overall positive number indicates that the male has been described as the torturer more often than the female in court proceedings. A negative value would indicate the opposite way. Similar analysis can be extended to other verbs such as , , or .
§ DESIGN CONSIDERATIONS
Adapting the and entailment frameworks to quantify gender inequality in our domain requires careful consideration of several aspects described in what follows.
§.§ Verbs for Target Sets
Traditionally, score is used to quantify gender or racial stereotypes. Majority of the elements present in those attribute sets would be nouns and adjectives (e.g., criminals, terrorists, doctors, police) <cit.> and seldom verbs <cit.>.
We are interested in understanding the action space of the two parties fighting a divorce case; we want to know if the court described that one party tortured or abused the other. Hence, verbs are a natural choice for our target set.
We inspect the list of high-frequency verbs in the corpus and narrow down to the following ten verbs: _unpleasant =
{,
,
,
,
,
,
,
, , }. A small subset of these words are already present in the list of unpleasant stimuli presented in <cit.>. We further compute the average valence score of these words as per the lexicon presented in <cit.>. We find the average valence score of _unpleasant is 2.7, comparable to the average valence score (2.16) of unpleasant stimuli presented in <cit.>.
Divorce being a bitterly fought family situation, we observe a sparse presence of pleasant verbs such as , , or in our corpus. Since infrequent words in the corpus do not have reliable embeddings <cit.>, in contrast with traditional applications of score, we choose the target set to be an empty set.
§.§ The Torturer and the Tortured
The attribute sets Å and as defined in the score represents the identifiers used for the plaintiff and defendant in our data (e.g., Å consisting of , , , and consisting of , , etc.). However, notice that score is agnostic about whether the identifier is the contributor or the receptor of target words. For example, torture does not happen in isolation; it requires a torturer and one who is tortured. Unlike nouns, verbs are typically associated with two entities – the subject and the object. To disambiguate between “the husband tortured the wife” and “the wife tortured the husband”, a word embedding needs to understand this nuance. Otherwise, the embedding is likely to place both the plaintiff and defendant identifiers equidistant to the verb.
To disambiguate these two situations, we run the corpus through the POS tagger <cit.> to find out the subject and object of the sentences and whether the statements are in active or passive voice. Based on this, we classify the subjects and objects as `male perpetrator', `female perpetrator', `male victim', or `female victim', in the sentences that has the target verbs. We replace these four cases with four unique words (denoted by , ,, and , respectively) so that those words do not occur anywhere else in any of the documents. We call this new dataset _replaced.
§ WORD EMBEDDING BASED ANALYSIS
We are interested in two research questions:
RQ 1: How does gender inequality manifest in divorce court proceedings with respect to unpleasant verbs in 𝒳?
RQ 2: Is our careful disambiguation of the torturer and the tortured necessary at all?
In order to answer these two questions, we run two sets of experiments with identical training configurations. First, we run experiments on _replaced using the target and attribute sets as defined in the previous section. We train the word embedding model 10 times and calculate the scores for each of the following two cases: when both genders are (a) perpetrators, i.e., when Å={}, ={}, and (b) victims, i.e., when Å={}, ={}. We use the default parameters for training our FastText <cit.> Skip-gram embedding with the dimension set to 100 for all word-embeddings in this paper. Second, we run a baseline experiment with the original text data without replacing them with the four unique words (_divorce) and use the attribute sets as Å={} and ={}. The number of runs and the embedding method are the same in both experiments. The results are shown in <Ref>.
As already described, a negative score indicates is more associated with the target set as compared to Å. Hence, if we look from the perspective of the victim, we find that women are more associated with the unpleasant verbs than men. In contrast, when viewed from the perpetrator's perspective, a positive score implies that men are more associated with the unpleasant verbs. Hence, our results indicate that in our corpus, women are more often the victims while men are more often the perpetrators.
Our baseline experiments that do not make any distinction between the perpetrator and the victim give a score close to zero indicating near-perfect gender equality. This inaccurate result, while highly surprising from a social science perspective, is not unexpected given how the original framework functions. The two entities (husband and wife) are present around the unpleasant verbs with nearly equal frequency. If the method does not make any distinction between the roles of victim and perpetrator, will give inaccurate results. We thus carefully use the score to elicit the correct
gender bias when applied to legal texts for our social science research question.
§ SOCIETAL INEQUALITY AND MODEL BIAS
Our word embeddings are computed from scratch while our next set of experiments relies on downstream applications built on top of large language models. Large language models (LLMs) are known to have a wide range of biases due to the train data <cit.> and extant literature has examined gender bias in the form of occupational stereotypes present in NLI systems <cit.>. We thus need to disentangle societal inequalities that are potentially reflected in our corpus and model biases that are potentially present in the NLP applications.
Essentially, for a premise/hypothesis pair ⟨𝒫,ℋ⟩, the NLI system estimates the probability P(ℋ |𝒫). However, how LLMs encode the probability P(ℋ) when the hypotheses primarily consists of the two genders (male and female) and a set of verbs is understudied. A thorough investigation first reveals that the masked word prediction probability of several well-known LLMs is sensitive to gender. We next present a measure to quantify gender bias sensitivity of NLI frameworks and present mitigating strategies. Finally, we use a bias-mitigated NLI system on our corpus and report findings.
§.§ Implicit Bias in Agent and Theme in LLMs
Unlike existing literature that primarily target occupational stereotypes to quantify and analyze gender bias <cit.>, we focus on a very basic unit in a sentence – the verbs. Following <cit.>, let in a sentence X verbs Y, X represent the agent and Y represent the theme.
Many verbs imply the relative authority levels between the agent and the theme. For example, in the sentence The football coach instructed the players to play a conservative game, the agent (the football coach) has more authority than the theme (the players). In contrast, the agent has less authority than the theme in the sentence The football coach honored the players' suggestion to play a conservative game. First proposed in <cit.>, the connotation relation of power captures this notion of power differential between an agent and a theme with respect to a given verb.
While the connotation relation of power has been analyzed in the context of gender inequality in movie scripts <cit.> and follow-on research focused on editorial fixes to remove bias <cit.>, little or no literature exists that documents the implicit gender bias present towards the agent and the theme when specific verbs are considered. This research is important and has a broader impact beyond our current social inference task. For instance, if an LLM encodes that it is less likely for a woman to inspire or guide someone than a man, this bias may percolate to downstream tasks leading to erroneous social conclusions when applied to large-scale data for other social inference tasks.
We use cloze tests to evaluate this implicit bias. A brief description of cloze test follows.
Cloze test: When presented with a sentence (or a sentence stem) with a missing word, a cloze task <cit.> is essentially a fill-in-the-blank task. For instance, in the following cloze task: In the , it snows a lot, is a likely completion for the missing word. Word prediction as a test of LLM's language understanding has been explored in <cit.>.
Bias Evaluation Framework:
We describe our proposed testing framework for gender bias. Let _𝑐𝑙𝑜𝑧𝑒 (w, 𝒮) denote the completion probability of the word w with a masked cloze task 𝒮 as input. For a given verb v, we consider the following four cloze tests:
* A [MASK] v a woman (denoted by v_womanAsTheme)
* A [MASK] v a man (denoted by v_manAsTheme)
* A man v a [MASK] (denoted by v_manAsAgent)
* A woman v a [MASK] (denoted by v_womanAsAgent)
In an ideal world where the LLM treats men and women equally, _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) and _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme) should be equal. However, our preliminary exploratory analysis indicates that is not the case. For example, when v is set to inspire, _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) is 0.20 whereas _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme) is 0.16. When we set v to guide, the gap widens – _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) is 0.71 whereas _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme) is 0.36.
Again, in an ideal world where the LLM treats men and women equally, _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsAgent) and _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsAgent) should be equal.
Let 𝒱 denote the set of all verbs listed in <cit.> where the agent has more power than the theme. Our overall measures of implicit bias are:
(a) (1/|𝒱|) ·( ∑_v ∈𝒱 (_𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) - . . _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme)) ), and (b) (1/|𝒱|) ·(∑_v ∈𝒱 (_𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsAgent) - . . _𝑐𝑙𝑜𝑧𝑒 (woman, v_womanAsAgent)) ).
Measure (a) quantifies bias_agent. A positive value indicates that the LLM encodes a man being in the position of agent likelier than a woman on expectation.
Measure (b) quantifies bias_theme. A positive value indicates that the LLM encodes a man being in the position of theme likelier than a woman on expectation. We investigate three well-known LLMs for this audit: <cit.>; <cit.>; and <cit.>. We consider 1,222 verbs listed in <cit.>. We also consider verbs in 𝒳_unpleasant for this study.
Table <ref> summarizes our gender bias audit of LLMs with respect to verbs implying more power to the agent than the theme. We first note that for both verb sets, bias_agent is substantially larger than bias_theme. This result indicates that men are considerably more likely to be considered as the agent when women is the theme and the verb implies that the agent has greater power than
the theme. We also note that the completions favor mildly men over women even for the theme, however, the values are closer to 0.
§.§ Implicit Bias in NLI Systems
We describe our approach to quantify model bias in our NLI framework specific to our task.
Consider we modify the sub-corpus 𝒟_torture to 𝒟_torture^flipped where the gender identifiers in each premise sentence are flipped to the equivalent identifier of the opposite gender. For instance, the premise The wife tortured the husband both mentally and physically will be modified as The husband tortured the wife both mentally and physically. Flipping gendered words to test bias through counterfactuals in the context of coreference resolution has been previously explored in <cit.>. We argue that if a premise in 𝒟_torture entails A man tortures a woman, the flipped premise in 𝒟_torture^flipped should entail A woman tortures a man instead in a gender-neutral NLI system. Hence the entailment gap for computed on 𝒟_torture should be equal in magnitude and opposite in polarity as the entailment gap computed on 𝒟_torture^flipped. The NLI system's (ℳ) overall bias score with respect to verbs present in 𝒳_unpleasant is thus computed as
NLI_bias(ℳ, 𝒳_unpleasant) = ∑_v ∈𝒳_unpleasantabs( (gap(𝒟_v, v) + gap(𝒟_v^flipped, v))/|𝒳_unpleasant|.
In simple words, for each verb, we compute the entailment gap (value_1) for the relevant sub-corpus and the flipped sub-corpus (value_2). We subtract value_2 from value_1 and take the absolute value of the sum. The bias score is the average value of this sum across all verbs: a score close to 0 indicates that the NLI system has a minimal bias, whereas larger values indicate greater bias.
Our baseline is an off-the-shelf NLI system from Allen NLP trained using (denoted by ℳ_base). We find that NLI_bias(ℳ_base, 𝒳_unpleasant) is 0.27 [We note that a bias-aware NLI variant from Allen NLP has a better starting point (bias score 0.20) than the base model. However, the bias-aware model exhibits slower convergence than the base model when we conduct our active learning steps as discussed in Section 7.3. With identical experimental setting, after iteration 3, the bias-aware model improves its bias score to 0.133.].
§.§ Bias Mitigation Via Inconsistency Sampling
Active Learning is a powerful and well-established form of supervised machine learning technique <cit.> characterized by the interaction between the learner, aka the classifier, and the teacher (oracle or annotator). Each interaction step consists of the learner requesting the teacher the label of an unlabeled instance sampled using a given sampling strategy and augmenting the data set with the newly acquired label. Next, the classifier is retrained on the augmented data set. This sequential label-requesting and re-training process continues until some halting condition is reached (e.g., exceeded annotation budget or the desired classifier performance). At this point, the algorithm outputs a classifier, and the objective for this classifier is to closely approximate the (unknown) target concept in the future. The key goal of active learning is to reach a strong performance at the cost of fewer labels.
Some of the well-known sampling methods include uncertainty sampling <cit.>, certainty sampling <cit.>, and density-based sampling <cit.>.
Beyond a static strategy, more complex strategies such as adapting strategy selection parameters based on estimated future residual error reduction or combining multiple sampling strategies to balance the label distribution in the procured data set have been explored in <cit.> and <cit.>, respectively.
Inconsistency Sampling. First introduced in Dutta et al. <cit.>, this sampling technique exploits the underlying logical structure of the ⟨ premise, hypothesis ⟩ space. For instance, a premise cannot both entail (or contradict) a given hypothesis and its negation. In our work, we extend this idea and exploit a ⟨ premise, hypothesis ⟩ space richer than Dutta et al. <cit.> for logical inconsistency.
Consider the premise/hypothesis pair Continuously her husband used to harass and torture her everyday/A man tortures a woman. We argue that if this premise entails the hypothesis (which it does), the modified premise/hypothesis pair with replacing every gendered word with the opposite gender – i.e., Continuously his wife used to harass and torture him everyday/A woman tortures a man – should also entail. If not, it signals a logical inconsistency. For each sampling iteration, we add 60 samples giving equal weightage to the verbs present in 𝒳_unpleasant.
Table <ref> summarizes our active learning results. For both models, ℳ_base and ℳ_bias-aware, we conduct three rounds of active learning using inconsistency sampling and stop when the performance improvement becomes indiscernible (≤ 0.01). All annotations are independently conducted by two annotators. Since legal documents are typically written in clear, unambiguous language, we observe a near-perfect agreement (Cohen's κ value 0.96). The remaining disagreements are resolved through a post-annotation adjudication step. Table <ref> indicates that with subsequent active learning steps, our NLI system exhibits lesser bias. Given that the maximum possible bias score is 2, we achieve substantial improvement in mitigating the bias.
Now that we are more confident that our model inferences are less sensitive to gender, we evaluate the societal bias present in our corpus.
Figure <ref> summarizes our text entailment results. Barring , for all other verbs, men are identified as perpetrators more often than women. We further note that verbs that indicate physical abuse, such as and , particularly stand out with larger values. The average entailment gap for verbs unambiguously indicating physical harm – , , , , and – is much higher (0.41) than verbs that may or may not indicate physical harm (0.19) such as , , , , and . A manual inspection of randomly sampled 200 ⟨ premise, hypothesis⟩ pairs aligns with our automated method's overall findings.
§ DISCUSSIONS AND LIMITATIONS
In this paper, we present the first-ever computational analysis (to our knowledge) of gender inequality in divorce court proceedings in India. Based on the documented allegations of parties involved in the divorce, our analyses indicate a striking gender inequality as described in these public records. While documented evidence of marital distress in India exists in social science literature, how such factors play out in divorce has limited understanding. Our study sheds light on a vulnerable and vulnerable and practically invisible community in India.
Methodologically, we identify and address several gaps and limitations of existing NLP techniques to quantify gender inequality. We believe our finding specific to legal text is new, and our method to address it is simple, effective, and intuitive. Casting the problem of quantifying gender inequality as a text entailment task is also new. Our results on text entailment results suggest that NLI can be a viable tool to computational social science researchers to analyze similar research questions (e.g., who gets the child custody can be estimated with hypotheses the husband gets the custody of the child and the wife gets the custody of the child). Moreover, our bias mitigation strategy exploiting a novel inconsistency sampling technique using counterfactuals holds promise.
Our work has the following limitations.
Sentence level processing: An important point to keep in mind, however, is that our analyses operate at the sentence level. If in a court proceeding, a sentence records that the plaintiff accuses the defendant of wrongdoing which the defendant denies in a subsequent sentence, how these two contradicting claims are resolved in the court cannot be inferred without language models that can handle document-level contexts. We believe our research will open the gates for investigation with newer-age LLMs that can handle broader contexts.
Archival limitation: The sparse presence of the North-Eastern region in our dataset is most likely due to archival limitation as some of these states record the highest rate of divorce <cit.>. Our study is also limited by the overall archival extent of .
Economic independence: Some of the court proceedings mention the litigants' occupations. We annotated randomly 100 sampled occupations for women.
While an overwhelming majority of the sampled occupations are homemakers, compared to World Bank Data on labor force participation of women in India (23%), 32% of the women are working women in our sampled occupations.
Economic independence and divorce merit a deeper exploration.
Out-of-court settlements, separation, abandonment: Finally, not all unhappy marriages end up in divorce and reach court for dissolution. Many out-of-court settlements happen. As documented in <cit.>, the number of separated women in 2011 is almost three times the number of divorced women. Since divorce is still looked at as a social stigma <cit.> and family institutions are highly valued in India, there could be many women who continue with their dysfunctional marriages while unhappy. The court does not know their stories.
§ ETHICAL STATEMENT
We work with public court records. Prior studies exist on Indian court proceedings <cit.>. We conduct aggregate analysis refraining from presenting any personally identifiable information in the paper. Hence, we do not see any ethical concern. Rather, we believe our findings and methods can be valuable to policymakers and social scientists.
A study on binary gender inequality runs the risk of oversimplifying gender, which we acknowledge lies on a spectrum. Same-sex marriage is yet not legal in India. Further nuances will be needed to extend our work to other cultures allowing same-sex marriages. We are also sensitive to previous studies that point out the potential harms of the erasure of gender and sexual minorities <cit.>.
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|
http://arxiv.org/abs/2307.04099v1 | 20230709052131 | GNP Attack: Transferable Adversarial Examples via Gradient Norm Penalty | [
"Tao Wu",
"Tie Luo",
"Donald C. Wunsch"
] | cs.LG | [
"cs.LG",
"cs.CR",
"cs.CV"
] |
Visible and infrared self-supervised fusion trained on a single example
Nati Ofir
August 12, 2023
=======================================================================
Adversarial examples (AE) with good transferability enable practical black-box attacks on diverse target models, where insider knowledge about the target models is not required. Previous methods often generate AE with no or very limited transferability; that is, they easily overfit to the particular architecture and feature representation of the source, white-box model and the generated AE barely work for target, black-box models. In this paper, we propose a novel approach to enhance AE transferability using Gradient Norm Penalty (GNP). It drives the loss function optimization procedure to converge to a flat region of local optima in the loss landscape. By attacking 11 state-of-the-art (SOTA) deep learning models and 6 advanced defense methods, we empirically show that GNP is very effective in generating AE with high transferability. We also demonstrate that it is very flexible in that it can be easily integrated with other gradient based methods for stronger transfer-based attacks.
Adversarial machine learning, Transferability, Deep neural networks, Input gradient regularization
§ INTRODUCTION
Deep Neural Networks (DNNs) are the workhorse of a broad variety of computer vision tasks but are vulnerable to adversarial examples (AE), which are data samples (typically images) that are perturbed by human-imperceptible noises yet result in odd misclassifications.
This lack of adversarial robustness curtails and often even prevents deep learning models from being deployed in security or safety critical domains such as healthcare, neuroscience, finance, and self-driving cars, to name a few.
Adversarial examples are commonly studied under two settings, white-box and black-box attacks. In the white-box setting, adversaries have full knowledge of victim models, including model structures, parameters and weights, and loss functions used to train
the models. Therefore, they can directly obtain the gradients of the victim models and seek adversarial examples by misleading the loss function toward incorrect predictions. White-box attacks are important for evaluating and developing robust models and serve as the backend method for many black-box attacks, but is limited in use due to its requirement of having to know the internal details of target models. In the black-box setting, adversaries do not need specific knowledge about victim models other than their external properties (type of input and output). Two types of approaches, query-based and transfer-based, are commonly studied for black-box attacks. The query-based approach attempts to estimate the gradients of a victim model by querying it with a large number of input samples and inspecting the outputs. Due to the large number of queries, it can be easily detected and defended. The transfer-based approach uses surrogate models to generate transferable AE which can attack a range of models instead of a single victim model. Hence it is a more attractive approach to black-box attacks.
This paper takes the second approach and focuses on designing a new and effective method to improve the transferability of AE. Several directions for boosting adversarial transferability have appeared. Dong et al. <cit.> proposed momentum based methods. Attention-guided transfer attack (ATA) <cit.> uses attention maps to identify common features for attacking. Diverse Input Method (DIM) <cit.> calculates the average gradients of augmented images. <cit.> generates transferable AE using an ensemble of multiple models.
Despite the efforts of previous works, there still exists a large gap of attack success rate between the transfer-based setting and the ideal white-box setting.
In this paper, we propose a novel method to boost adversarial transferability from an optimization perspective. Inspired by the concept of “flat minima” in the optimization theory <cit.>
which improves the generalization of DNNs, we seek to generate AE that lie in flat regions where the input gradient norm is small, so as to “generalize” to other victim models that AE are not generated on. In a nutshell, this work makes the following contributions:
* We propose a transfer-based black-box attack from a new perspective that seeks AE in a flat region of loss landscape by penalizing the input gradient norm.
* We show that our method, input gradient norm penalty (GNP), can significantly boost the adversarial transferability for a wide range of deep networks.
* We demonstrate that GNP can be easily integrated with existing transfer-based attacks to produce even better performance, indicating a highly desirable flexibility.
§ METHOD
Given a classification model f(x): x ∈𝒳→ y ∈𝒴 that outputs a label y as the prediction for an input x, we aim to craft an adversarial example x^* which is visually indistinguishable from x but will be misclassified by the classifier, i.e., f(x^*) ≠ y. The generation of AE can be formulated as the following optimization problem:
max _x^*ℓ(x^*, y), s.t. x^*-x_p ≤ϵ,
where the loss function ℓ(·, ·) is often the cross-entropy loss, and the ł_p-norm measures the discrepancy between x and x^*. In this work, we use p=∞ which is commonly adopted in the literature. Optimizing Eq. (<ref>) needs to calculate the gradient of the loss function, but this is not feasible in the black-box setting. Therefore, we aim to create transferable AE on a source model yet can attack many other target models.
We develop a new method to boost adversarial transferability from a perspective inspired by “flat optima” in optimization theory. See Fig. <ref>. If an AE is located at a sharp local maximum, it will be sensitive to the difference of decision boundaries between the source model and target models. In contrast, if it is located at a flat maximum region, it is much more likely to result in a similar high loss on other models (which is desired).
Thus, we seek to generate AE in flat regions. To this end, we introduce a gradient norm penalty (GNP) term into the loss function, which penalizes the gradient norm of the loss function with respect to input. The reason is that flat regions are characterized by small gradient norms, hence penalizing the gradient norm will encourage the optimizer to find an AE that lies in a flat region. We thus enhance the adversarial transferability since a minor shift of decision boundary will not significantly change the loss value (prior work has shown that different networks often share similar decision boundaries).
§.§ Baseline Attacks
GNP is a very flexible method in that it can be easily incorporated into any existing gradient based method to boost its strength. We consider the following existing, gradient based attacks to demonstrate the effect of GNP.
Later in sec:experiments, we will also show how GNP works effectively on state-of-the-art transfer-based attacks as well.
Fast Gradient Sign Method (FGSM). FGSM <cit.> is the first gradient-based attack which crafts an AE x^adv by attempting to maximize the loss function J(x^adv, y; θ) with a one-step update:
x^adv=x+ϵ·sign(∇_x ℓ(x, y; θ)),
where ∇_x J(x, y; θ) is the gradient of loss function with respect to x, and sign(·) denotes the sign function.
Iterative Fast Gradient Sign Method (I-FGSM). I-FGSM extends FGSM to an iterative version:
x_t+1^adv = x_t^adv + α·sign(∇_x_t^advℓ(x_t^adv, y; θ)),
x_0^adv = x,
where α=ϵ / T is a small step size and T is the number of iterations.
Momentum Iterative Fast Gradient Sign Method (MI-FGSM). MI-FGSM <cit.> integrates a momentum term into I-FGSM and improves transferability by a large margin:
g_t+1 = μ· g_t + ∇_x_t^advJ(x_t^adv, y; θ)/∇_x_t^advJ(x_t^adv, y; θ)_1,
x_t+1^adv = x_t^adv + α·sign(g_t+1),
where g_0 = 0 and μ is a decay factor.
§.§ GNP Attack
As explained in sec:method, we aim to guide the loss function optimization process to move into a flat local optimal region.
To this end, we introduce GNP to penalize large gradient norm, as
L(x, y) = ℓ(x, y)-λ∇_x ℓ(x, y)_2
where ℓ(·) is the original loss function of the source model, and the regularization term is our GNP, which encourages small gradient norm when finding local maxima.
For gradient based attacks (e.g., FGSM, I-FGSM, MI-FGSM, etc.), we need to calculate the gradient
of the new loss (<ref>). To simplify notation, we omit y in the loss function since we are calculating gradient with respect to x. Using the chain rule, we have
∇_x L(x)=∇_xℓ_(x)-λ∇_x^2 ℓ_(x) ∇_xℓ_(x)/∇_xℓ_(x)
This equation involves the calculation of Hessian matrix H = ∇_x^2 ℓ_(x). This is often infeasible because of the curse of dimensionality (such a Hessian matrix in DNNs tends to be too large due to the often large input dimension). Therefore, we take the first-order Taylor expansion together with the finite difference method (FDM) to approximate the following gradient:
∇_x L_(x+rΔx)≈∇_xℓ_(x)+H rΔx
where Δx=∇_x ℓ(x)/∇_xℓ(x), and r is the step length to control the neighborhood size. Thus we obtain the regularization term of (<ref>) as:
H∇_xℓ_(x)/∇_xℓ_(x)≈∇_xℓ(x+r ∇_xℓ_(x)/∇_xℓ_(x))-∇_xℓ(x)/r
Inserting (<ref>) back into (<ref>), we obtain the gradient of the regularized loss function as:
∇_x L(x)=(1+β) ∇_xℓ_(x) -β∇_xℓ_(x+r ∇_xℓ_(x)/∇_xℓ_(x))
where β=λ/r is the regularization coefficient. We summarize the algorithm of how GNP is integrated into I-FGSM in Algorithm <ref>, but I-FGSM can be replaced by any gradient based attack.
§ EXPERIMENTS
§.§ Experiment Setup
Dataset and models. We randomly sample 5,000 test images that can be correctly classified by all the models, from the ImageNet <cit.> validation set. We consider 11 SOTA DNN-based image classifiers: ResNet50 <cit.>, VGG-19 <cit.>, ResNet-152 <cit.>, Inc v3 <cit.>, DenseNet <cit.>, MobileNet v2 <cit.>, SENet <cit.>, ResNeXt <cit.>, WRN <cit.>, PNASNet <cit.>, and MNASNet <cit.>. Following the work in <cit.>, we choose ResNet50 as the source model and the remining 10 models as target models.
Implementation Details. In experiments, the pixel values of all images are scaled to [0, 1]. The adversarial perturbation is restricted by 3 scales ϵ=4/255,8/255,16/255. The step length is set as r=0.01 and regularization coefficient β=0.8, we run 100 iterations for all attacks and evaluate model misclassification as attack success rate.
§.§ Experimental Results
§.§.§ Integration with baseline attacks
We first evaluate the performance of GNP by integrating it with baseline attacks including I-FGSM and MI-FGSM. The results are shown in tab:1. We use a pre-trained ResNet50 as the source model and evaluate the attack success rate (ASR) of the generated AE on a variety of target models under different scales of perturbation ϵ. GNP achieves significant and consistent improvement in all the cases. For instance, taking the average ASR of all the 10 target models under perturbation ϵ = 8/255, GNP outperforms I-FGSM and MI-FGSM by 26.51% and 13.67%, respectively. In addition, the improvements of the attack success rates on a single model can be achieved by a large margin of 33.06%.
§.§.§ Integration with existing transfer-based attacks
Here we also evaluate the effectiveness of GNP when incorporated into other transfer-based attacks such as DIM <cit.> and TIM <cit.>. The results are given in tab:2 and show that DIM+GNP and TIM+GNP are clear winners over DIM and TIM alone, respectively. Specifically, DIM+GNP achieves an average success rate of 91.95% under ϵ = 16/255 for the 10 target models, and TIM+GNP outperform TIM by a large margin of 16.28% under ϵ = 8/255. We note that we only present the integration of GNP with two typical methods here, but our method also apply to other more powerful gradient-based attack methods.
§.§.§ Attacking “secured” models
For a more thorough evaluation, we also investigate how GNP will perform when attacking DNN models that have been adversarially trained (and hence are much harder to attack). We choose three such advanced defense methods to attack, namely, JPEG <cit.>, R&P <cit.> and NRP <cit.>. In addition, we choose another three ensemble adversarially trained (AT) models, which are even harder than regular AT models, and attack them: Inc-v3_ens3, Inc-v3_ens4 and IncRes-v2_ens1 <cit.>. We craft AE on the ResNet50 surrogate model with ϵ=16/255, and use DIM+TIM as the “backbone” to apply GNP. The results are presented in tab:3, where we can see that GNP again boosts ASR significantly against the six “secured” models, achieving consistent performance improvements of 11.46–14.37%.
§.§ Ablation Study
We conduct ablation study on the hyper-parameters of the proposed GNP attack, i.e., step length r and regularization coefficient β. Since r represents the radius of neighborhood that is flat around current AE, a larger r is preferred; on the other hand, setting it too large will increase the approximation error of Taylor expansion and thus mislead the AE update direction. The β is to balance the goal of fooling the surrogate model and finding flat optima. fig:ablation reports the results of our ablation study, where ASR is averaged over 10 target models (excluding the source ResNet50) attacked by I-FGSM + GNP with ϵ=8/255. We observe that adding the GNP regularization term clearly improves performance (as compared to β=0) and the performance gain is rather consistent for β in a wide range of 0.6–1.6. The step length r does not affect the performance gain too much either, and r=0.01 seems to be the most stable. Thus, the ablation study reveals that GNP is not hyper-parameter sensitive and works well in a variety of conditions.
§ CONCLUSION
In this paper, we have proposed a new method for improving the transferability of AE from an optimization perspective, by seeking AE located at flat optima. We achieve this by introducing an input gradient norm penalty (GNP) which guides the AE search toward flat regions of the loss function. This GNP method is very flexible as it can be used with any gradient based AE generation methods. We conduct comprehensive experimental study and demonstrate that our method can boost the transferability of AE significantly.
This paper focuses on untargeted attacks, but GNP can be rather easily applied to targeted attacks as well, by making a small change to the loss function. We plan to have a thorough investigation in future work.
IEEEbib
|
http://arxiv.org/abs/2307.13116v1 | 20230712082737 | Pathway: a fast and flexible unified stream data processing framework for analytical and Machine Learning applications | [
"Michal Bartoszkiewicz",
"Jan Chorowski",
"Adrian Kosowski",
"Jakub Kowalski",
"Sergey Kulik",
"Mateusz Lewandowski",
"Krzysztof Nowicki",
"Kamil Piechowiak",
"Olivier Ruas",
"Zuzanna Stamirowska",
"Przemyslaw Uznanski"
] | cs.LG | [
"cs.LG",
"cs.AI",
"cs.DC"
] |
Authors in alphabetical order. Corresponding author: [email protected].
[email protected]
Pathway.com
Paris
France
We present Pathway, a new unified data processing framework that can run workloads on both bounded and unbounded data streams. The framework was created with the original motivation of resolving challenges faced when analyzing and processing data from the physical economy, including streams of data generated by IoT and enterprise systems. These required rapid reaction while calling for the application of advanced computation paradigms (machine-learning-powered analytics, contextual analysis, and other elements of complex event processing). Pathway is equipped with a Table API tailored for Python and Python/SQL workflows, and is powered by a distributed incremental dataflow in Rust. We describe the system and present benchmarking results which demonstrate its capabilities in both batch and streaming contexts, where it is able to surpass state-of-the-art industry frameworks in both scenarios. We also discuss streaming use cases handled by Pathway which cannot be easily resolved with state-of-the-art industry frameworks, such as streaming iterative graph algorithms (PageRank, etc.).
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Pathway: a fast and flexible unified stream data processing framework for analytical and Machine Learning applications
Przemysław Uznański
August 12, 2023
======================================================================================================================
§ INTRODUCTION
Traditionally, data processing systems were designed either for high throughput batch computations, or for low latency streaming processing. However, modern data applications often demand low latencies at high data throughputs. One solution is the lambda architecture <cit.>, which calls for running two similar workloads: a batch one for exact computations on historical data and a streaming one used to patch the batch results with latest data. Alternatively, aiming to avoid architecture complexity, it is also possible to rely on data processing frameworks which unify batch and streaming computations.
The new data processing framework which we describe in this paper, Pathway, has a unified runtime suitable for running both streaming and batch tasks. Its design results from the need to perform certain types of real-time analytics workloads, which we considered in the logistics and supply chain vertical (see Section <ref>) but arguably representative of a wider range of industry data. These workloads call for a contextual data analysis, sometimes entering into the real-time machine learning space, in addition to giving significant attention to out-of-order data point arrival in event streams. They also typically require the reconciliation of numerous event streams, some of which may carry contradictory (erroneous) information.
Pathway is a data processing framework with a Python API and a reactive data processing engine with a tunable batch size which allows it to be dynamically adjusted for a desired throughput vs latency trade-off. In this paper, we describe its features and provide benchmark results, comparing Pathway to leading batch and streaming data processing systems on a classical analytical benchmark, namely a word counting task implemented using groupby, and a fundamental iterative graph processing algorithm, namely PageRank. We demonstrate that Pathway is capable of achieving throughput outperforming state-of-the-art general purpose batch engines, while being able to respond with latencies better than state-of-the-art streaming systems. At the same time, Pathway succeeds in handling in streaming mode types of iterative and contextual workloads, such as PageRank on a changing graph, which to our knowledge are not supported by any generally industrialized system with a Dataframe/Table API programming layer. Overall, the system performance is owed to a combination of design choices around mapping between tabular syntax and actual key-value data organization, the performance of the underlying differential dataflow assembly for key-value data, the designed operator primitives, and inter-operator optimizations made in the transpilation process.
The paper is structured as follows. In Section <ref>, we outline the original motivation for the creation of Pathway originating from industry use cases. In Section <ref>, we provide a historical overview of some of the major approaches to dataflow-based streaming data processing. An overview of the Pathway framework, together with code examples, is put forward in Section <ref>. In Section <ref> we provide results of benchmarks of Pathway, Spark, Flink, and other frameworks for wordcount (streaming) and PageRank (batch, streaming, and backfilling). Section <ref> contains some concluding remarks and directions of work in progress.
§ INITIAL INDUSTRY MOTIVATION
We initially designed Pathway as a data processor able to accommodate the needs of our real-time analytical platform. We were working with major clients in logistics and supply chain, primarily in international trade (third-party logistics, containerized maritime trade) and postal services. We were creating deployments for perpetual workloads, with a focus on geospatial events data representing events recorded by moving assets. This included IoT data from physical tracker devices, GPS data collected from applications deployed in operators’ mobile phones, location scans recorded at depots and warehouses, and manual operator entries. Physical aspects considered included containers, shipments, trucks, human operators, and vehicles in passenger transportation. A single data enrichment pipeline was deployed and intended for use across multiple use cases in the organization (see Fig. <ref> for an illustration), most immediately in the context of physical process observability and monitoring (real-time control towers), anomaly-detection, and forecasts (Estimated Times of Arrival). This fits into the spirit of building cross-organizational “data products” which are often sponsored by multiple users across the organization <cit.>. A discussion of some of the business objectives behind specific deployments undertaken by La Poste (French Postal Services), DB Schenker (a major global freight forwarder), and others can be found on Pathway’s website <cit.>.
A characteristic shared across most of the deployments we observed, and perhaps a generally-accepted property of the industry verticals concerned, was the nature of the event streams. While the majority of data volumes was arriving with a latency of up to several seconds from measurement to system ingress, a significant part of the data was nonetheless arriving delayed or revised (corrected) within a time window of several hours. This stemmed from numerous causes, from network connectivity issues of devices to manual correction of incorrect or incomplete manual data entries. Overall, the type of data treated gives rise to the following feasibility thesis about process observability design in general: preliminary conclusions (insights and alerts) can be drawn from the incoming data within seconds of data points’ arrival, however, delayed data points can also lead to a change in detected anomalies and changes of “digital twin” representation of the traced processes, even up to 48 hours later.
In terms of requirements on outcomes, the considered analytics deployments operate across multiple time scales. Certain types of urgent alerts (such as physical assets veering off-course, unplanned door-opening alerts, etc.) need to be raised and dealt with within seconds. Alerts related to transport quality monitoring (temperature changes of perishables, etc.) or changes to congestion in routing and depots typically need to be raised within minutes. Some information related to changes of process (e.g., detection of new key points of interest) can be recomputed with a significantly large delay, up to 24 hours. This creates room for a certain trade-off between real-time and minibatch computation.
At the same time, for the analytics data pipeline we were providing to our clients, full batch recomputation of the entire pipeline was not a feasible option. Many insights relied on contextual computation. For example, in the absence of data labels, obtaining insights related to correct transport monitoring conditions (temperature, shocks, etc.) is often only feasible through analysis of “typical” values observed for similar transports in the past. Also, the attribution of certain data anomalies, such as a GPS tracker on an asset leaving a prescribed depot area at an incorrect time, to a specific probable cause (e.g., sensor measurement aberration due to GPS noise, versus, actual transport anomaly) is highly dependent on the context of other measurements made in a similar or comparable area, or by the same or comparable measurement device in the past.
Overall, the data pipelines considered relied on data-driven insights which were fed typically upwards of 1 year of historical data as a reference frame for applied machine learning models. The computational data pipelines despite significant optimization could take upwards of 24 hours to perform a complete recomputation on the multi-core machines allocated to them by the clients. This made our move to streaming / minibatch analytics pipelines driven by a double motivation. First, we needed to reduce latency for the time-critical use cases. Secondly, we needed to introduce pipeline incrementality to optimize excessive recomputation in view of the available computational resources. A final need was related to quickly accommodating changes to pipeline code, notably, being able to integrate fixes to bugs and data problems, without having to wait more than a day to obtain effects.
For some of the less time-critical and less computationally demanding operations in the pipeline, non-incremental batch computation remained an option. In this case, our expectation was to share logic used by both these batch and real-time procedures, to avoid some of the issues of lambda architecture.
Finally, the engine powering the pipeline was required to be deployable both in client data centers or with major cloud providers, without cloud provider lock-in.
These aspects were the first motivation in our design of the Pathway data processing framework.
§ RELATED FRAMEWORKS, BATCH, AND STREAMING SYSTEMS
MapReduce <cit.> demonstrated how many high-throughput, high-reliability batch computing jobs may be implemented using a common pattern of chaining two user defined operations: a mapper and a reducer, while handling at the framework level the scheduling of computations, their restarts and proactive control over stragglers. It influenced all following data processing systems and defined the main desiderata for systems: decoupling storage from compute and preserving timeliness and reliability of computations in the presence of failing machines.
Flume <cit.> introduced computation graphs on top of MapReduce, allowing deployment of complex multistep data processing pipelines expressed using simple operations on large collections of data. Then, Spark <cit.> introduced computation graphs over resilient in-memory datasets, RDDs, for fast and reliable batch computations. Spark pipelines can be written in Java, Python and SQL. Spark was also extended to handle simple cases of streaming data <cit.>, implemented using the RDD model conceptually as an infinite sequence of machines holding batches of a table which come online one-by-one. Independently, Spark GraphX <cit.> extension generalizes the Pregel <cit.> model for batch computations on graph data.
A major trouble related to stream processing frameworks is state management: beyond the simplest cases, stream elements are not processed independently from one another. Instead, a stream processor maintains a state and updates it with each new consumed stream element. The MillWheel system <cit.> introduced a model for persistent stream processing backing the state to an external scalable database, such as BigTable <cit.> or Spanner <cit.>. Concurrently, the Flink <cit.> framework uses barrier snapshotting to maintain consistent snapshots of the system operation. However, streaming systems must also deal with the problem of producing consistent outputs under data late arrivals. The Dataflow <cit.> model highlighted the problem and presented practical ways to trade correctness for response latency by introducing a notion of watermarks and activation triggers which can be used to precisely control how an application reacts to late data.
The need to provide real-time answers has motivated the Lambda <cit.> architecture. Introduced by Apache Storm <cit.> creators, it advocated combining a batch system that processed all historical data in an exact way with a speed layer that provided approximate answers to real-time implemented using an event streaming system. Many systems took the idea further, introducing Streaming and Batch APIs similar enough to be interchangeable based on the needs. The Beam project provides a compiler for computation graphs into either streaming or batch processing backends. Apache Flink maintained batch and streaming engines with similar capabilities, with the notable difference of iterations supported only by the batch layer. Recently, the batch engine was abandoned in favor of running the streaming engine in a batch runtime context which, however, lacks iterative computation support. In this context, we note that Pathway enforces and enables strict parity between batch and streaming computations, by using a unified execution engine.
A powerful incremental data processing framework was developed by the Naiad <cit.> team and later continued in Rust as a pair of projects, Timely and Differential Dataflow <cit.>. They propose to generalize tracking progress in a distributed computational graph using partially ordered clocks (Timely Dataflow), and adding to this the capacity of working with deltas (Differential Dataflow). These capacities are also to varying degrees exploited by projects transpiling to Differential Dataflow from different API’s, such as those based on SQL <cit.> or Datalog <cit.>.
Pathway is also inspired by deep learning frameworks: Theano <cit.>, Tensorflow <cit.> and PyTorch <cit.>. Like Pathway, they are libraries for the Python programming language, betting on Python becoming the language for data wrangling and artificial intelligence. The lessons learned from Tensorflow’s story across its versions 1 and 2 were behind a lot of design decisions in Pathway’s dataflow graph-building, as well as Pathway’s decision to adhere to Python as the primary supported API. At the same time, Pathway more closely follows PyTorch’s approach of being able to interact with a partially (dynamically built) computation graph during early experiments.
§ THE PATHWAY FRAMEWORK
Pathway is a Python-based data processing framework which allows expressing data transformations. Pathway was made to be easy and fast. The implementation has two layers - a runtime engine written in Rust and a Python layer handling computation graph (dataflow) building and optimizing. Pathway code can be written and tested interactively (even in Jupyter notebooks) with the computation graph being built in the background. This allows for quick prototyping needed in data science tasks. The same code can then be run on streaming data sources by disabling eager computation and instead handing the computation graph to the runtime engine.
In any deployment - on either streaming data sources or or in interactive mode, Pathway calls into the same Rust runtime. This allows us to obtain the needed computational efficiency while preserving the fast-paced development of Python code, and also preserving consistency of results across different modes of deployment.
In terms of API, a Table is the primary object for expressing data transformations in Pathway. A Table is a collection of columns that share an underlying set of identifiers. Internally, the Pathway engine operates on columns, which are a data storage with a homogenous type. Columns are indexed with identifiers.
In terms of usability, Tables are directly comparable to Dataframes in Pandas and PySpark, or Tables in Flink’s Table API. From a user perspective, this allows to interpret table transformations as happening on static data, with static columns and tables, while keeping in mind that the expressed computation supports dynamic data. That is, inserts, deletions and modification of input data points are automatically propagated through the dataflow, resulting in updates to outputs.
Pathway's approach is meant to eliminate most concerns about impact of system processing time and out-of-order data arrival on computational outcomes. Bounded stream workloads can in principle be replayed in Pathway, obtaining repeatable (identical) final results in output tables each time. This comes
subject to a number of evident assumptions - such as not calling in user-defined code external state processing functions or API's which do not behave deterministically, which depend on wall clock time, or on data processing order. Code written natively in Pathway is transparent to these concerns, allowing for predictable operation with clean code logic, and easier testing in CI/CD pipelines.
Functions defined at data row level may be expressed in Pathway using standard map (apply) and flatmap syntax. A notable extension of Pathway’s syntax is that of transformer classes which take a declarative (ORM) view of computed data tables. They allow for the expression of logic with inter-row dependencies at row-level, permitting e.g. recursive search over rows of a data table representing a graph or other data structure. This general concept is reminiscent of Pregel-like interfaces.
A syntax restriction of Pathway with respect to local batch frameworks such as Pandas is that offset-based row indexing is not supported by Pathway. Instead, Pathway provides fast support for local iteration over sorted indexes (most efficient over columns arriving in almost-sorted order, such as event time columns). This restriction is logical from the point of view of deploying code in a streaming environment, where offsets in data ordering have little meaning, particularly for out-of-order data. It also seems logical for batch workloads which are meant to shard / scale.
Pathway code transpiles from Python to a high-level dataflow graph which corresponds roughly to the code logic defined by the user. This high-level dataflow then transpiles to an internal dataflow assembly. In a precompilation phase, any SQL expressions which were placed within the Python code via Pathway’s SQL API <cit.> are also first decomposed with a parse tree and converted into the high-level dataflow graph.
In terms of the assembly layer, with respect to incrementally maintaining the results of iterative algorithms, we originally found the logic of the Differential Dataflow project to be closest to satisfying our needs. At the same time, the approach did not provide the data abstraction layer we needed nor interoperability with other systems or data sources. Currently, Pathway code transpiles to a dataflow assembly in Rust relying on a modified subset of Differential Dataflow. A lot of Pathway’s dataflow assembly consists of custom operators, built on top of a modified LSM tree implementation, with extensions allowing for fast handling of event contexts in sharded time-sorted event streams.
Within the scope of the transformative syntax at a Table level, the correspondence between Pathway’s Table operators and those in its dataflow assembly layer is in general many-to-many, and also affected by optimization settings. The recursion-friendly transformer class syntax at data row level is also transpiled in a special way to the assembly layer.
In the process of deployed code execution, Pathway stays mostly in its Rust layer. Callbacks through bindings from Rust to the actual Python interpreter occur only when Pathway’s transpiler is unable to eliminate them through one of several applied approaches (i.e., as a fallback of last resort). The scale of Python-related efficiency issues has consequently turned out limited, and so we have so far not needed to deploy Pathway with GIL-free versions of Python.
Pathway is equipped with a connector layer meant for easy use from Python. It provides a mix of input/output interfaces using built-in connectors in the Rust/C layer (for interfacing with Kafka, database CDC over Debezium, file storage formats), as well as configurable Python layer connectors (covering e.g. REST API and websockets). Convenience wrappers around these connectors may be used to perform “asynchronous” callbacks to external API’s. While dataflow-driven, from a usability perspective, the Pathway framework provides a hybrid dataflow/event-driven feel in the connectivity layer, providing a way to describe triggers on events that occur in the data sources, such as new data arrival or timer expiration.
Pathway’s engine predictably relies on uniform workers (each comprising i/o and computational threads), each performing the same workflow, with data sharding. We remark that in enterprise deployments, Pathway’s dataflow is logically extended by a data persistence layer, a control layer based on Kubernetes, and functionalities related to space optimization which are outside the scope of this work; a typical deployment scheme is presented in Fig. <ref> in the Appendix.
Changes to Pathway's output Tables are then propagated to down-stream systems through Pathway's output connectors, configured in the same Python code. In the case of commercial deployments of Pathway, output tables may also serve to provide consistent data snapshots to outside systems through an SQL server layer as shown in Fig. <ref>.
Unlike most streaming frameworks, Pathway provides streaming consistency guarantees stronger than eventual consistency, and avoids approximate watermarking. For the simple case of a system with a single, non-sharded input data source, with input messages considered atomic (transactional), the user has full control over the stream progress (offsets) for which outputs must be computed by Pathway. This happens by way of COMMIT-type control messages injected through inputs. More generally, Pathway treats in a rigorous way multiple distributed data sources, each with its own timestamped control messages, and harmonizes them into a single output data source which follows an internal Pathway data clock; we omit the details from this paper.
§.§ Pathway code examples
§.§.§ Simple Wordcount
The code below demonstrates a complete example for a word counting task:
[language=Python,breaklines=true]
import pathway as pw
# Kafka settings
rdkafka_settings =
"bootstrap.servers": "address:9092",
...
# Kafka connector fetches json inputs on the "words" topic
words = pw.kafka.read(
rdkafka_settings,
topic_names=["words"],
value_columns=["word"],
format="json",
# This setting controls the Pathway batch size
autocommit_duration_ms=1000
)
# Actual wordcount computation
result = words.groupby(this.word).reduce(
this.word,
count=pw.reducers.count(),
)
# Kafka connector writes back computed word counts
pw.kafka.write(result, rdkafka_settings, topic_name="word_counts", format="json")
# Launch the computation
pw.run()
The code essentially performs four operations. First, it configures how input data is accessed by Pathway. In this case, we use a Kafka connector with JSON message encoding. The autocommit duration determines data batching and controls the tradeoff between system throughput and latency. Next, the actual computation is defined, using a groupby-reduce construct. The third step indicates how Pathway will send its results - again we stream the data to a Kafka topic. The first three steps build a computation graph which encodes all operations that will be performed during execution. The final line of the file starts the computation and enters an infinite loop processing the unbounded data stream.
§.§.§ PageRank
We demonstrate below a basic implementation of a PageRank computation in Pathway, taken from the documentation <cit.>.
[language=Python,breaklines=true]
def pagerank(edges, steps: int = 5) -> pw.Table[Result]:
in_vertices = edges.groupby(id=edges.v).reduce(degree=0)
out_vertices = edges.groupby(id=edges.u).reduce(degree=pw.reducers.count(None))
degrees = pw.Table.update_rows(in_vertices, out_vertices)
base = out_vertices.difference(in_vertices).select(flow=0)
ranks = degrees.select(rank=6_000)
for step in range(steps):
outflow = degrees.select(
flow=pw.if_else(
degrees.degree == 0, 0, (ranks.rank * 5) // (degrees.degree * 6)
))
inflows = edges.groupby(id=edges.v).reduce(
flow=pw.reducers.int_sum(outflow.ix[edges.u].flow)
)
inflows = pw.Table.concat(base, inflows)
ranks = inflows.select(rank=inflows.flow + 1_000)
return ranks
The PageRank routine takes as an input a single table of edges, with two columns `u` and `v` containing pointers (hashed vertex labels) to respective endpoints, and returns table indexed by vertices, with the respective number of PageRank surfers computed.
In the implementation we see operations that transform columns, eg: which is used for row-wise transformation, used for aggregation (from edges to vertices), or operations which allow for manipulation of sets of rows in tables (called universes) like or .
Pathway uses as a basic construct for expressing operations on columns. Anatomy of expression is as follows: basic constructs are references to columns, accessed via python attributes of tables, i.e. and consts. Basic arithmetic and logic operators applied on expressions are building blocks for higher order expressions, and are used to describe complex vectorized operations.
The indexing operator demonstrates Pathway’s capabilities to work with pointers. The .ix operator denotes vectorized dereference, i.e. a join between a key expression and ids of a table. Pointer support facilitates algorithmic thinking during development, internally it is implemented using join, which is of course also fully supported by Pathway.
Pathway uses strong typing of its columns, expressions and tables. Specifically, the type of data stored in each column is tracked. Additionally, row keys (universes) are tracked and used in the type system.
§ BENCHMARKS
We report performance of Pathway on streaming and batch tasks which are representative of workloads we want to support: online streaming tasks and graph processing tasks. We evaluate the graph processing task in three modes: first, as a batch computation. Next, as an incremental online computation that should auto-update its results while streaming changes to the graph. Finally, a mixed batch-online mode we call backfilling which evaluates the ability of the engine to switch from batch to online. During backfilling the engine first computes the solution on a large data set in batch mode, then has to start responding in real time to streaming updates. The backfilling scenario simulates e.g. recomputing results after a change to the algorithm.
§.§ Experiment design
All code needed to reproduce the benchmarks is publicly available at our GitHub repository[<https://github.com/pathwaycom/>].
All experiments were run on dedicated machines with: 12-core AMD Ryzen 9 5900X Processor, 128GB of RAM and SSD drives. For all multithreaded benchmarks we explicitly allocate cores to ensure that threads maximally share L3 cache. This is important, as internally the CPU is assembled from two 6-core halves, and thread communication between halves is impacted. For this reason we report results on up to 6 cores for all frameworks.
We run all experiments using Docker, enforcing limits on used CPU cores and RAM consumption.
§.§ Streaming benchmark: wordcount
The benchmark task is a simple variant of the Wordcount benchmark in which each line of the input contains a single word, and the goal is to maintain for each word the total number of occurrences in the input stream. In particular, we do not require the code to split the sentences into words, nor to be case insensitive. Instead, we focus on comparing input/output efficiency and the performance of groupby-count operations.
We compare all tested frameworks using the same benchmarking harness which collects all relevant statistics. Test runs are orchestrated using docker compose, which manages all computations: the Kafka service, the data producer and result gathering sink, as well as the wordcount computation to be tested. We manually assign non-overlapping sets of CPU cores to each service. Last, we rely on timestamps assigned by Kafka (using the LogAppendTime option) to compute output latencies.
The streamer is a single-thread Rust binary which simulates a bursty stream with a predefined mean throughput. We have noticed that some frameworks respond with a high latency to the initial messages they produce, but after a while converge to a steady-state performance. To eliminate benchmarking the transient behaviors we employ a burn-in period during which the message throughput is gradually increased. We then discard these events from latency statistics computation. The input Kafka topic is not partitioned.
§.§.§ Test scenario
For our usual test scenario, we use 76 million words taken uniformly at random from a dictionary of 5000 random 7-lowercase letter words. We split the dataset into two parts: we use 16 million words as some kind of burn-in period, and we include towards the final readouts only the latencies of the remaining 60 million words. Each experiment is repeated 5 times; the median of all runs is reported.
Since many of the tested frameworks employ minibatching, and guarantee only eventual consistency, we must accordingly define latency. First, we match each entry of the input stream with the earliest value in the output stream that has ‘correct’ count - if there is no exact match, we take the first entry in the output stream that has a larger count. We then compute the latency as the difference of Kafka broker timestamps of matched messages. This way of matching messages produces a matching that is optimistic for systems under consideration. Even if a benchmarked framework processes messages out-of-order, our matching is optimistic and minimizes maximum reported latency. Therefore, we believe the proposed latency definition allows a fair evaluation of all systems using a black-box approach.
§.§.§ Benchmarked systems and their setup
In our experiments, we tested several frameworks for stream processing. Each of those frameworks has multiple parameters and variants that allow a user to adjust the performance of the framework to the task at hand. Below, we briefly describe our configuration, and explain why we choose it for each of the frameworks.
Pathway Setup
For Pathway we control the minibatch size using the connector autocommit parameter (we have tested 5ms, 10ms, 20ms and 100ms the range is expanded to allow comparison with the two Flink setups) and set the number of threads to match the allocated core count.
Flink Setup
We consider two Flink setups:
Flink defaults, which are implemented using the Scala stream API and which process each message separately,
Flink minibatching in which we configure the runtime to process minibatches of messages (similarly to KafkaStreams, Spark, and Pathway), implemented using Flink Scala Table API.
The two variants provide a control of the throughput-latency tradeoff: the default setup provides better latency at low throughputs, while minibatching obtains maximum sustainable throughput.
We set the parallelism parameter to match the number of allocated cores. The minibatching setup controls batch size using the mini-batch.allow-latency parameter, whose best settings chosen for comparison were 20ms and 100ms. We have compared Flink performance under the cluster, and single-machine multithreaded configuration and run all tests in the multithreaded configuration which was faster.
Kafka Streams Setup
We tune two parameters: the size of a batch (using autocommit frequency parameter set to 20ms and 100 ms) that is ingested by Kafka Streams and parallelism. Since Kafka Streams benefits from multithreaded computations only when reading from multi-partitioned topics, we increase its parallelism instead using
the number of replicas set through docker compose.
Finally, we set the enableObjectReuse flag to speed Flink operations by disabling copies of objects made by default for code safety.
Spark Structured Streaming
Spark supports two streaming modes: Structured Streaming and Continuous Streaming. However, at the time of preparing the benchmarks Continuous Streaming didn’t support groupby aggregations and we have restricted ourselves to benchmark Structured Streaming.
Similarly to other frameworks, we configure parallelism (using spark-submit –master local[k]) and batch size (setting .trigger(Trigger.ProcessingTime(s"$pTime milliseconds")) to 20ms or 100ms).
§.§.§ Wordcount benchmark results
The Wordcount task is submodular: with large batch sizes reduce the amount of work needed and improve throughput while increasing latency. Thus the task nicely demonstrates the tension between throughput and latency of streaming systems.
We present experimental results of the observed latency / throughput curve in Fig. <ref>.
Out of the four tested solutions, Flink and Pathway obtain results on the Pareto front, clearly dominating Spark Structured Streaming and Kafka Streams. Pathway clearly dominates the default Flink setup in terms of sustained throughput, and dominates the Flink minibatching setup in terms of latency for all of the throughput spectrum we could measure. Actually, for most throughputs, Pathway also achieves lower latency than the better of the two Flink setups.
§.§ Batch benchmark: PageRank
We benchmark the total time to complete a batch PageRank computation on the LiveJournal social network dataset graph, which contains 4,847,571 nodes and 68,993,773 edges.
We then compare implementations in Pathway, Flink, and Spark. We rely on an equivalent, idiomatic implementation of PageRank for all three frameworks - noting that In Spark we benchmark several code variants for PageRank: two different flavors of the reference implementation (RDD code, Spark SQL code), and also an implementation proposed in Spark documentation examples, and finally Spark’s GraphX implementation, which are not strictly equivalent.
For all frameworks the input is encoded in JSON format. The output is streamed to /dev/null in order to reduce the influence of non-essential IO operations. The result of the benchmark is the total time spent between the start and the end of the PageRank program. Each experiment is repeated 5 times; the median of all runs is reported.
Pathway setup
For Pathway we use the implementation provided in its official documentation and described above in section <ref>. We report the results of two versions of Pathway engine: 1) the publicly available package, and 2) a build with more aggressive optimizations scheduled for general availability upon extensive testing (the benchmark release).
Flink setup
For Flink, we implement the same idiomatic logic into Flink’s Scala Table API. The resulting code is slightly more verbose than in Pathway, due to the lack of join-based operators in Flink allowing updating tables more conveniently. Nevertheless, the implementation is analogous to Pathway’s allowing for fair comparison.
Spark setup
Spark provides GraphX, a dedicated library for graph processing. We included the recommended GraphX PageRank routine in the comparison. In an effort to benchmark similar operations as for the other frameworks, we also implemented the same logic as for Flink and Pathway using the RDD and SQL API's. We have found the SQL API to be about 3 times faster, which we attribute to optimizations enabled by using a more declarative specification of computations; subsequently, we do not present the results for the RDD API. Finally, we also benchmarked a PageRank Implementation provided in the Spark documentation.
§.§.§ Batch PageRank Results
We report the batch PageRank results in Table <ref>. The fastest performance is achieved by the Spark GraphX implementation and the more aggressively-optimized Pathway build. The formulation (and syntax) of the GraphX algorithm is different from the others. Performing an apples-to-apples comparison of performance of equivalent logic in Table APIs, Pathway is the fastest, followed by Flink and Spark.
§.§ Streaming benchmark: PageRank
We now compare two variants of computing PageRank on a dynamically changing dataset. In the first scenario, called minibatch streaming, we start with an empty graph and add edges in batches of size 1000.
In the second scenario, called backfilling, the system first processes a large batch which contains a significant fraction of the full graph. Then remaining edges are added in batches of size 1000. This scenario tests the ability of the systems to recompute in batch the results as necessary e.g. after a code change, but keeping all state necessary to resume later streaming operation.
The streaming benchmark demands more RAM memory than the batch case - all streaming operations must keep their state in memory for the whole duration of the computation. Thus, we report results on subsets of the live journal dataset containing 400k and 5M edges. For comparison, we also provide timings of batch runs on these reduced datasets.
In both variants we focus on the total runtime of tested systems.
We evaluate only two systems on the streaming PageRank task: Pathway and Flink. We don’t test Kafka Streams because it was suboptimal on the streaming wordcount task. Moreover, no Spark variant supports such a complicated streaming computation: GraphX doesn't support streaming, Spark Structured Streaming doesn't allow chaining multiple groupby’s and reductions, and Spark Continuous Streaming is too limited to support even simple streaming benchmarks.
Pathway Setup
Pathway runs exactly the same code as in the batch section. Pathway allows external control of batch size by injecting explicit “COMMIT” control messages into the data stream. We use this mechanism.
Flink Setup
For Flink the code for the batch and streaming mode is mostly the same, thanks to Flink’s unified Table API. However, some changes were needed as some table operations are only available in batch mode: we have replaced the “minus” operator using leftjoin+filter and we have enforced task specific batching as described in the Appendix <ref>. In scenarios where Flink was unable to perform as expected, we relaxed the batching requirement and attempted a variety of setups which could possibly lead Flink’s streaming engine to completion of the task, in any way.
§.§.§ Streaming updates results
We report the results in the Table <ref>. We see that while both systems are able to run the streaming benchmark, Pathway maintains a large advantage over Flink. It is hard to say whether this advantage is “constant” (with a factor of about 50x) or increases “asymptotically” with dataset size. Indeed, extending the benchmarks to tests on larger datasets than those reported in Table <ref> is problematic as Flink’s performance is degraded by memory issues.
§.§.§ Backfilling
This scenario greatly reduces the amount of intermediate results that must be computed.
We once again reduce the dataset size to accommodate memory requirements.
As shown in Table <ref>. Pathway again offers superior performance, completing the first of the datasets considered approximately 20x faster than Flink. The first large batch is processed by Pathway in times comparable to the pure batch scenario. This makes the backfilling scenario very practical. In this way, Pathway preserves the ease of defining and updating batch pipelines, while being able to use them in the streaming context as well. For backfilling on the complete LiveJournal dataset, Flink either ran out of memory or failed to complete the task on 6 cores within 2 hours, depending on the setup.
§ CONCLUSIONS
Motivated by applications in context-sensitive real-time analytics for the industry, we have put forward Pathway - a unified engine which can switch mid-way from batch processing to streaming. This enables, e.g., data backfilling at speed.
As confirmed by the outcomes of all of the benchmarks we have performed so far on standard tasks, Pathway deployments are able to consistently achieve performance better than the compared state-of-the-art frameworks with Table/Dataframe API’s. At the same time, Pathway extends the scope of algorithms on dynamically changing data which may be approached with the convenience of expressing logic through a programming interface focused around manipulation of tables and table rows in Python - potentially unlocking new use cases not previously considered in such streaming frameworks.
It is interesting to reflect on the scope of algorithms and models which can be described in Pathway and benefit from efficient recomputation following data changes. As discussed herein, iterative algorithms on changing data (such as PageRank for a changing graph) fall into this category. A different case concerns event-streams, where changes to the streamed data replay only a relatively recent part of the head of the stream. Convenient handling of such cases of partially frozen data at a syntactic level is foreseen in future revisions of Pathway.
We also foresee next steps around tighter integration of Pathway with Python libraries and API's, to demonstrate how to fully leverage the power of the Python ecosystem in Pathway. Concrete examples include easy-to-setup connectors for external systems with push-API's (especially in monitoring and alerting use cases), and interfacing with Machine Learning libraries suited for working with online data, including time series.
4mm
ACM-Reference-Format
§ CONSUMING MINI-BATCHES OF A GIVEN SIZE IN STREAMING MODE IN FLINK
In order to achieve commits every 1000 edges we used Flink’s minibatch optimization. To this end we had to configure table.exec.mini-batch.allow-latency and table.exec.mini-batch.size parameters (both are required by Flink).
Initially, we tried to set mini-batch.size=1000, and set a large allow-latency. This however, resulted in updates occurring much more frequently than every 1000 edges. It seems that the mini-batch.size configuration applies to every aggregation operator (and not input), rendering it useless for the PageRank algorithm which performs some aggregations in each iteration.
Therefore, we started tuning the mini-batch.allow-latency setting. Flink can work with processing time or event time. It turns out that this processing mode affects the mini-batching mode (ProcTimeMiniBatchAssignerOperator vs RowTimeMiniBatchAssginerOperator, see Flink’s source code). We therefore aimed to set event time for the k-th edge to be equal to k with a watermark strategy for ascending timestamps. We set “mini-batch.allow-latency” to 1000 and “mini-batch.size” to something large in order to make it irrelevant – we used 10^8.
This however did not work on its own, as Flink’s planner was automatically choosing mode to use processing time, as our PageRank application has only unbounded aggregations and global joins (i.e. we don’t have any windowing in the logic). To circumvent this issue, we added a dummy operator at the beginning of the pipeline which makes an interval join on input edges with self. This forced Flink's planner to use event time (referred to also as RowTime) in minibatching.
|
http://arxiv.org/abs/2307.05846v1 | 20230711233748 | Assessing the calibration of multivariate probabilistic forecasts | [
"Sam Allen",
"Johanna Ziegel",
"David Ginsbourger"
] | stat.ME | [
"stat.ME",
"stat.AP"
] |
Time resolved eye diagrams to exploit hidden high energy branches in a nonlinear wideband vibration energy harvester
Andreas Amann
August 12, 2023
====================================================================================================================
Rank and PIT histograms are established tools to assess the calibration of probabilistic forecasts. They not only check whether an ensemble forecast is calibrated, but they also reveal what systematic biases (if any) are present in the forecasts. Several extensions of rank histograms have been proposed to evaluate the calibration of probabilistic forecasts for multivariate outcomes. These extensions introduce a so-called pre-rank function that condenses the multivariate forecasts and observations into univariate objects, from which a standard rank histogram can be produced. Existing pre-rank functions typically aim to preserve as much information as possible when condensing the multivariate forecasts and observations into univariate objects. Although this is sensible when conducting statistical tests for multivariate calibration, it can hinder the interpretation of the resulting histograms. In this paper, we demonstrate that there are few restrictions on the choice of pre-rank function, meaning forecasters can choose a pre-rank function depending on what information they want to extract concerning forecast performance. We introduce the concept of simple pre-rank functions, and provide examples that can be used to assess the location, scale, and dependence structure of multivariate probabilistic forecasts, as well as pre-rank functions that could be useful when evaluating probabilistic spatial field forecasts. The simple pre-rank functions that we introduce are easy to interpret, easy to implement, and they deliberately provide complementary information, meaning several pre-rank functions can be employed to achieve a more complete understanding of multivariate forecast performance. We then discuss how e-values can be employed to formally test for multivariate calibration over time. This is demonstrated in an application to wind speed forecasting using the EUPPBench post-processing benchmark data set.
§ INTRODUCTION
It is standard practice for operational weather centres to issue forecasts that are probabilistic. Such forecasts typically take the form of an ensemble of possible weather scenarios, allowing weather centres to quantify the uncertainty inherent in their predictions. However, despite the unequivocal utility of ensemble forecasting, there is no guarantee that the issued ensemble forecasts are reliable, or calibrated, in the sense that they align statistically with the corresponding observations. To determine whether or not forecasts can be trusted, methods are required to analyse forecast calibration.
Although several notions of forecast calibration exist for real-valued outcomes <cit.>, it is common in practice to assess whether forecasts are probabilistically calibrated. Probabilistic calibration of ensemble predictions can be visualised using rank histograms <cit.>. Given a set of ensemble forecasts and observations, rank histograms display the ranks of the observations when pooled among the corresponding ensemble members, thereby assessing whether the observations and the ensemble members are exchangeable. These graphical diagnostic tools are useful in practice since they not only assess calibration, but they also reveal what deficiencies (if any) are present in the forecasts. As such, rank histograms have become an integral component of probabilistic weather forecast evaluation.
Several extensions of rank histograms have been proposed to evaluate the calibration of probabilistic forecasts for multivariate outcomes. These multivariate rank histograms introduce a so-called pre-rank function that condenses the multivariate forecasts and observations into univariate objects, from which a standard rank histogram can be constructed <cit.>. Proposed extensions differ in the choice of pre-rank function: <cit.> and <cit.> suggested using the minimum spanning tree of the multivariate ensemble members and observation as a pre-rank function; <cit.> introduced a pre-rank function based on a multivariate ranking of the ensemble members and observation; <cit.> proposed two alternative approaches that leverage the average rank of the observation across the individual dimensions; while <cit.> recently argued that proper scoring rules are designed to summarise the information contained in the multivariate forecast and observation into a single value, and they therefore provide a canonical choice for a pre-rank function when testing for multivariate calibration.
Several of these choices have been reviewed and compared using simulated forecasts and observations <cit.>. These comparisons illustrate how the interpretation of the multivariate rank histogram depends on the choice of pre-rank function, and that different pre-rank functions are more adept at identifying different types of mis-calibration in the forecasts. The authors of these studies therefore recommend that multiple pre-rank functions are used to construct multivariate rank histograms, to obtain a more complete understanding of how the multivariate forecasts behave.
However, despite the attention they have received in the literature, multivariate rank histograms are relatively rarely employed in practice. For example, weather centres regularly issue ensemble forecast fields over relevant spatial domains, and, although these forecast fields are inherently multivariate, their calibration is rarely assessed beyond evaluating univariate calibration at the individual locations or grid points. In the univariate case, the rank histogram shows the relative position of the observation among the ensemble members. In the multivariate case, existing pre-rank functions generate multivariate rank histograms with different interpretations, and practitioners may be uncertain as to which pre-rank function(s) they should apply when evaluating their forecasts. The pre-rank function is typically designed to preserve as much information as possible when condensing the multivariate forecasts and observations into univariate objects, leading to a multivariate rank histogram with a less intuitive interpretation. In particular, forecast systems with contrasting biases can often result in a similarly-shaped histogram.
In this paper, we argue that the main purpose of rank histograms is to identify the deficiencies in probabilistic forecasts by providing a graphical visualisation of forecast calibration. In order to achieve this effectively in a multivariate context, it is imperative that the pre-rank function is straightforward to interpret. We generalise and apply the arguments of <cit.> and <cit.> that any function that transforms multivariate vectors to univariate values can be used as a pre-rank function, which facilitates flexible assessments of forecast calibration. The forecaster can choose the pre-rank functions depending on what information they want to extract from their forecasts. For example, if interest is on the dependence structure of the multivariate forecast distribution, then a pre-rank function could be chosen that quantifies this dependence structure; if interest is on extreme events, then the pre-rank function could quantify how extreme the forecast or observation is, and so on.
We can formally test whether or not a prediction system is calibrated by checking whether its (multivariate) rank histogram is flat. <cit.> recently compared approaches to achieve this using popular measures of histogram flatness. Here, we advocate an alternative approach based on e-values <cit.>, which provide a dynamic alternative to p-values when conducting statistical hypothesis tests. While classical tests require that the evaluation period is fixed in advance, e-values generate statistical tests that are valid sequentially. This makes them particularly relevant in sequential forecasting settings <cit.>, allowing us to test forecast calibration sequentially over time without compromising type-I-error guarantees. We additionally discuss appropriate methods to address the problem of multiple testing that arises when several pre-rank functions are used to assess multivariate calibration.
In the following section, we introduce rank histograms, both in a univariate and multivariate setting. Section <ref> discusses existing pre-rank functions that have been proposed in the literature, and introduces possible alternatives that assess particular features of multivariate forecasts. These pre-rank functions are employed in a simulation study in Section <ref>. Section <ref> outlines how e-values can be employed to sequentially test for forecast calibration, and Section <ref> presents a case study in which multivariate rank histograms and e-values are used to assess the calibration of gridded wind speed ensemble forecasts over Europe. Section <ref> concludes. code to implement the proposed multivariate histograms in practice is available at <https://github.com/sallen12/MultivCalibration>.
§ RANK HISTOGRAMS
In this paper, we restrict attention to ensemble forecasts, since multivariate weather forecasts are almost exclusively in this form. However, the proposed framework readily applies to continuous forecast distributions, and this is treated in detail in the Appendix. An ensemble forecast with M members is a collection of possible scenarios 𝐱_1, …, 𝐱_M∈^d for the future outcome 𝐲 = 𝐱_0 ∈^d. An ensemble forecast can be interpreted as a multivariate probabilistic forecast by considering the empirical distribution of 𝐱_1, …, 𝐱_M as the predictive distribution.
To assess the calibration of ensemble forecasts for real-valued outcomes (d = 1), we can record the rank of each observation 𝐲∈ among the corresponding ensemble members 𝐱_1, …, 𝐱_M∈ for a large number of forecast cases, and check whether all ranks occur with the same frequency (up to sampling variation). This is typically achieved by displaying the distribution of the ranks in a histogram <cit.>. Formally, the (randomised) rank of a real number z_0∈ amongst z_0, … , z_M∈ is defined as
rank(z_0; z_1, …, z_M) = 1 + ∑_i=1^M{z_i < z_0} + W ∈{1, …, M + 1},
where denotes the indicator function, and W is zero if N = #{i = 1, …, m | z_0 = z_i} is zero, and is uniformly distributed on {1, …, N } otherwise.
In the univariate case, an ensemble forecast is probabilistically calibrated if its rank histogram is flat. If the rank histogram is not flat, then its shape often provides additional information regarding how the forecasts are mis-calibrated: a ∪-shaped histogram suggests the observations are frequently either above or below all ensemble members, implying the forecasts are under-dispersed; a ∩-shaped histogram implies that the forecasts are over-dispersed; and a triangular histogram suggests that the forecasts tend to either over- or under-predict the outcome, indicative of a systematic forecast bias. Due to this straightforward interpretation of the rank histogram's shape, they are now well-established when evaluating operational weather forecasts.
Several attempts have been made to emulate this behaviour when assessing the calibration of multivariate ensemble forecasts. Since the notion of a rank is not well-defined on ^d, it is customary to introduce a so-called pre-rank function
ρ: ^d×^d ×…×^d_M times→
that is invariant under permutations of the last M-arguments. The pre-rank function converts the multivariate observations and ensemble members to univariate objects, from which a standard rank histogram can be constructed <cit.>. That is, the calibration of the multivariate ensemble forecast 𝐱_1, …, 𝐱_M can be assessed by considering the univariate rank of the transformed observation ρ(𝐲,𝐱_1, …, 𝐱_M) within the transformed ensemble members ρ(𝐱_1,𝐲,𝐱_2, …,𝐱_M), …, ρ(𝐱_M, 𝐲, 𝐱_1, …, 𝐱_M-1). A multivariate rank histogram is obtained by repeating this for a large number of forecast cases, and displaying the relative frequency that ρ(𝐲,𝐱_1, …, 𝐱_M) assumes each possible rank. If the ensemble members and the observation are exchangeable, the resulting rank histogram is uniform, and we say that the multivariate ensemble forecasts are calibrated with respect to the pre-rank function ρ.
The simplest way to construct an admissible pre-rank function is to choose ρ such that it does not depend on the last M arguments. We term such pre-rank functions simple pre-rank functions, and omit the last M arguments in their definition
ρ : ^d→.
Simple pre-rank functions therefore transform the multivariate forecasts and observations to univariate summary statistics. The calibration of multivariate ensemble forecasts can analogously be assessed by ranking the transformed observation ρ(𝐲) within the transformed ensemble members ρ(𝐱_1), …, ρ(𝐱_M), and displaying the resulting ranks within a histogram. This approach can also readily be adapted to assess the calibration of forecasts on an arbitrary outcome space Ω by using a pre-rank function ρ: Ω→. While most pre-rank functions introduced in the literature are not simple, we argue that simple pre-rank functions are often more intuitive in practice, since they can easily be designed to focus evaluation on specific aspects of the multivariate forecasts.
§ PRE-RANK FUNCTIONS
§.§ Existing pre-rank functions
To construct a canonical extension of the univariate rank histogram, <cit.> introduced a multivariate rank as a pre-rank function:
ρ_mv(𝐱_0,𝐱_1,…,𝐱_M) = ∑_m=0^M{𝐱_m≼𝐱_0},
where 𝐱_m≼𝐱_0 signifies that x_m,j≤ x_0,j for all j = 1, …, d with 𝐱_m = (x_m,1, …, x_m,d) ∈^d for m = 0, …, M. For example, a pre-rank of M+1 for 𝐲 = 𝐱_0 corresponds to a vector that exceeds the elements of the ensemble 𝐱_1,…,𝐱_M in all dimensions, whereas a low pre-rank suggests that 𝐲 = 𝐱_0 is not larger than the ensemble members in all dimensions. In high dimensions, it is often the case that few elements of 𝐱_0,…,𝐱_M are comparable with respect to ≼, resulting in most elements receiving the same pre-rank. Randomisation of these pre-ranks then trivially yields a flat rank histogram.
Recognising this, <cit.> introduced two alternative pre-rank functions that are more robust to the dimensionality. The average rank pre-rank function is defined as
ρ_av(𝐱_0,𝐱_1,…,𝐱_M) = 1/d∑_j=1^drank(x_0,j;x_1,j,…,x_M,j).
The interpretation of the average rank is similar to the multivariate rank, but it typically leads to fewer ties between elements of S and therefore requires less randomisation, making it more applicable in higher dimensions. <cit.> also proposed the band-depth pre-rank function:
ρ_bd(𝐱_0,𝐱_1,…,𝐱_M) = 1/d∑_j=1^d[ M + 1 - rank(x_0,j;x_1,j,…,x_M,j) ] [ rank(x_0,j;x_1,j,…,x_M,j) - 1 ].
This representation assumes that there are no ties between x_0,j, x_1,j, …, x_M,j for j = 1, …, d, though a more general formula exists for when this assumption does not hold <cit.>. The band-depth pre-rank function measures the centrality of 𝐱_0 among 𝐱_0, 𝐱_1, …, 𝐱_M, with a large pre-rank suggesting that 𝐱_0 is close to the centre, and a small pre-rank indicating that 𝐱_0 is an outlying scenario.
<cit.> and <cit.> alternatively proposed using the inverse length of the minimum spanning tree of the set 𝐱_1, …, 𝐱_M as a pre-rank function for 𝐱_0. This pre-rank function also measures the centrality of 𝐱_0, resulting in multivariate rank histograms with a similar interpretation to those generated using the band-depth pre-rank function. For concision, this minimum spanning tree approach is omitted from the applications in the following sections.
Finally, <cit.> recently introduced pre-rank functions based on proper scoring rules. Proper scoring rules are functions that take a probabilistic forecast and an observation as inputs, and output a single numerical value that quantifies the forecast accuracy, thereby allowing competing forecast systems to be ranked and compared <cit.>. <cit.> argue that, by design, proper scoring rules condense the information contained in the forecasts and observations into a single value, and they therefore provide an appealing choice of pre-rank function when assessing multivariate forecast calibration. For example, a pre-rank function can be derived using the energy score, arguably the most popular scoring rule when evaluating probabilistic forecasts for multivariate outcomes <cit.>:
ρ_es(𝐱_0, 𝐱_1, …, 𝐱_M) = 1/M∑_m=1^M𝐱_m - 𝐱_0 - 1/2 M^2∑_m=1^M∑_k=1^M𝐱_m - 𝐱_k,
where · denotes the Euclidean distance in ^d. This pre-rank function measures the distance between 𝐱_0 and the ensemble members 𝐱_1, …, 𝐱_M. A low pre-rank therefore indicates that 𝐱_0 is similar to the ensemble members, whereas outlying values will receive higher pre-ranks. As <cit.> remark, the latter term in this pre-rank function does not depend on 𝐱_0, and could therefore be removed without changing the resulting ranks. Alternative multivariate scoring rules could also readily be used in place of the energy score.
§.§ Generic pre-rank functions
The pre-rank functions listed above all depend non-trivially on both 𝐱_0 and 𝐱_1, …, 𝐱_M. That is, the function that condenses the multivariate observations and ensemble members into univariate objects depends itself on these observations and forecasts (hence the term “pre-rank”). This is not problematic if the pre-rank functions are invariant to permutations of 𝐱_0, 𝐱_1, …, 𝐱_M. In this section, we argue that it is often more intuitive to employ simple pre-rank functions that depend only on 𝐱_0. We demonstrate how this approach allows us to target particular aspects of the multivariate forecasts when assessing calibration.
As is standard when evaluating multivariate forecasts, many of the pre-rank functions discussed herein require that the different dimensions are on the same scale. If this cannot be assumed, then the forecasts and observations should be standardised prior to evaluation. This standardisation becomes part of the pre-rank function, so it should either depend entirely on past data or be permutation invariant. For example, one can use the past climatological mean and standard deviation along each dimension, or the mean and standard deviation of the observation and ensemble members. The interpretation of the multivariate rank histograms will generally change depending on whether one has standardised or not, since dimensions on a larger scale will typically have more influence on the results; we are then checking calibration with respect to a different pre-rank function.
If the goal is to visualise the (mis-)calibration of the multivariate forecasts, then the most important aspect of the pre-rank function is that it leads to multivariate rank histograms that are interpretable. Forecasters should therefore employ pre-rank functions that are capable of extracting the most relevant information from their forecasts. This does not rule out the pre-rank functions introduced above, but it highlights that practitioners need not limit themselves to these approaches when assessing multivariate calibration in practice.
On the other hand, if the goal is to conduct a formal statistical test for multivariate calibration, in the sense that the ensemble members and the observation are exchangeable, interpretability of pre-rank functions may not be the most important aspect. We do not consider tests for a global null hypothesis of multivariate calibration in this paper, but we comment on how such tests can be constructed in principle in Section <ref> based on several pre-rank functions and e-values.
When evaluating multivariate forecasts, it is common to focus on the location, the scale, and the dependence structure of the forecasts. Hence, rather than choosing one pre-rank function that simultaneously tries to assimilate these different elements, we propose separate pre-rank functions that assess each aspect individually; it has repeatedly been acknowledged that several pre-rank functions should be used to obtain a more complete understanding of multivariate forecast performance, so it makes sense that these pre-rank functions focus on distinct features of the multivariate forecasts.
For example, a simple pre-rank function to assess the forecast's location might take the mean of the d elements in the multivariate vector:
ρ_loc(𝐱_0) = 𝐱̅_0 = 1/d∑_j=1^d x_0,j.
This pre-rank function can readily be applied to the observed outcome and each ensemble member, and the rank of the transformed observation can then be calculated. This pre-rank is easier to calculate than the existing approaches listed above. The interpretation of the resulting rank histogram is simple: if ρ_loc(𝐲) frequently attains a high (low) rank among the pre-ranks of the corresponding ensemble members, then the multivariate histogram will appear triangular, suggesting that the multivariate forecasts are negatively (positively) biased when predicting the mean, or location, of the observed vector. As in the univariate case, a ∪-shaped (∩-shaped) histogram suggests that the ensemble forecasts are under-dispersed (over-dispersed) when predicting the observed mean.
Similarly, if we want to assess how well our probabilistic forecasts can predict the scale, or dispersion of the multivariate observations, we could take the variance or standard deviation of the d elements as a simple pre-rank function:
ρ_sc(𝐱_0) = s_𝐱_0^2 = 1/d∑_j=1^d( x_0,j - 𝐱̅_0) ^2.
This is valid even if the number of dimensions is small: we are not using this to estimate some unknown population variance, but rather as a simple measure of dispersion in the multivariate vector 𝐱_0. The interpretation of the resulting multivariate rank histogram is analogous to above, but with the spread of the forecasts over the multivariate domain as the target variable, rather than the mean.
Assessing the dependence structure is less trivial, since most measures of dependence, such as correlation coefficients, are estimated from a time series of multivariate observations. The pre-rank function, on the other hand, should take only a single realisation 𝐱_0∈^d as an input. However, tools do exist to quantify the dependence in multivariate vectors. For example, the variogram at a chosen lag h ∈{1, …, d - 1} measures the variation between dimensions separated by lag h, with a small value suggesting strong dependence between these dimensions. A variogram-based pre-rank function can therefore be derived to measure the dependence between different dimensions, such as
ρ_dep(𝐱_0; h) = -γ_𝐱_0(h)/s_𝐱_0^2,
where
γ_𝐱_0(h) = 1/2 (d - h)∑_j = 1^d - h |x_0,j - x_0,j+h|^2
is an empirical variogram at lag h that only requires knowledge of the multivariate vector 𝐱_0. The negative sign ensures that a larger value of ρ_dep(𝐱_0; h) indicates a larger dependence, in keeping with the interpretation of ρ_loc and ρ_sc above. Hence, if ρ_dep(𝐲; h) is consistently large compared to ρ_dep(𝐱_1; h), …, ρ_dep(𝐱_M; h), then the ensemble members under-estimate the dependencies between dimensions separated by lag h, whereas the opposite suggests that the ensemble forecasts over-estimate the dependence at lag h. By measuring dependence using the empirical variogram, we implicitly assume that the lag between dimensions is meaningful, such as in a time series setting. This can be generalised further using adapted notions of spatial distances and bins <cit.>.
This pre-rank function depends on the choice of a lag h at which to calculate the empirical variogram. <cit.> construct a variogram-based proper scoring rule based on the quantity
∑_i=1^d∑_j=1^d w_i, j |x_0,i - x_0,j|^2,
for nonnegative weights w_i,j, arguing that it provides an effective means to evaluate a multivariate forecast's dependence structure. This quantity could also be used as a dependence pre-rank function, circumventing the choice of a specific lag h. However, it would require selecting a matrix of nonnegative weights. If these weights are all the same, then this weighted combination will be proportional to the variance used to define ρ_sc above. This reflects the intrinsic difficulty in distinguishing between errors in the forecast spread and in the forecast dependence structure from only a single multivariate observation: if variables are highly dependent on each other, then the spread of the vector should be small. Dividing the empirical variogram by s_𝐱_0^2 means this dependence pre-rank function focuses more on the dependence structure and is less sensitive to errors in the scale.
§.§ Pre-rank functions for spatial fields
The three pre-rank functions in the previous section are simple examples that condense the multivariate forecasts and observations into univariate summary statistics. Alternative pre-rank functions can also be employed in practice and we illustrate this by considering spatial field forecasts. Here, a spatial field is an element 𝐱 in a domain ^p × q, where each point on the p × q grid represents a separate dimension; for ease of notation, we write 𝐱 in place of 𝐱_0 in this section. The location and scale pre-rank functions can readily be applied to spatial fields by `unravelling' 𝐱 to obtain a vector of length d = p × q. To employ the dependence pre-rank function, however, we must use an empirical variogram that is a function of spatial lags (see below). We discuss additional pre-rank functions that are tailored to the evaluation of spatial field forecasts.
One such example was introduced recently by <cit.>, who used the fraction of threshold exceedances (FTE) as a pre-rank function when assessing probabilistic precipitation fields. The FTE measures the proportion of values in the spatial field that exceed some threshold of interest, and the corresponding pre-rank function is
ρ_FTE(𝐱; t) = 1/p q∑_i=1^p∑_j=1^q{ x_i,j > t },
for some t ∈. The FTE is well-known in the context of spatial forecast evaluation, since it forms the basis of the fraction skill score <cit.>. By employing it as a pre-rank function, we can assess to what extent the forecasts reliably predict in how many dimensions the threshold t will be exceeded, giving us an idea of forecast calibration when predicting more extreme outcomes. If all components of all ensemble members and the outcome are below t, which will often be the case for very large thresholds, then the corresponding pre-ranks will be all be zero and the resulting rank of the observation will be determined at random. As discussed by <cit.>, these pre-ranks contain no information, meaning these instances can be omitted from the rank histogram without changing its interpretation. The FTE pre-rank function can also be applied to multivariate vectors 𝐱∈^d, by replacing the double summation in Equation <ref> with a summation over j from 1 to d.
Alternative, well-established geostatistical principles could similarly be leveraged within this framework to assess other characteristics of multivariate forecast distributions. For example, while the empirical variogram captures the variation between elements of the multivariate vector that are separated by a given lag, another aspect to consider is whether this variation changes depending on the direction between the elements. That is, whether or not the variogram is isotropic. A variogram is said to be isotropic if it depends only on the distance between elements of the multivariate vector, and not on the direction between them. By introducing a pre-rank function that measures the isotropy of an empirical variogram, we can assess to what extent the multivariate ensemble forecasts reproduce the degree of (an)isotropy present in the observed outcomes.
Several statistical tests have been proposed to assess whether or not the assumption of isotropy is valid when modelling spatial fields <cit.>. Here, we propose a pre-rank function inspired by the test statistic discussed in <cit.>. Let ℐ = {1, …, p}×{1, …, q } represent a set of grid points, and define the empirical variogram of a field 𝐱∈^p × q at multivariate lag 𝐡∈{0, …, p - 1}×{0, …, q - 1} as
γ_𝐱(𝐡) = 1/2|ℐ(𝐡)|∑_𝐣∈ℐ(𝐡) |x_𝐣 - x_𝐣+𝐡|^2,
where ℐ(𝐡) = {𝐣∈ℐ : 𝐣 + 𝐡∈ℐ}, meaning the sum is over all dimensions (i.e. grid points) that are separated by the multivariate vector 𝐡. Equation <ref> can readily be adapted to use this spatial variogram.
The isotropy pre-rank function considered here is then
ρ_iso(𝐱; h) = - {[ γ_𝐱((h, 0)) - γ_𝐱((0, h))/γ_𝐱((h, 0)) + γ_𝐱((0, h))]^2 + [ γ_𝐱((h, h)) - γ_𝐱((-h, h))/γ_𝐱((h, h)) + γ_𝐱((-h, h))]^2}.
This pre-rank function quantifies the squared distance between the variogram in the horizontal direction 𝐡 = (h, 0) and the vertical direction 𝐡 = (0, h), plus the squared distance between the variogram in the two diagonal directions 𝐡 = (h, h) and 𝐡 = (-h, h). This thus considers two separate orthogonal directions, and determines how much the variogram changes between these directions. Again, the minus sign just ensures that a higher pre-rank indicates a higher measure of isotropy. A pre-rank close to zero therefore suggests that the variogram does not change between the pairs of directions, whereas a lower value of this pre-rank function suggests that the variogram depends on the direction between elements of the field, indicative of anisotropy.
The denominators in Equation <ref> help to standardise the variogram differences, recognising that the variogram will typically be larger as the lag increases. In statistical tests, this standardisation is often performed using estimates of the covariance matrix of the variogram at different lags, typically obtained using expensive resampling methods. This standardisation ensures that the squared difference between variograms at different lags asymptotically follows a multivariate normal distribution, facilitating the introduction of a chi-squared test statistic with which to test for isotropy <cit.>. However, this is not required when defining a pre-rank function, and hence we choose an alternative standardisation that is interpretable and straightforward to implement in practice.
It is most common to use unit lags (i.e. h = 1) within tests for isotropy <cit.>, though alternative pairs of lags could also be employed in Equation <ref>. It is straightforward to extend this pre-rank function so that it considers multiple lags simultaneously, for example by summing ρ_iso(𝐱; h) over different values of h. As with the variogram-based pre-rank function in Equation <ref>, this avoids the selection of a single lag, but typically introduces additional hyperparameters. Several alternative summary measures also exist to describe the isotropy, such as the anisotropy ratio <cit.>. These measures are typically more complicated to estimate than Equation <ref>, often requiring some assumptions about the underlying data generating process, though they may also provide informative pre-rank functions.
§ SIMULATION STUDY
§.§ Multivariate Gaussian
We revisit the simulation study of <cit.> to illustrate how the multivariate rank histograms behave when there are errors in the location, scale, and correlation structure of the forecast distributions. Suppose the observations are drawn from a multivariate normal distribution with mean vector μ = 0, and covariance matrix Σ for which
Σ_i, j = σ^2exp( - |i - j|/τ), i, j = 1, …, d.
The parameter σ^2 > 0 controls the variance of the observations along each dimension, while τ > 0 determines how quickly the correlation decays as the distance between the dimensions increases. In this sense, there is assumed to be an ordering of the variables, as is typically the case in a time series or spatial setting. We set d = 10, σ^2 = 1, and τ = 1. Analogous conclusions are also drawn from other configurations.
For each observation, M = 20 ensemble members are drawn at random from a mis-specified multivariate normal distribution. We consider six possible mis-specifications, corresponding to under- and over-estimation of the mean vector μ, scale parameter σ^2, and correlation parameter τ. The parameter configurations corresponding to these six scenarios are listed in Table <ref>. Corresponding multivariate rank histograms are displayed in Figure <ref>.
The multivariate and average pre-rank functions are insensitive to changes in the scale, whereas the band-depth pre-rank function, in measuring the centrality of the observation among the ensemble members, distinguishes well between under- and over-dispersed forecast distributions. On the other hand, when errors are present in the mean of the forecast distributions, the band-depth pre-rank function is unable to differentiate between under-prediction and over-prediction, in contrast to the multivariate and average ranks.
As <cit.> remark, the energy score pre-rank function also quantifies the centrality of the observation among the ensemble members. However, in almost all cases, the energy score pre-rank function results in a multivariate rank histogram of the same shape, with the observation receiving a higher pre-rank than the ensemble members. In this case, we are comparing the energy score obtained by the ensemble forecast when the observation 𝐲 occurs, to the score obtained when each of the ensemble members 𝐱_m (m = 1, …, M) occurs as the observation; it is not too surprising that the score is generally largest when the observation is not a member of the ensemble, resulting in these negatively skewed histograms. Hence, although this pre-rank function leads to powerful tests for multivariate calibration, the corresponding rank histograms are unable to distinguish between different types of forecast errors. This could perhaps be circumvented by treating the observation as an additional ensemble member when calculating Equation <ref>, in which case this pre-rank function should behave similarly to the band-depth pre-rank function.
Since none of the existing pre-rank functions can distinguish between all three types of forecast error, <cit.> and <cit.> recommend that several pre-rank functions are implemented. This motivates the use of specific pre-rank functions that are designed to focus on each aspect of forecast performance individually. As desired, the location pre-rank function clearly discriminates between errors in the mean of the forecast distribution, the scale pre-rank function between errors in the scale, and the dependence pre-rank function between errors in the correlation of the multivariate forecasts. The dependence pre-rank function is implemented here with lag h=1. The scale and dependence pre-rank functions are insensitive to biases in the mean, though the location pre-rank function also slightly detects errors in the scale and dependence structure. As discussed, while the dependence pre-rank function is insensitive to forecast dispersion errors, it is difficult to construct a pre-rank function that can identify errors in the scale of the forecast distribution, without also being insensitive to errors in the correlation structure.
§.§ Gaussian random fields
Consider now the case where the observations are spatial fields. In particular, suppose they are realisations of a Gaussian random field on a regular 30 × 30 grid. We assume the grid is standardised such that the distance between adjacent grid points is one unit. This extends the previous example to a higher dimensional setting in which there is additionally spatial structure present in the data. As before, the observations are drawn from a zero-mean random field with an exponential covariance function such that the covariance between two locations 𝐢 and 𝐣 on the grid is
σ^2exp( - ||𝐢 - 𝐣||/τ), 𝐢, 𝐣∈{1, …, 30}×{1, …, 30},
which is governed by a scale parameter σ^2 = 1 and correlation parameter τ = 1.
Ensembles of size M=20 are similarly drawn from a mis-specified Gaussian random field, and the calibration of these multivariate ensemble forecasts is again assessed under several possible mis-specifications. In this case, results are presented for errors in the scale, correlation, and isotropy of the forecast distributions. The multivariate rank histograms corresponding to biases in the forecast mean are similar to those in Figure <ref> (not shown).
The errors in the scale and correlation structure are analogous to those listed in Table <ref>. The exponential covariance function used to generate the observations depends only on the distance between the different dimensions, and not the direction between them. This observation-generating process is therefore isotropic. To generate forecasts that misrepresent the isotropy in this process, geometric anisotropy is introduced by rescaling the grids in the vertical direction by a factor of 1.25. A second scenario is also considered whereby the observation fields are generated using this approach, with the forecasts obtained using the (isotropic) covariance function in Equation <ref>.
For concision, the multivariate rank and energy score pre-rank functions have been omitted from this higher-dimensional simulation study: as discussed, in high dimensions, the multivariate rank trivially leads to a flat rank histogram, since most pre-ranks are determined at random; the energy score pre-rank function behaves as in the previous simulation study, and fails to distinguish between the different types of forecast errors. Results are also presented for the FTE and isotropy pre-rank functions introduced in Section <ref>. The FTE pre-rank function is employed with a threshold of t=1, roughly equal to the 85th percentile of a standard normal distribution.
These multivariate rank histograms are displayed in Figure <ref>. The average pre-rank function fails to identify errors in the scale and isotropy, whereas the band-depth rank differentiates well between under- and over-estimation of the scale and dependence in the forecast distributions. Unsurprisingly, the location pre-rank function is largely insensitive to any of the forecast errors, since it is designed to focus on the mean of the predictive distributions. The scale pre-rank function is highly sensitive to errors in the variance and dependence structure, while the dependence pre-rank function (at unit multivariate lag) is able to detect errors in the correlation structure. The FTE pre-rank function is insensitive to the scale of the forecast distributions, though it does distinguish between errors in the correlation structure.
Finally, none of these pre-rank functions can identify errors in the isotropy of the forecast distributions. The measure of isotropy given in Equation <ref> appears to yield a pre-rank function that is very sensitive to these errors, and is therefore capable of informing the forecaster when the forecast fields misrepresent the isotropy of the true variogram. This is true both when the observed fields are generated from an isotropic process but the forecast fields are not, and also vice versa. This pre-rank function is insensitive to all other types of forecast errors, meaning it focuses uniquely on the isotropy.
§ MONITORING CALIBRATION WITH E-VALUES
To demonstrate the behaviour of the different pre-rank functions, the simulation studies in the previous section consist of forecasts that are independent and identically distributed. The performance of weather forecasts, however, generally varies over time: for example, in different seasons, weather regimes, or due to updates in numerical weather prediction models. Since these conditional biases may cancel each other out such that they are not visible in the rank histograms, it is additionally useful to monitor how forecast calibration evolves over time.
While several metrics have been proposed to quantify the flatness of a rank histogram <cit.>, we advocate an approach based on e-values, which simultaneously provide a sequentially valid test for forecast calibration <cit.>. E-values provide a dynamic alternative to p-values when conducting statistical hypothesis tests, and while classical tests require that the evaluation period is fixed in advance, e-values generate statistical tests that are valid sequentially, i.e. under optimal stopping. An e-value for a given null hypothesis is a nonnegative random variable E such that [E] ≤ 1 when the null hypothesis is true. Realisations of E that are less than one therefore support the null hypothesis, whereas larger e-values provide evidence against the null hypothesis.
When testing for the flatness of a rank histogram, a suitable null hypothesis is that the rank of the observation is uniformly distributed on the set of possible ranks {1, …, M + 1}. <cit.> propose monitoring calibration using the e-values
E_t = (M + 1) p_A(R_t),
where R_t is a random variable denoting the rank of the observation among the ensemble members at time t ≥ 1, and p_A(r) is the probability of rank r ∈{1, …, M + 1} occurring under the alternative hypothesis. The alternative distribution p_A can be estimated from the ranks observed prior to the current time. This can be done empirically, but if the number of ensemble members is even moderately large, <cit.> suggest modelling p_A using a beta-binomial distribution, whose parameters are estimated sequentially using maximum likelihood estimation based on the previously observed ranks.
Essentially, if a rank r has already been observed more often than the expected frequency under the null hypothesis, then p_A(r) should be larger than 1/(M + 1), resulting in a value E_t > 1. This recognises that the rank histogram is becoming less similar to a flat histogram, giving evidence against the null hypothesis. Conversely, if r has previously been observed less often than expected, then E_t will be smaller than one, recognising that by observing r the rank histogram becomes closer to a flat histogram.
In a sequential setting, if E_t is an e-value conditional on the information available at time t-1, for each t ≥ 1, then the product e_t = ∏_i=1^t E_i represents the cumulative evidence for or against the null hypothesis up until time t, and is itself an e-value. Theoretical arguments then justify rejecting the null hypothesis at time t and significance level α if e_t≥ 1/α, and this can be monitored over time without having to fix the sample size in advance. This is equivalent to rejecting the null hypothesis if 1/e_t≤α; that is, the reciprocal of an e-value constitutes a conservative p-value.
The calibration of forecasts with a lead time k of one time unit can readily be assessed using this approach. For lead times k > 1, the situation is more complicated and we refer to <cit.> for theoretical details. In summary, we can aggregate k sequences of e-values that are separated by lag k using the average product
e_t = 1/k∑_j=1^k∏_i ∈𝒦_j(t) E_i,
where 𝒦_j(t) = {j + sk : s = 0, 1, …, t-1, j + sk ≤ t } is the set of indices between j and t that are separated by lag k. <cit.> demonstrate that if this aggregated sequence is scaled by the constant (e log(k))^-1 (with e = exp(1)), then e_t will exceed 1/α with probability at most α. Hence, when testing whether or not a forecast system is calibrated at lead time k > 1, the null hypothesis of calibration can be rejected at time t and significance level α if e_t≥ e log(k)/α.
By visualising e_t as a function of time, it becomes easy to identify periods where evidence against calibration is accumulating. Figure <ref> presents examples for the forecasts considered in the following section. An increasing e-value indicates a period of mis-calibration, whereas a declining e-value suggests that either the forecasts are calibrated, or the type of mis-calibration has changed. While the e-value does not specify what type of mis-calibration is present in the forecasts, a time series of e-values can be used to identify periods worth analysing in further detail <cit.>.
E-values allow us to test whether multivariate forecasts are probabilistically calibrated with respect to a chosen pre-rank function. When several pre-rank functions are used to assess different facets of calibration, one can compute e-values for each pre-rank function, but care must be taken when hypotheses of calibration are rejected since this introduces a multiple testing problem. The simplest approach to account for multiple testing is to apply a Bonferroni correction; that is, instead of rejecting the null hypothesis when the e-value surpasses the threshold 1/α or e log(k)/α, for forecasts of lag 1 and lag k > 1 respectively, a hypothesis is only rejected if the e-value surpasses the threshold ℓ/α or ℓ e log(k)/α, respectively, where ℓ is the number of pre-rank functions considered. As with p-values, there are more powerful methods to correct for multiple testing than the Bonferroni correction, and an alternative procedure for e-values is proposed in <cit.>.
If one aims to construct powerful tests for the global null hypothesis of exchangeability of the multivariate ensemble members and the observation, it appears sensible to use a limited number of pre-rank functions that provide complementary information, and to combine the resulting e-values into one e-value by predictable mixing; we refer readers to <cit.> and <cit.>, where predictable mixing of test martingales is discussed and is referred to as so-called betting strategies. The same reasoning applies in the case of general multivariate predictive distributions; here, the null hypothesis would be that the forecasts are auto-calibrated, see Appendix. We do not explore this proposal in this paper since we want to emphasise understanding deficiencies in the calibration of multivariate ensemble forecasts, and this cannot be achieved using one global test for calibration.
§ CASE STUDY
§.§ Data
Section <ref> demonstrates how the different pre-rank functions behave when evaluating forecasts in an idealised framework. In this section, we apply the same pre-rank functions to evaluate the calibration of gridded ensemble forecasts for European wind speeds. We consider 10m wind speed forecasts and observations taken from the European Meteorological Network's (EUMETNET) post-processing benchmark dataset <cit.>, which has recently been introduced to provide a “common platform based on which different post-processing techniques of weather forecasts can be compared.” The dataset is also canonical when illustrating forecast evaluation methods.
The EUPPBench dataset contains daily forecasts issued by the European Center for Medium-range Weather Forecasts' (ECMWF) Integrated Forecasting System (IFS) during 2017 and 2018. Twenty years of reforecasts are also available for every Monday and Thursday in this two year period, which are generated by the same forecast model but with a smaller number of ensemble members (11 instead of 51). We restrict attention here to the 20 years of reforecasts at a lead time of five days, meaning we work with ensembles of size M=11.
These forecasts are compared to ERA5 reanalyses <cit.>, which provide a best guess for the observed wind speed fields. The forecasts and observations are on a regular longitude-latitude grid that covers a small domain in central Europe (2.5-10.5E, 45.75-53.5N). The grid has a horizontal resolution of 0.25^∘, roughly corresponding to 25 kilometers, and is therefore comprised of 33 distinct longitudes and 32 latitudes. This domain is displayed in Figure <ref>.
When forecasting surface weather variables such as wind speed, the ensemble forecasts issued by numerical weather models are typically subject to systematic biases. To remove these biases, it is common to statistically post-process the numerical model output. Statistical post-processing methods use a historical archive of forecasts and observations to learn and then remove systematic errors in the raw ensemble forecasts. Since the goal of this paper is to illustrate the utility afforded by multivariate rank histograms, we employ a simple, well-known univariate post-processing scheme to re-calibrate the wind speed ensemble forecasts at each grid point separately, and then compare competing approaches to convert these univariately post-processed forecasts into spatially-coherent probabilistic forecast fields. Readers are referred to <cit.> for a thorough overview of statistical post-processing methods.
To post-process the IFS ensemble forecasts at each grid point, we employ a standard ensemble model output statistics (EMOS) approach <cit.>, in which the future wind speed is assumed to follow a truncated logistic distribution <cit.>. The predictive distributions are truncated below at zero, meaning positive probability is assigned only to positive wind speeds. The location parameter of the truncated logistic distribution is a linear function of the ensemble mean forecast at the same time and location, and the scale parameter is similarly a linear function of the ensemble standard deviation. The parameters of this post-processing model are estimated using the first 15 years of reforecasts, and the resulting predictions are then assessed using the remaining 5 years. This results in 3135 forecast and observation fields for model training, and 1045 pairs for verification.
This univariate post-processing method re-calibrates the wind speed forecasts at each individual grid point. However, in doing so, the multivariate dependencies present in the raw ensemble forecast are lost. To obtain forecast distributions that have a realistic multivariate dependence structure, it is common to extract evenly-spaced quantiles from the univariate post-processed distributions, and then reorder these quantiles according to a relevant dependence template. The most popular approach to achieve this is ensemble copula coupling <cit.>. ECC is an empirical copula approach that uses the raw ensemble forecasts issued by the numerical weather models as a template to reorder the post-processed forecast distributions. ECC is known to be a straightforward and effective approach to reinstall the dependencies present in the numerical ensemble forecasts, and is frequently implemented in operational post-processing suites.
For comparison, we additionally compare the resulting forecast fields to those obtained using an alternative copula-based reordering scheme, namely the Schaake Shuffle <cit.>. Like ECC, the Schaake Shuffle reorders evenly-spaced quantiles from the post-processed forecast distributions according to some dependence template. In contrast to ECC, the Schaake Shuffle uses a random selection of past multivariate observations to construct the dependence template, rather than the raw ensemble forecast. Although the Schaake Shuffle can leverage an arbitrary number of previous observations in this dependence template, only eleven observations are used here, so that the resulting ensemble forecasts have the same number of members as those generated using ECC. Further details regarding these two approaches can be found in <cit.> and references therein. Several variants of both ECC and the Schaake Shuffle have also recently been proposed, but we restrict attention here to the two most widely-used implementations; <cit.> find that these extensions do not provide significant benefits.
The calibration of the forecast fields obtained from these two post-processing methods is compared to the calibration of the raw numerical model output. An example of the forecast fields generated using the three methods is presented in Figure <ref>.
§.§ Results
Firstly, consider the univariate calibration of the competing forecasting methods. Figure <ref> displays the univariate rank histograms corresponding to the raw ensemble forecasts (i.e. the IFS reforecasts) before and after undergoing post-processing. The ranks are aggregated across all 1056 grid points. Although we describe two different multivariate post-processing methods, these differ only in how they reorder the univariate post-processed forecasts, and thus result in the same univariate rank histograms. Figure <ref> illustrates that the raw ensemble forecasts are slightly under-dispersed, with the observed wind speed falling outside the range of the ensemble members more often than would be expected of a calibrated forecast. The simple post-processing approach corrects these dispersion errors.
Figure <ref> displays the corresponding multivariate rank histograms. The same pre-rank functions are employed as in Section <ref>. The dependence and isotropy pre-rank functions are implemented with unit lag, and the FTE pre-rank function uses a threshold t=6ms^-1, roughly corresponding to the 90^th percentile of the wind speeds in the training data across all grid points. The average and location pre-rank functions suggest that the raw ensemble forecasts are relatively well-calibrated when predicting the mean wind speed over the domain. However, the corresponding band-depth histogram indicates that these forecasts do not reliably capture the centrality of the observation among the ensemble members. The IFS forecast fields also appear to reliably predict the variance of the observed wind speed fields, as well as the number of threshold exceedances, but they severely under-estimate the dependence between adjacent grid points, and over-estimate the measure of isotropy in the ERA5 reanalyses.
Consider now the post-processed forecasts that are reordered according to ECC. The multivariate rank histogram corresponding to the band-depth pre-rank function is much closer to uniform. While this may suggest that these post-processed forecasts improve upon the multivariate calibration of the IFS output, we can use targeted pre-rank functions to identify remaining sources of mis-calibration in these forecasts. In particular, post-processing using EMOS and ECC does not correct the errors in the dependence structure that manifest in the IFS forecast fields, and the resulting forecasts also over-estimate the measure of isotropy at unit lag. Aside from the band-depth pre-rank function, the multivariate rank histograms for the ECC forecasts exhibit the same patterns as those for the raw IFS forecasts. The Schaake Shuffle performs similarly, though while these forecasts also under-estimate the dependence between adjacent grid points, they do so to a lesser degree than IFS and ECC.
These patterns are reinforced by e-values. Figure <ref> displays the cumulative e-values when assessing the calibration of the forecast systems with respect to three of the chosen pre-rank functions. E-values are displayed for the band-depth, scale, and dependence pre-rank functions. In all cases, a “burn-in” period of 100 forecast-observation pairs is required from which to estimate p_A when calculating the e-values. The three forecasting methods issue calibrated predictions of the scale of the ERA5 reanalyses fields. For the band-depth pre-rank function, there is quickly sufficient evidence to conclude that the multivariate rank histogram corresponding to the raw ensemble forecasts is not uniform. The post-processing methods, on the other hand, appear probabilistically calibrated with respect to the band-depth pre-rank function. For the dependence pre-rank function, as suggested by the multivariate rank histograms, there is sufficient evidence at the 5% level to suggest that none of the multivariate forecasts are calibrated. Hence, although post-processing offers improvements upon the raw IFS forecasts, there is still vast potential to improve upon these baseline post-processing methods. Moreover, we expect that the mis-calibration of all forecasting methods would be more pronounced if the forecasts were compared to station observations rather than reanalysis fields.
Since multivariate forecast calibration is assessed using multiple pre-rank functions, one might ask what constitutes a good set of pre-rank functions? A collection of pre-rank functions will be most useful when the individual pre-rank functions provide complementary information. To assess this, it is useful to have a measure of dependence between the rank of ρ(𝐲) for different choices of the pre-rank function ρ. If the dependence is strong, then it suggests that one of the pre-rank functions provides redundant information.
Table <ref> contains the correlations between the ranks of ρ(𝐲) among ρ(𝐱_1), …, ρ(𝐱_M) for the various choices of ρ. Results are shown for the raw ensemble members, though very similar correlations are obtained for the two post-processing methods. There is a strong positive correlation between the average rank and location pre-rank functions, which both assess the mean behaviour of the spatial fields, and also the FTE pre-rank function. There is also strong positive correlation between the scale, dependence, and FTE pre-rank functions. The location pre-rank function is strongly correlated with the scale pre-rank function in this example, suggesting that errors when predicting the average wind speed over the spatial domain are linked to errors when predicting the variation in wind speeds over the domain. The band-depth and isotropy pre-rank functions, on the other hand, exhibit relatively low correlations with the other pre-rank functions, making them particularly useful in this application.
§ CONCLUSION
For probabilistic forecasting systems to be as useful as possible, the forecasts they issue must be calibrated, in the sense that they are statistically consistent with what actually materialises. In practice, the calibration of univariate ensemble forecasts is typically assessed using rank histograms. Ensemble forecasts for multivariate outcomes can be evaluated using multivariate rank histograms, which use a so-called pre-rank function to transform the observations and ensemble members into univariate objects, prior to constructing a standard rank histogram. In this paper, we highlight that there is considerable flexibility in the choice of pre-rank function, meaning practitioners can choose pre-rank functions on a case-by-case basis, depending on what information they wish to extract from their forecasts. In particular, while previously proposed pre-rank functions depend on the forecast and observed outcome, we argue that this need not be the case. Instead, simple pre-rank functions can be employed that more directly target specific aspects of the multivariate forecasts.
We introduce generic pre-rank functions that can focus attention to the location, scale, and dependence structure of the multivariate forecast distributions, allowing each of these components to be assessed individually. The resulting histograms are straighforward to interpret, making it easier to assess what systematic deficiencies exist in the forecasts. We also propose suitable pre-rank functions to assess the calibration of probabilistic spatial field forecasts, which are regularly issued by operational weather forecasting centres. Although we focus here on spatial forecast fields as an example, the arguments presented herein apply to other multivariate forecasts, and future work could consider relevant pre-rank functions in other multivariate forecasting settings, such as time series forecasting.
A long-standing challenge when evaluating spatial field forecasts is to design verification tools that value realistic-looking forecast fields, i.e. fields that do not violate any physical laws. If we could quantify how realistic a weather field is, then these measures of realism could be used as pre-rank functions in the framework discussed herein. Doing so would not only reveal whether our forecast fields are realistic, but, if not, the corresponding multivariate rank histograms should additionally allow us to identify how our forecasts are unrealistic.
The pre-rank functions we introduced are used to compare ensemble fields obtained from a near-operational ensemble prediction system before and after having undergone statistical post-processing. These results help to understand not only how the post-processed forecasts improve upon the raw model output, but also what deficiencies these forecasts still exhibit. In recognising the limitations of the predictions, it becomes easier to remove them, resulting in more accurate and reliable forecasts in the future. While the multivariate post-processing frameworks employed in this study are commonly applied in operational post-processing suites, the multivariate rank histograms in Figure <ref> suggest that these forecast still exhibit significant biases, particularly related to the dependence between the wind speed at nearby grid points. An interesting avenue for future work would therefore be to compare these results to those obtained using state-of-the-art machine learning models that have recently been introduced for multivariate post-processing <cit.>.
The general framework outlined herein involves identifying univariate characteristics of multivariate objects, and evaluating the multivariate forecasts via their ability to predict these characteristics. This approach has also been proposed when constructing multivariate scoring rules <cit.>. <cit.> illustrate that this framework can essentially be interpreted as a weighting of the scoring rule, where the transformation determines which outcomes are to be emphasised when calculating forecast accuracy. Weighted versions of these multivariate rank histograms could also be employed to target particular multivariate outcomes when assessing forecast calibration, as outlined by <cit.>. Note, however, that the transformations used within scoring rules need not be interpretable, unlike the pre-rank function used to construct multivariate rank histograms, and hence some transformations that are suitable when calculating forecast accuracy may not be when interest is on forecast calibration.
§ ACKNOWLEDGEMENTS
The authors are grateful to the Swiss Federal Office of Meteorology and Climatology (MeteoSwiss) and the Oeschger Centre for Climate Change Research for financially supporting this work. Sebastian Arnold, Jonas Bhend, Alexander Henzi, and Marc-Oliver Pohle are also thanked for fruitful discussions during the preparation of this manuscript.
apalike
§ APPENDIX
§.§ Continuous probabilistic forecast distributions
Suppose we are interested in forecasting a univariate random variable Y, and let F be a forecast for Y in the form of a cumulative distribution function. The forecast is said to be probabilistically calibrated if its probability integral transform (PIT)
Z_F = F(Y-) + V[F(Y) - F(Y-)]
follows a standard uniform distribution, where V is a standard uniform random variable that is independent from Y and F, and F(Y-) = lim_x ↑ Y F(x) <cit.>. The motivation behind this definition is the result that Y ∼ F if and only if Z_F follows a standard uniform distribution. The probabilistic calibration of a forecast system (rather than a single forecast) can be assessed by treating the forecast F as random.
A stronger notion of calibration for a forecast system is auto-calibration, ℒ(Y | F) = F, where ℒ denotes the law, or distribution. That is, the conditional distribution of Y given the (random) forecast F is equal to F. Auto-calibration is a strictly stronger requirement than probabilistic calibration <cit.>. However, in contrast to probabilistic calibration, auto-calibration generalises without any adaptations to multivariate predictions and outcomes.
In practice, the probabilistic calibration of a forecast system can be assessed by calculating the PIT values corresponding to a sequence of forecasts and observations. These PIT values can then be displayed in a histogram to check whether or not they resemble a sample from a standard uniform distribution. These PIT histograms generalise rank histograms, and their interpretation is analogous to the interpretation of rank histograms discussed in Section <ref>.
Suppose now that we wish to predict a multivariate random variable 𝐘 that takes values in ^d, for d>1. Given a simple pre-rank function ρ: ^d→ and a multivariate forecast distribution F, F is said to be probabilistically calibrated with respect to the pre-rank function ρ if
Z_F_ρ = F_ρ(ρ(𝐘)-) + V[F_ρ(ρ(𝐘)) - F_ρ(ρ(𝐘)-)]
follows a standard uniform distribution, where V is a standard uniform random variable, and F_ρ is the univariate cumulative distribution function induced by F and ρ, i.e. F_ρ(x) = (ρ(𝐗) ≤ x) for 𝐗∼ F and x ∈. This definition of multivariate calibration is equivalent to assessing whether the forecast distribution is probabilistically calibrated when predicting the univariate random variable ρ(𝐘).
The following simple but powerful result gives a theoretical justification for considering probabilistic calibration with respect to pre-rank functions.
Suppose that the multivariate forecast distribution F is auto-calibrated for 𝐘∈^d. Then F is probabilistically calibrated with respect to any simple pre-rank function ρ:^d →.
Let x ∈ [0,1]. Then
(Z_F_ρ≤ x) = 𝔼[𝔼[{Z_F_ρ≤ x}| F ]]
= 𝔼[x] = x,
where we used auto-calibration for the second equality.
Multivariate PIT histograms can analogously be defined to assess the calibration of continuous multivariate forecast distributions. The pre-rank functions listed in Section <ref> can equally be applied to continuous multivariate forecast distributions but the distribution F_ρ may have to be approximated by simulation.
<cit.> demonstrates how pre-rank functions for ensemble forecasts depending on M+1 arguments (instead of just the first argument) can be generalised to the case where the forecast is an arbitrary predictive distribution over ^d. We do not detail the arguments here since we advocate simple pre-rank functions depending on the first argument only.
|
http://arxiv.org/abs/2307.07411v1 | 20230710121834 | Detecting LLM-Generated Text in Computing Education: A Comparative Study for ChatGPT Cases | [
"Michael Sheinman Orenstrakh",
"Oscar Karnalim",
"Carlos Anibal Suarez",
"Michael Liut"
] | cs.CL | [
"cs.CL",
"cs.CY"
] |
Detecting LLM-Generated Text in Computing Education]Detecting LLM-Generated Text in Computing Education: A Comparative Study for ChatGPT Cases
[email protected]
University of Toronto Mississauga
Mississauga
Canada
[email protected]
0000-0003-4930-6249
Maranatha Christian University
Bandung
Indonesia
[email protected]
0000-0002-6012-932X
Escuela Superior Politécnica del Litoral
Guayaquil
Ecuador
[email protected]
0000-0003-2965-5302
University of Toronto Mississauga
Mississauga
Canada
Due to the recent improvements and wide availability of Large Language Models (LLMs), they have posed a serious threat to academic integrity in education. Modern LLM-generated text detectors attempt to combat the problem by offering educators with services to assess whether some text is LLM-generated. In this work, we have collected 124 submissions from computer science students before the creation of ChatGPT. We then generated 40 ChatGPT submissions. We used this data to evaluate eight publicly-available LLM-generated text detectors through the measures of accuracy, false positives, and resilience. The purpose of this work is to inform the community of what LLM-generated text detectors work and which do not, but also to provide insights for educators to better maintain academic integrity in their courses. Our results find that CopyLeaks is the most accurate LLM-generated text detector, GPTKit is the best LLM-generated text detector to reduce false positives, and GLTR is the most resilient LLM-generated text detector. We also express concerns over 52 false positives (of 114 human written submissions) generated by GPTZero. Finally, we note that all LLM-generated text detectors are less accurate with code, other languages (aside from English), and after the use of paraphrasing tools (like QuillBot). Modern detectors are still in need of improvements so that they can offer a full-proof solution to help maintain academic integrity. Further, their usability can be improved by facilitating a smooth API integration, providing clear documentation of their features and the understandability of their model(s), and supporting more commonly used languages.
<ccs2012>
<concept>
<concept_id>10003456.10003457.10003527</concept_id>
<concept_desc>Social and professional topics Computing education</concept_desc>
<concept_significance>500</concept_significance>
</concept>
<concept>
<concept_id>10002951.10003317.10003347.10003355</concept_id>
<concept_desc>Information systems Near-duplicate and plagiarism detection</concept_desc>
<concept_significance>500</concept_significance>
</concept>
<concept>
<concept_id>10002951.10003317.10003338.10003341</concept_id>
<concept_desc>Information systems Language models</concept_desc>
<concept_significance>500</concept_significance>
</concept>
</ccs2012>
[500]Social and professional topics Computing education
[500]Information systems Near-duplicate and plagiarism detection
[500]Information systems Language models
[
Michael Liut
August 12, 2023
===================
§ INTRODUCTION
In academia, a way to encourage students utilizing all learning opportunities and experiences is to properly maintain academic integrity in the courses <cit.>. Students need to complete any exams and assessments with their best effort. Further, they need to actively engage with the instructors (and tutors).
Although Artificial Intelligence (AI) can foster education <cit.>, it might be misused to breach academic integrity. Paraphrasing tools <cit.> and code obfuscation tools <cit.> for example, are misused to cover up evidence for plagiarism (a breach of academic integrity about copying one's work and reusing it without proper acknowledgment <cit.>).
Misuse of AI chatbots with large language models (LLM) <cit.> such as ChatGPT[<https://openai.com/blog/chatgpt>] is another trending threat for breaching academic integrity. Students can complete exams or assessments with limited effort, resulting in questionable performance; it is unclear whether the learning objectives are actually met. The misuse can be considered as contract cheating (i.e., getting help in exchange for mutual incentives <cit.>) since AI chatbots provide responses in exchange for additional user data. However, considering AI responses are generated based on other people's textual data without proper acknowledgment, we believe it is more justifiable to consider the misuse as plagiarism.
While checking student work for plagiarism, instructors are often aided by automated detectors. A number of detectors have been developed to detect whether a work is a result of LLM. Two of them are GPT-2 Output Detector <cit.> and Giant Language model Test Room (GLTR) <cit.>. Nevertheless, due to the recency of misuse of AI chatbots, Computing educators might have limited information about publicly available detection detectors. Further, it is challenging to choose the most suitable detector for their teaching environment. To the best of our knowledge, there are no empirical studies comparing the detectors in terms of effectiveness.
In response to the aforementioned gaps, we investigate LLM-generated text detectors and formulate the following research question (RQ): “How effective are LLM-generated text detectors?”
It is clear that there is a need in the community to understand if the currently available detectors are able to detect LLM-generated content <cit.> and what there reliability is.
As an additional contribution, we also report our experience in using the LLM-generated text detectors. It might be useful for readers interested in employing those detectors in their classrooms.
§ RELATED WORK
This section discusses common breaches of academic integrity in computing education and misuse of AI to breach academic integrity.
§.§ Common Breaches of Academic Integrity
Academic integrity encourages students to act honestly, trustworthy, respectfully, and responsibly in learning[<https://lo.unisa.edu.au/course/view.php?id=6751&section=6>]. <cit.> lists five common breaches of academic integrity in computing education: plagiarism, collusion, contract cheating, exam cheating, and research fraud. It is important to inform students about instructors' expectations about academic integrity in their courses <cit.> and penalize those who breach academic integrity.
Plagiarism happens when ideas, words, or even code is reused without proper acknowledgment and permission to the original author(s) <cit.>.
It is commonly identified with the help of automated detectors <cit.> such as Turnitin[<https://www.turnitin.com/>], Lichen <cit.>, MOSS[<https://theory.stanford.edu/ aiken/moss/>], and JPlag <cit.>. Any submissions with high similarity will be investigated and if they are indeed a result of misconduct, the students will be penalized <cit.>.
Nevertheless, identifying plagiarism is not always straightforward; some perpetrators disguise their act with automated paraphrasing <cit.>, essay spinning <cit.> or code obfuscation <cit.>. The automated detectors should be resilient to common disguising practices in addition to being effective and efficient.
GPlag <cit.> and BPlag <cit.> for examples, focus on content semantic while measuring similarity among submissions.
<cit.> and <cit.> developed detectors that detect substantial changes among consecutive saves.
<cit.> developed a detector that is automatically integrated to a programming workspace to record any code edits.
Collusion is also about reusing ideas, words, or code without proper acknowledgment. However, the original author(s) is aware about the matter and somewhat allows it <cit.>. Typically, this occurs when two or more students work closely beyond reasonable levels of collaboration <cit.>. Collusion can be identified in the same manner as plagiarism with the help of automated detectors. Similar submissions are reported by the detectors and then manually investigated by the instructors; students whose submissions are indeed a result of misconduct will be penalized.
Contract cheating occurs when third parties are paid to complete student assessments <cit.>. The third parties can be professional companies or even their colleagues. Contract cheating is quite challenging to identify as the third parties tend to know how to evade detection. It is only identifiable when the writing style and the quality of the submission is substantially different to those of the student's prior submissions. To expedite the identification process, instructors can either use the help of authorship identification detectors <cit.> such as Turnitin Authorship Investigate[<https://help.turnitin.com/MicroContent/authorship-investigate.htm>] <cit.> or check contract cheating sites <cit.>.
Exam cheating happens when some students have unfair advantages in the exams <cit.>. The advantages can vary from concealed notes during exams, leaked exam questions, to impersonation (i.e., an individual switch places with a student to take the exam). Exam cheating can be identified via careful investigation on the whole process of the exams. Sometimes, such identification can be aided with online proctoring systems <cit.> (e.g., Proctorio[<https://proctorio.com/>] and ProctorExam[<https://proctorexam.com/>]) or local monitoring tools (e.g., NetSupport[<https://www.netsupportschool.com/>]).
Research fraud means reporting research results without verifiable evidence <cit.>. It can be data fabrication (i.e., generating artificial data to benefit the students) or data falsification (i.e., updating the data so that it aligns with the students' desired findings). Both are parts of research misconduct[<https://grants.nih.gov/policy/research_integrity/definitions.htm>] and they can happen in research-related assessments. Research fraud can be identified via careful investigation on the whole process of research. Due to its complex nature, such misbehaviour is manually identified on most cases. However, instructors can get some help from source metadata <cit.> and automated image manipulation detection <cit.>.
§.§ Misuse of AI
AI substantially affects education <cit.>. It improves student learning experience via the help of intelligent tutoring systems <cit.> and personalized learning materials <cit.>. AI expedites the process of providing feedback <cit.>, identifying breaches of academic integrity <cit.>, maintaining student retention <cit.>, learning programming <cit.>, creating programming exercises <cit.>, and recording attendance <cit.>.
Advances in AI might also be misused for breaching academic integrity.
Paraphrasing tools <cit.> which are intended to help students learn paraphrasing are misused to cover up plagiarism.
Code generators like GitHub Copilot <cit.> which are intended to help programmers in developing software are misused to complete programming tasks that should be solved independently.
Code obfuscation tools <cit.> which are intended to secure code in production are misused to disguise similarities in copied code submissions.
AI chatbots <cit.>, especially those with Large Language Model (LLM) <cit.> are intended to help people searching information, but they are misused to unethically complete exams[<https://edition.cnn.com/2023/01/26/tech/chatgpt-passes-exams/index.html>] and assessments[<https://theconversation.com/chatgpt-students-could-use-ai-to-cheat-but-its-a-chance-to-rethink-assessment-altogether-198019>].
LLM is derived from Language Model (LM), a statistical model at which each sequence of words are assigned with a probability <cit.>. Per query or question, the response is generated by concatenating sequences of words that have high probability with the query or the question.
ChatGPT is a popular example of LLM. The tool is developed by OpenAI, a non-profit American research laboratory on top of GPT-3, a LLM with deep learning to generate human-like text. The tool relies on reinforcement and supervised learning to further tune the model.
A number of automated detectors have been developed to help instructors identifying AI misuses for breaching academic integrity. In the context of plagiarism and collusion, automated detectors nullify common alterations that can be done without understanding the content <cit.> and remove contents that are not evident for raising suspicion <cit.>.
In dealing with misuses of AI chatbots, a few automated detectors are developed under the same way as the chatbots via pretrained model, but dedicated to detect AI-generated texts. GPT-2 Output Detector <cit.> and GLTR <cit.> are two of the examples.
§ METHODOLOGY
This section discusses how the research question stated in the introduction would be addressed and our preliminary work to discover publicly available LLM-generated text detectors.
We collected historical assignment data dating back to 2016 from two publicly funded research-focused institutions, one in North America and one in South America. The data collected was from upper-year undergraduate computer science and engineering students.
We analyzed a total of 164 submissions (124 were submitted by humans, 30 were generated using ChatGPT, and 10 were generated by ChatGPT and altered using the Quillbot paraphrasing tool) and compared them against eight LLM-generated text detectors. This results in a total of 1,312 prediction results.
Of the 164 submissions, 134 were written in English (20 of which were generated by a LLM, and another 10 which were LLM-generated and paraphrased) and 20 were written in Spanish (10 of which were AI-generated). The submissions were collected between 2016 and 2018 (prior to the release of ChatGPT), and were made in “databases”, “networking”, and a “final thesis project” course. These courses were specifically selected as they are upper-year computer science major courses that touch on a mix of systems and theory (databases and networking), as well as technical writing in computer science with a programming/development component (final thesis project). The students in these courses were primarily in a computer science major. It should also be noted that Spanish was selected as an alternative language to analyse because it is one of the world's most popular languages, and some of the authors have experience writing and evaluating technical material in this language.
The assessments analyzed in this study (see Table <ref>) are taken from three undergrad courses. The first course is a databases course offered to third-year computer science students in their first or second semester. It is a mix of database theory and practical systems application. There are 101 paper submissions from this course which involved a final assessment where students wrote a report analyzing two industry players and their use of databases and data centers, this was written in English.
The second course is a networking course offered to third-year computer science students in their second semester. It is a mix of theoretical concepts and practical system application. There are 13 paper submissions from this course which involved an exam question where students explain how they would implement the NOVEL-SMTP and NEO-SMTP email protocols using only UDP, this was written in English.
The third course is a final thesis project course offered to fourth-year computer science students throughout their final year of study (across both semesters). It is meant to bridge theory and practice to develop something that can be used/implemented in the real world. There are 10 paper submissions from this course which involved improving computing systems and engineering processes in their local community, this was written in Spanish.
Due to the character limitations, data below 1,000 characters was excluded and data above 2,500 characters was truncated to the last complete sentence. This ensures the input data fits within the range of all detectors. As many LLM-generated text detection platforms have a 2,500 character maximum, to ensure fairness across platform, we used 2,500 characters as our upper-bound.
LLM-generated texts were created with the help of ChatGPT[<https://openai.com/blog/chatgpt>], a popular LLM. The handouts were parsed to prompts by removing irrelevant information (course code, deadlines, submission instruction) so the prompts only contain the core requirements of the task. These prompts were then fed into ChatGPT to generate a solution to the assignment.
It should be noted, the authors mined through over 2,000 submissions in programming, data structures and algorithms, and compilers courses, however, the submission data varied too much for the content to easily be extracted and analyzed for detectors. Often due to a lack of context after removing any code. The selected submissions were purely writing-based and did not involve coding components, they did in some cases discuss theoretical concepts in computer science.
Finally, all of the detectors were tested in April 2023.
§.§ Discovering Publicly Available LLM-generated Text Detectors
Publicly available LLM-generated text detectors were discovered from January to February 2023 from social media (i.e., Twitter, Facebook, and blogs), online news, and previous literature on LLM-generated text detection (GPT-2, GLTR). Public interest in LLM-generated text detectors followed the release of GPTZero which went viral on January, 2023. After GPTZero, many other companies launched their own LLM-generated text detectors.
A number of LLM-generated text detectors were discovered but we limited this study to LLM-generated text detectors that appear to offer proprietary solutions to LLM-generated text detection. We found that some LLM-generated text detectors are likely to be replicas of open source work (GPT-2) and hence we excluded such detectors from the study.
We identified eight such publicly available LLM-generated text detectors, as shown in Table <ref>. Two of them (GPT-2 Output Detector and GLTR) are featured with technical reports <cit.>.
GPT-2 Output Detector <cit.> is a LLM-generated text detector based on the RoBERTa large pretrained model <cit.>. RoBERTa is a transformers model trained on a large corpus of raw English data. The GPT-2 Output Detector starts with the pre-trained ROBERTA-large model and trains a classifier for web data and the GPT-2 output dataset. The GPT-2 Detector returns the probability that an input text is real on GPT-2 text with accuracy of 88% at 124 million parameters and 74% at 1.5 billion parameters <cit.>. The detector is limited to the first 510 tokens, although there are extensions that extend this limit <cit.>.
GLTR <cit.> is a detector that applies statistical methods to detect GPT-2 text. The model is based on three simple tests: the probability of the word, the absolute rank of a word, and the entropy of the predicted distribution. This detector shows an interface where each word is highlighted along with a top-k class for that word.
The GLTR detector does not provide quantifiable overall probability that a text is AI-generated. To make a fair comparison between GLTR and other detectors, we define a detector on top of GLTR to make probability predictions using the normal distribution. We compute an average μ and a standard deviation σ over a sample dataset of 20 human and 20 ChatGPT submissions. The results were μ = 35.33, and s = 15.68. We then used those results to normalize a prediction by computing the standard score of a data point x using x - μ/s. This score is sent as input to the sigmoid function to obtain a probability prediction.
GPTZero was the first detector <cit.> to claim to detect ChatGPT data. The original version of the detector used two measures: perplexity and burstiness. Perplexity refers to a measurement of how well GPT-2 can predict the next word in the text. This appears similar to the way the GLTR detector works <cit.>. The second measure is burstiness: the distribution of sentences. The idea is that humans tend to write with bursts of creativity and are more likely to have a mix of short and long sentence. The current version of GPTZero gives four classes of results. Table <ref> shows how different classes are interpreted as probability. GPTZero claims an 88% accuracy for human text and 72% accuracy for AI text for this detector <cit.>.
AI Text Classifier is OpenAI's 2023 model fine tuned to distinguish between human-written and AI-generated text <cit.>. The model is trained on text generated from 34 models from 5 different organization. The model provides 5 different categories for the results based on the internal probabilities the model provides. Table <ref> shows how different classes are interpreted as probability. The interpretations are based on the final category, not the internal model. Usage of this classifier requires at least 1,000 characters.
GPTKit uses an ensemble of 6 other models, including DistilBERT <cit.>, GLTR, Perplexity, PPL, RoBERTa <cit.>, and RoBERTa (base). The predictions of these models are used to form an overall probability that a text is LLM-generated. However, the exact weight used for each of the detectors is unclear. The detector claims an accuracy of 93% based on testing on a dataset of 100K+ responses <cit.>.
CheckForAI claims to combine the GPT-2 Output Detector along with custom models to help limit false readings <cit.>. The detector also supports account sign up, history storage, and file uploads. The detector provides four classes to compute the probability of text, as shown in Table <ref>. This detector is currently limited to 2,500 characters.
CopyLeaks offers products for plagiarism and AI content detection targeted broadly for individuals, educators, and enterprises. The detector highlights paragraphs written by a human and by AI. CopyLeaks also claims detection across multiple languages, including Spanish (tested in this paper). CopyLeaks claims an accuracy of 99.12% <cit.>. The detector is currently available publicly <cit.>.
Originality.AI is a detector targeted for content publishers. The detector is available through a commercial sign-up page <cit.> with a minimum fee $20. We received research access for analysis of the detector. The detector comes with API access and a number of additional features for content creators. A self-proclaimed study by Originality on ChatGPT suggests that the detector has an accuracy of 98.65% <cit.>.
We did not impose a systematic approach <cit.> to discover publicly available LLM-generated text detectors. Most of the detectors are recent and cannot be easily found on the internet or academic papers. A systematic approach might cover fewer results.
§.§ Addressing the RQ: Effectiveness of LLM-generated text detectors
A detector is only worthy of use if it is reasonably effective. We addressed the RQ by comparing detectors listed in Table <ref> under three metrics: accuracy, false positives, and resilience. Instructors prefer to use detectors that are reasonably accurate, reporting a minimal number of false positives, and are resilient to disguises.
Accuracy refers to how effective the detectors are in identifying LLM-generated texts. We present all accuracy results using two measures of accuracy, as we have found that using only one measure may mislead about some aspect of the results.
The first method (averages) takes the average prediction each detector across a dataset. As discussed in the discovery section, each detector either provides a probability that a text is LLM-generated or a category that represents such a probability. We apply our category to AI conversion tables to obtain a probability for each detector. These probabilities are averaged for the final results.
The second method (thresholds) is calculated as the proportion of correctly-classified LLM-generated texts. These are measured as the number of texts that correctly receive above or below a 50% score out of the total number of texts. This measure is strict, so a prediction of 50% is always considered to be incorrect.
False positives are original submissions that are suspected by LLM-generated text detectors. Fewer false positives are preferred. For this metric, we collected student submissions before the release of ChatGPT (2019) and measured their degree of originality with the detectors. Any suspected submissions (originality degree less than 50%) were expected to be false positives.
Resilience refers to how good LLM-generated text detectors are in removing disguises. Some students might disguise their LLM-generated texts to avoid getting caught. QuillBot <cit.> is a paraphrasing tool capable of paraphrasing text. The tool uses Artificial Intelligence to reword writing. We paraphrased 10 ChatGPT submissions through QuillBot and measured the results.
It is worth noting that measuring effectiveness of LLM-generated text detectors is time consuming and labour intensive. Further, some detectors are not supported with API integration; the authors needed to manually copy and paste each test case.
§.§ Summarizing our experience using the LLM-generated text detectors
We also report our experience in using the LLM-generated text detectors. Several aspects are considered: intuitiveness, clarity of documentation, extendability, variety of inputs, quality of reports, number of supported LLM-generated languages, and pricing.
§ RESULTS
This section discusses our findings from addressing the research question and our experience using LLM-generated text detectors.
§.§ Addressing the RQ: Effectiveness of LLM-generated Text Detector
Table <ref> shows accuracy of each detector across human and ChatGPT data using the threshold method. The data shows CopyLeaks to be the most accurate LLM-generated text detector, with an accuracy of 97.06%. CopyLeaks is followed by the GPT-2 Output Detector/CheckForAI (96.62%), GLTR (88.73%), GPTKit (87.50%), OpenAI's Detector (77.37%), and GPTZero (49.69%).
Table <ref> shows the results using averages instead of thresholds. The results show CopyLeaks to provide the best probabilities (99.53%), followed by CheckForAI (96.56%), the GPT-2 Output Detector (96.29%), GPTKit (82.09%), OpenAI's Detector (82%), OriginalityAI (76.63%), GLTR (65.84%), GPTZero (64.47%).
The data in Tables <ref> and <ref> are both normally distributed, verified using the Shapiro-Wilk and Kolmogorov-Smirnov tests. Thus, no correction needed to be applied. Overall, from the t-tests (Table <ref>: t = 1.67 and p = 0.116, Table <ref>: t = 1.154, p = 0.268, both with 14 degrees of freedom) we did not find significant differences in the accuracy of LLM-generated text detectors between human and ChatGPT data.
Table <ref> shows the false positive results on the human data from the databases and network assignments. GPTKit is the only detector that managed to achieve no false positives across the entire set of human submissions. This is followed by CopyLeaks (1), the GPT-2 Output Detector/CheckForAI (2), OpenAI's detector (6), OriginalityAI (7), GLTR (20), and finally GPTZero (52).
A further investigation of GPTKit, which appears to be the the best detector for avoiding false positives, shows that this detector is still prone to false positives. While none of our original test samples appeared more than 50% fake, we found that some submissions score up to 37% fake from GPTKit. In some cases, removing the last paragraph(s) from these submissions led to a false positive. Figures <ref> and <ref> show such a case. We note that in this case the output of GPTKit also shows that the detector merged separate paragraphs into a single one. This unexpected merge may contribute to the problem.
Table <ref> shows results of 10 ChatGPT papers before and after the Quillbot paraphraser. The results are measured using overall accuracy. The GLTR detector was the most resilient, with none of the predictions changing. It is worth noting that the overall weighted result of GLTR also decreased by 10%, although the change did not effect the accuracy. In contrast, the rest of the detector saw a significant drop following the transformation of Quillbot.
Figures <ref> and <ref> show an example of a ChatGPT data point that went from 98% before Quillbot to 5% after Quillbot on Originality.
Tables <ref> and <ref> show results from the capstone course data, written using Spanish. We found that CopyLeaks and the AI Text Classifier tend always output fake predictions on AI data. In contrast, the GPT-2 Output Detector, GPTZero, CheckForAI, GLTR, GPTKit, and Originality tend to output human predictions.
The data in Tables <ref> and <ref> are both normally distributed, verified using the Shapiro-Wilk and Kolmogorov-Smirnov tests. Thus, no correction needed to be applied. Overall, from the t-tests (Table <ref>: t = 1.766 and p = 0.099, Table <ref>: t = 1.862, p = 0.084, both with 14 degrees of freedom) we did not find significant differences in the accuracy of LLM-generated text detectors between human (Spanish text) and ChatGPT (Spanish text) data.
The GLTR detector shows an interesting mild success with Spanish data. The average top-k score on human data was 104, while the average top-k score on ChatGPT data was 85. When we changed the implementation of GLTR to set a mean of a 94.5 top-k score, GLTR managed to achieve the highest accuracy of 65% on Spanish text.
§.§ Our experience using the LLM-generated text detectors
Generally, many LLM-generated text detectors are intuitive to use. Similar with many online similarity detectors for identifying text plagiarism <cit.>. They have a web-based interface where a user can paste the text they want to check its originality. GPTZero and CheckForAI allow their users to upload a document instead.
While there are a number of LLM-generated text detectors, only two of them have their technical reports publicly available (GPT-2 Output Detector <cit.> and GLTR <cit.>). This is possibly due to at least two reasons. First, technical reports might be misused by some individuals to trick the detectors. Second, some detectors are commercial.
Most LLM-generated text detectors do not facilitate API integration. GPTZero, GPTKit, OriginalityAI, CopyLeaks provide such a feature with a fee. Without API integration, it is challenging to integrate the detectors to existing teaching environments, especially learning management system. LLM-generated text detectors are unlikely to be independently used as the task is labor intensive.
As many of the detectors are commercial, their code is not publicly available. This might complicate instructors to further develop the detectors to fit their particular needs. The only open source detectors are the GPT-2 Output Detection and GLTR.
The detectors are also limited in the input formats they support. Most of them only allow raw text pasted in a form, making them difficult to automate. The PDF parsers that we attempted to use often parsed in an incorrect order and had a tendency to include unwanted characters. We had to write custom scripts to parse the text in a format that translates all information to text.
Detection results are challenging to interpret. Detectors attempt to combat this problem by highlighting content that is more likely to be AI-generated. Table <ref> shows the highlighting support each detector provides. Highlighting is provided on either a paragraph, sentence, or a word basis.
While highlighting does seem to mitigate some barriers, we found that the highlighting feature can still be misleading. This was particularly evident in GPTZero, which highlighted 52 human submissions as either possibly or entirely AI-generated. Figure <ref> shows a sample human report where some sentences were highlighted as more likely to be written by AI. It is unclear what makes the highlighted text more likely be written by AI than the other sentences.
In terms of output quality, it seems like the detectors are limited in their ability to export results. Nevertheless, some detectors were more effective than others. We provided screenshots of GPTKit, GPTZero, and Originality in this report since they provided more detailed results and it was easier to screenshot the results along with the text in contrast to the other detectors. It was more challenging to show full results of other detectors as they did not allow side-by-side results.
Most LLM-generated text detectors only support English as the language of LLM-generated text. While one can still send text in other languages, the results do not appear meaningful as we previously showed.
As many LLM-generated text detectors are commercial and they are relatively new, there appear to mostly individual pricing options. GPTZero CopyLeaks, for instance, have business pricing. GPTZero currently has a subscription plan for business users for $19.99USD per month.
These detectors might be far less useful for instructors living in countries with weak currency; the pricing options are only available in USD.
§ DISCUSSION
The current state of LLM-generated text detectors suggests that they are not yet ready to be trusted blindly for academic integrity purposes or as reliable plagiarism detectors such as Turnitin, MOSS, or JPlag. Our study demonstrates that detectors under-perform compared to the GPT-2 Output Detector and GLTR, which are older and freely available detectors from 2019.
At first glance, it appears that LLM-generated text detectors are fairly accurate with human data being correctly detected ∼89.59%[this percentage is the average accuracy for human data using Tables <ref> and <ref>.] while the average accuracy for ChatGPT-generated data is substantially lower; ∼77.42%[this percentage is the average accuracy for ChatGPT-generated data using Tables <ref> and <ref>.]. Upon deeper inspection, it is apparent that the number of potential false positives can lead to a wide array of issues, especially if being trusted for plagiarism detection at educational institutions.
Delving further, when a paraphraser (in this case, QuillBot) is utilized the average accuracy is slightly reduced for human data
∼89.02%[this percentage is the average accuracy for human data using Tables <ref> and <ref>.] but this substantially reduces the accuracy of ChatGPT-generated data ∼49.17%[this percentage is the average accuracy for ChatGPT-generated data using Tables <ref> and <ref>.]. This means that in more than half of all cases, ChatGPT-generated data cannot correctly be identified by these detectors. Though, some detectors perform better than others (e.g., GLTR), it is still a serious concern for users of these detectors.
Additionally, once non-English languages are introduced, these detectors are easily exacerbated. We investigate submissions made in Spanish and see that the average accuracy for human data lowers to an average of ∼70.99% [this percentage is the average accuracy for human data using Tables <ref> and <ref>.], and ChatGPT-generated data reduces to an abysmal ∼17.50%[this percentage is the average accuracy for ChatGPT-generated data using Tables <ref> and <ref>.]. Though only Spanish was investigated, it introduces the need for additional research into alternative languages (non-English).
Presently, all LLM-generated text detectors struggle with languages other than English, code, and special symbols, resulting in fairly inaccurate results. As a point of clarity, it would be ideal for these detectors to explicitly state their limitations and aim to produce human predictions in such cases.
In terms of usability, LLM-generated text detectors need some improvements. Although they are intuitive to use and generate acceptable reports, many of them are not well documented at a technical level, some do not have APIs making them more difficult to integrate into local and larger systems (e.g., Learning Management Systems), and the support of these detectors is limited. Furthermore, some of these detectors require processing fees.
From our results, LLM-generated text detectors appear to lack in understandability. We are aware that all of these detectors leverage similar large language models for detection purposes. However, they might differ in terms of their technical implementation, parameters, pre-trained data, etc. These are unlikely to be revealed since most of the detectors are for commercial-use and, thus, proprietary. While some detectors highlight sentences that are more likely to be AI-generated (Table <ref>), the results produced by the detectors are not clear enough for users of these detectors.
§ THREATS TO VALIDITY
Our study has several threats to validity:
* The findings of the study reflect detector results that are accurate as of April 2023. The detectors are volatile, and owners of these detectors could update their models. Results could change based on updates to LLM-generated text detectors.
* Accuracy, false positives, and resilience were arguably sufficient to represent effectiveness. However, additional findings can be obtained by considering other effectiveness metrics.
* The data sets were obtained from two institutions; one uses English as the operational language while another uses Spanish. This means that the findings might not be generalizable to other institutions, especially those with different operational languages.
* While we believe that the data sets are sufficient to support our findings, we acknowledge that more data sets can strengthen the findings.
§ CONCLUSION
This paper examines eight LLM-generated text detectors on the basis of effectiveness. The paper shows that while detectors manage to achieve a reasonable accuracy, they are still prone to flaws and can be challenging to interpret by the human eye. Ultimately, LLM-generated text detectors, while not yet reliable for academic integrity or plagiarism detection, show relatively accurate results for human-generated data compared to ChatGPT-generated data. However, false positives are a significant concern, especially when used for plagiarism detection in educational institutions. When a paraphrasing tool like QuillBot is employed, the accuracy decreases for both human and ChatGPT-generated data. Additionally, the detectors struggle with non-English languages, resulting in even lower accuracy. It is crucial for these detectors to acknowledge their limitations and aim for improved performance in various language contexts.
§.§ Future Work
Future detectors could attempt to incorporate a combination of metrics along with their accuracy for AI detectors. A combination of many factors along with the accuracy and false positive rates may give educators better insights into the predictions. This could include text-based features such as burstiness and repetition as well as AI-learned features such as probabilities. These detectors could further be fine-tuned for specific domains to improve their reliability.
Additionally, there is a fundamental need to have accurate and understandable LLM-generated text detectors available for educators to combat against the rising concern of academic integrity due to these publicly available LLMs. It is also important for the researchers to contact the creators of these detectors to better understand the related issues and needs of the end users, but also to facilitate a deeper conversation about the functionality and correctness of their instruments.
Finally, there is an apparent need to investigate the use of non-English languages using these detectors as large language models, like the one(s) used by ChatGPT, can produce content in languages other than English.
ACM-Reference-Format
|
http://arxiv.org/abs/2307.07478v1 | 20230714170002 | Modal analysis on quantum computers via qubitization | [
"Yasunori Lee",
"Keita Kanno"
] | quant-ph | [
"quant-ph"
] |
Modal analysis
on quantum computers via qubitization
Yasunori Lee[
<[email protected]>
]
and
Keita Kanno[
<[email protected]>
]
QunaSys,
Bunkyo, Tokyo, Japan
Natural frequencies and normal modes are basic properties of a structure
which play important roles in analyses of its vibrational characteristics.
As their computation reduces to solving eigenvalue problems,
it is a natural arena for application of quantum phase estimation algorithms,
in particular for large systems.
In this note, we take up some simple examples of (classical) coupled oscillators and show how the algorithm works
by using qubitization methods based on a sparse structure of the matrix.
We explicitly construct block-encoding oracles along the way,
propose a way to prepare initial states,
and briefly touch on a more generic oracle construction for systems with repetitive structure.
As a demonstration, we also give rough estimates of
the necessary number of physical qubits and actual runtime it takes
when carried out on a fault-tolerant quantum computer.
empty
§ INTRODUCTION AND SUMMARY
The ultimate target of quantum computation is to solve problems
which are practically intractable by classical computation
due to their size and/or computational time.
Among these problems is computing
eigenvalues
of a large matrix,
where there is a quantum algorithm called the quantum phase estimation (QPE) <cit.>
which is exponentially efficient compared to any known classical algorithms
(e.g. conjugate gradient method)
in terms of both space and time (gate) complexity.[
Strictly speaking,
one needs to prepare a suitable initial quantum state
to compute a specific eigenvalue,
which in general cannot be done efficiently and ruins the exponential advantage.
Here we assume that a desirable initial state can be easily prepared,
which is the case for the problems in this note (→ Sec. <ref>).
]
This task is of great importance as it covers a vast range of applications,
including in particular quantum chemistry/physics problems of
computing the energy of large systems given their Hamiltonians.
To figure out whether the desired QPE can actually be carried out within reasonable time,
one must go beyond asymptotic scaling and estimate concrete numbers of qubits and quantum gates (especially non-Clifford ones)
needed to implement the algorithm in a fault-tolerant manner.
To the authors' knowledge, the program along this line was initiated in a full-fledged form by <cit.>,
employing Lie-Trotter-Suzuki decomposition
and resulting in costs small enough to raise hopes but motivating further reductions.
In the past few years,
a novel powerful method called qubitization <cit.> has been developed, and
utilizing it significantly reduced the number of necessary non-Clifford gates (at the expense of a moderate number of qubits),
see e.g. <cit.>.
At the time of writing, however,
almost all the literature on application of qubitization-based QPEs have focused on quantum chemistry/physics problems,
relying on decomposition of a target matrix (i.e. Hamiltonian) into a linear combination of unitaries (LCU).
In this note,
we take up yet another important class of eigenvalue problems, namely modal analysis,
and see how quantum phase estimation algorithms can be applied for toy problems of (classical) coupled oscillators.
Modal analysis is important since the knowledge of natural frequencies and normal modes of a structure
is indispensable to avoid unwanted resonances leading to its collapses or failures,
which makes it appear in various engineering scenes <cit.>.
Notably, in the simplest cases, analyses reduce to eigenvalue problems of a sparse matrix,
and thus one can draw on qubitization relying not on a decomposition into LCU but on a sparse structure of the matrix,
which gives them a somewhat different flavor compared to the existing analyses.[
This should not be confused with so-called sparse qubitization
which is actually LCU-based qubitization,
where the (part of) original Hamiltonian is made “sparse” by truncating small coefficients.
]
Although there are some previous studies <cit.> which seem to be in a similar vein at first sight,
their approach is top-down and motivated by generic well-structured matrices themselves,
while our work is bottom-up and more problem-oriented, which makes the direction orthogonal to some extent.
The toy problems themselves are quite trivial and would provide hardly any practical benefit,
but the resource estimation at least confirms the exponential advantage of quantum computation over its classical counterpart,
and the authors hope that this work would serve as a step toward analyses of realistic uses in more generality:
For example, the oracle construction introduced in this note can be applied to
efficient simulations of exponentially many coupled oscillators described by <cit.>.
The rest of the note is organized as follows.
In Section <ref>, we give a brief overview of how to block-encode a generic sparse matrix into a unitary matrix and how to qubitize it.
In Section <ref>, we consider concrete models of (classical) coupled oscillators and explicitly construct the unitary in terms of quantum circuits, starting from those of underlying oracles.
In Section <ref>, we propose a way to prepare the input initial state of a QPE algorithm.
Finally in Section <ref>, we roughly estimate total resources required to carry out the desired QPE.
Appendix <ref> provides elementary-gate-level implementations of various circuit components appearing in the main part.
§ REVIEW OF QUBITIZATION OF SPARSE MATRICES
The problem of interest here is to compute the eigenvalues λ of a given N× N Hermitian matrix H.
In order to feed the matrix H to a quantum computer, one has to somehow embed it in a unitary matrix U_H.
One naive embedding is just to take U_H=e^icH where c is some constant.
Although this is conceptually simple,
actual implementation as a quantum circuit is often costly, making its practical use prohibitive.
There is another embedding called block encoding
where U_H is an M× M unitary matrix (M ≥ N)
and is reduced to H using a certain state vector |ψ⟩ of size M-N as
⟨ψ| U_H |ψ|=⟩ H.
This type of embedding is in fact not unique, and it turns out <cit.> that
among them is an especially nice one U_H^∗ such that
U_H^∗(
|λ⟩|ψ⟩± i|orthogonal⟩/√(2))
=
e^∓ iarccosλ(
|λ⟩|ψ⟩± i|orthogonal⟩/√(2))
where H|λ⟩ = λ|λ⟩ and (⟨ψ|⟨λ|)|orthogonal⟩ = 0.
Qubitization
is a method to generate U_H^∗ from a block encoding U_H,
and by applying QPE to U_H^∗, one can extract the desired information on λ
with far fewer computational costs compared to the naive embedding.
Fortunately,
systematic ways of (efficient) block encoding are known
for several types of matrices with special properties or structures.
Previous studies on cost estimation have mostly took up those with representations in a
linear combination of unitaries,
appearing in quantum chemistry/physics problems as a result of Jordan-Wigner transformation of underlying fermionic Hamiltonians.
However, there is another nice class of matrices with such systematic block encoding,
namely d-sparse matrices[
Here we follow <cit.> and adopt the symbol d (presumably after density).
] with at most d non-zero entries in each row (and/or column).[
Of course one can call any N× N matrix “N-sparse,”
but the computational advantage arises only for sparse enough matrices i.e. d ≪ N.
(The same is true for the case of LCU;
any matrix can be written in a form of LCU,
but the whole algorithm can be carried out efficiently only for those with a small norm.)
]
In the following, we introduce two special oracles O_F, O_H
and review how the block encoding and the qubitization for d-sparse matrices are realized using them.
§.§ Block encoding
One way to block-encode an N× N Hermitian matrix H
is by using two oracles O_F and O_H <cit.> (see also <cit.>) such that
for x∈{1,…,N} and i∈{1,…,d},
O_F |x,i⟩=|x,y_i⟩
where y_i ∈{1,…,N} is the position (i.e. column index) of the i-th non-zero element in the x-th row, and
for x,y∈{1,…,N},
O_H |x,y⟩|z⟩=|x,y⟩|z⊕ H_xy⟩
where (z is any number and) H_xy is the (x,y)-element of H, hereafter assumed to be (real) non-negative,[
The restriction is to avoid subtleties regarding signs and phases arising from the square roots in Eq. (<ref>);
for a completely general treatment, see <cit.>.
This indeed holds for the examples in Sec. <ref>.
]
and ⊕ denotes a bitwise XOR.
Let
a_1, a_2 denote ancillary single-qubit registers and
a_s, s denote ancillary and signal ⌈log_2 N⌉-qubit registers respectively.
Our initial target is a unitary operator
U_H = U_2^† U_1
where two unitaries U_1, U_2 are such that
[ U_1 |0⟩_a_1|0⟩_a_2|0⟩_a_s|x'⟩_s = 1/√(d)∑_y'(
√(H_x'y'/i,jmax H_ij)|0⟩_a_1 +
√(1-H_x'y'/i,jmax H_ij)|1⟩_a_1)
|0⟩_a_2|y'⟩_a_s|x'⟩_s; U_2 |0⟩_a_1|0⟩_a_2|0⟩_a_s|x⟩_s = |0⟩_a_11/√(d)∑_y (
√(H_xy/i,jmax H_ij)|0⟩_a_2 +
√(1-H_xy/i,jmax H_ij)|1⟩_a_2)
|x⟩_a_s|y⟩_s ]
and the summations are over y (resp. y') whose corresponding matrix elements H_xy (resp. H_x'y') are non-zero.
This unitary U_H block-encodes the original matrix H as
⟨0|_a_1⟨0|_a_2⟨0|_a_s⟨x|_s U_H
|0⟩_a_1|0⟩_a_2|0⟩_a_s|x'⟩_s
=
1/d ·i,jmax H_ij∑_y∑_y'√(H_xy H_x'y')⟨x|y'|⟨%s|%s⟩⟩y|x'_
= H_xx'
i.e.
⟨0|_a_1⟨0|_a_2⟨0|_a_s U_H
|0⟩_a_1|0⟩_a_2|0⟩_a_s∼
H.
Starting from a state |0⟩_a_1|0⟩_a_2|0⟩_a_s|x'⟩_s|0⟩_v with an additional register v of suitable size,
one can easily verify that U_1 can be realized for example by sequentially acting
* some gates (e.g. Hadamard gates H^⊗log_2 d when d is a power of two),
making an equal-superposition state 1/√(d)∑_i=1^d |0⟩_a_1|0⟩_a_2|x',i⟩_s,a_s|0⟩_v,
* O_F, turning the state into 1/√(d)∑_i=1^d |0⟩_a_1|0⟩_a_2|x',y'_i⟩_s,a_s|0⟩_v,
* O_H, loading the values of matrix elements as
1/√(d)∑_i=1^d |0⟩_a_1|0⟩_a_2|x',y'_i⟩_s,a_s|H_x'y'_i⟩_v,
* a controlled rotation,
leading to
1/√(d)∑_y'(
√(H_x'y'/i,jmax H_ij)|0⟩_a_1 +
√(1-H_x'y'/i,jmax H_ij)|1⟩_a_1)|0⟩_a_2|x',y'⟩_s,a_s|H_x'y'⟩_v,
* O_H again, uncomputing the state and returning the v register to |0⟩_v
(as k ⊕ k = 0 holds for any integer k), and thus achieving the desired state as in Eq. (<ref>).
Note that the controlled rotation can actually be done in-place, making the v register unnecessary and simplifying Step 3 – 5.
Similarly, starting from a state |0⟩_a_1|0⟩_a_2|0⟩_a_s|x⟩_s|0⟩_v,
one can check that U_2 can be realized by sequentially acting
* some gate, making an equal-superposition state 1/√(d)∑_i=1^d |0⟩_a_1|0⟩_a_2|x,i⟩_s,a_s|0⟩_v,
* O_F, turning the state into 1/√(d)∑_i=1^d |0⟩_a_1|0⟩_a_2|x,y_i⟩_s,a_s|0⟩_v,
* O_H, loading the values of matrix elements as
1/√(d)∑_i=1^d |0⟩_a_1|0⟩_a_2|x,y_i⟩_s,a_s|H_xy_i⟩_v,
* a controlled rotation, leading to
|0⟩_a_11/√(d)∑_y (
√(H_xy/i,jmax H_ij)|0⟩_a_2 +
√(1-H_xy/i,jmax H_ij)|1⟩_a_2)|x,y⟩_s,a_s|H_xy⟩_v,
* O_H again, uncomputing the state,
* SWAP gates swapping a_s and s registers, achieving the desired state as in Eq. (<ref>),
the only difference to U_1 being (which qubit to rotate and) the additional swaps at the end.
§.§ Qubitization
With the unitary U_H thus constructed at hand, one can generate a nice block encoding U_H^∗ called the walk or iterate operator.
Following <cit.>,
let us add another qubit and consider a unitary Ũ|0⟩⟨0|⊗ U_H + |1⟩⟨1|⊗ U_H^†.
Together with a NOT operation on the additional qubit X (|0⟩⟨1| + |1⟩⟨0|) ⊗ I_a ⊗ I_s,
the unitary U_H^∗ is defined to be
U_H^∗(
2|+⟩|0⟩_a_1|0⟩_a_2|0⟩_a_s⟨+|⟨0|_a_1⟨0|_a_2⟨0|_a_s⊗ I_s - I ⊗ I_a ⊗ I_s
) XŨ
which can be realized as
[scale=1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
|1⟩_sgn Z Z
1 1 H -1 H
a_1 [wires=8, nwires=4,7]U_H [wires=8, nwires=4,7]U_H^† -1
a_2 -1
[wires=3]a_s -1
⋮ ⋮ ⋮
-1
[3]s
⋮ ⋮
;
with yet another qubit (at the top in the circuit diagram) identifying a potential sign flip resulting from the reflection operation.
Recalling Eq. (<ref>), one can then carry out an ordinary QPE against this unitary to compute the phase arccosλ,
from which the desired eigenvalue λ of the original Hermitian matrix H can be immediately extracted.
§ APPLICATION TO MODAL ANALYSIS
Armed with the theory of qubitization, let us see how it is actually worked out.
The target is a system of coupled oscillators with which we are very familiar as a basic model of various phenomena in Nature.
As will be shown below, the analyses often involve solving eigenvalue problems of sparse matrices,
and thus is within a scope of the method introduced in the previous section.
For the sake of simplicity, we always take the matrix size to be N=2^n.
§.§ Settings
We will be considering a system of N=2^n point masses, linearly (i.e. in one dimension) and periodically connected by springs:
at (-4.5,0) ⋯;
(-4.0,0) – (-3.8,0);
(-3.3,0) circle [radius=0.5] node[above=0.6]m_i-1;
(-2.8,0) – (-2.6,0);
[
snake=coil,
segment amplitude=8,
segment length=4
] (-2.6,0) – node[below=0.5] k_i-1 (-0.6,0);
(-0.6,0) – (-0.5,0);
(0,0) circle [radius=0.5] node[above=0.6]m_i;
(0.5,0) – (0.7,0);
[
snake=coil,
segment amplitude=8,
segment length=4
] (0.7,0) – node[below=0.5] k_i (2.7,0);
(2.7,0) – (2.8,0);
(3.3,0) circle [radius=0.5] node[above=0.6]m_i+1;
(3.8,0) – (4.0,0);
[
snake=coil,
segment amplitude=8,
segment length=4
] (4.0,0) – node[below=0.5] k_i+1 (6,0);
at (6.5,0) ⋯;
[line width=1.5pt, ->] (-5.3,-1.66) – (7.3,-1.66) node[right] position;
(0,-1.51) – (0,-1.81) node[below=0.2] x_i;
(3.3,-1.51) – (3.3,-1.81) node[below=0.2] x_i+1;
(-3.3,-1.51) – (-3.3,-1.81) node[below=0.2] x_i-1;
Here, m_i's denote masses and k_i's denote spring constants.
The equation of motion for the i-th mass reads
m_i ẍ_i=k_i(x_i+1-x_i)-k_i-1(x_i-x_i-1)
(with x_0 x_N and x_N+1 x_1 etc.),
and the natural frequencies (resp. normal modes) are given as eigenvalues (resp. eigenvectors) of an N× N matrix
H^original =
(
[ -k_N+k_1/m_1 k_1/m_1 0 ⋯ 0 k_N/m_1; k_1/m_2 -k_1+k_2/m_2 k_2/m_2 0 ⋯ 0; 0 k_2/m_3 -k_2+k_3/m_3 ⋮; ⋮ []-10⋱ 0; 0 -k_N-2+k_N-1/m_N-1 k_N-1/m_N-1; k_N/m_N 0 ⋯ 0 k_N-1/m_N -k_N-1+k_N/m_N ]).
For the toy problems examined in this note,
we restrict ourselves to matrices (<ref>) whose diagonal elements are all equal.
This allows us to completely separate the diagonal part and the off-diagonal part and to forget about the former.
As a result, target matrices H's we want to diagonalize are 2-sparse (i.e. d=2) with very simple structure, namely
H_xy{[ ≠ 0 (|x-y| =1 mod 2^n),; = 0 (otherwise).; ].
Note that these simplicities are just for demonstration;
our methods can be easily generalized to treat matrices with e.g.
* non-identical diagonal elements
* non-periodic boundary conditions
* wider “band” (which in particular incorporate k (≥ 2)-dimensional cases with more complicated couplings between multiple coordinates described by kN× kN matrices)
as will be described along the way.
§.§ Adder
One of the fundamental components in implementation of the oracles O_F, O_H is an adder.
Given two states |x⟩, |y⟩ encoding (binary representations of) integers x,y modulo 2^n in a computational basis,
the (required) function of the adder is to return a state |x+y⟩ encoding (x+y) modulo 2^n.
In this note, we adopt the adder proposed by <cit.>,
whose I/O is schematically represented as follows
[scale=1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=3]|x⟩_x [wires=6, nwires=2,5] adder [wires=3]|x⟩_x
⋮ ⋮
[wires=3]|y⟩_y [wires=3]|x+y⟩_y
⋮ ⋮
;
and consumes 2n Toffoli gates (plus a single ancillary qubit which is omitted above).
For the details of implementation, see Appendix <ref>.
§.§ Oracle O_F
For the matrices (<ref>), non-trivial computations involving the oracle O_F are limited to[
The index origin is (shifted by one from the expressions in Sec. <ref> and) set to zero for convenience.
]
O_F|x,0⟩ =|x,x-1⟩,
O_F|x,1⟩ =|x,x+1⟩.
Recalling that flipping all bits of an integer y leads to
(2^n-1)-y ≡ (-y-1) modulo 2^n,
this O_F can be implemented in a rather straightforward manner
employing suitable ancillary qubits encoding “|-1⟩” as
[scale=1] at (-8,0) [row sep=20pt,between origins, column sep=10pt]
[wires=3]|x⟩_x [wires=10, nwires=2,5,6,9]O_F [wires=3]|x⟩_x
⋮ ⋮
[wires=4]|i⟩_y [wires=4]|x± 1⟩_y
⋮ ⋮
[wires=3]|0⟩_a [wires=3]|0⟩_a
⋮ ⋮
;
at (-4.7,0) =;
[scale=1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=7, nwires=2,5] adder
⋮ ⋮
[4]|x+2i⟩ [wires=7, nwires=2,6] adder [4]|x+2i-1⟩
⋮ ⋮
1
-1
⋮ ⋮
;
with each register x,y,a consisting of n qubits.
Here, the two CNOT gates in front of the first adder doubles the input i,
mapping |i=0⟩_y ↦|0⟩_y and |i=1⟩_y ↦|2⟩_y, respectively.[
In general, one can use another qubit to first somehow check whether 0 ≤ i≤ d-1 actually holds and output the result to the qubit,
and then apply the gates additionally controlled on the qubit, if necessary.
For small enough d, brute-force multi-controlled NOT gates would do the job without ruining efficiency.
]
Also, note that the results of the addition are always output to the y register,
meaning in particular that the second adder is placed “upside down.”
Since it contains two adders of n-bit integers,
the circuit as a whole consumes 2 × 2n = 4n Toffoli gates with (3n+1) qubits,
where the second adder recycles the ancillary qubit of the first adder (which is omitted in the diagram).
§.§ Oracle O_H
When non-zero H_xy's of the matrices (<ref>) have simple dependence on a difference (x-y),
a naive way to implement the oracle computation O_H|x,y⟩|z⟩ = |x,y⟩|z⊕ H_xy⟩
is to first read out (x-y) and then bitwise-XOR the corresponding element H_xy.
The former can be realized for example by adding x and (-y-1),
and the latter can be realized by (hard-coded) bitwise NOT gates controlled on the difference,
i.e. by placing NOT gates acting on qubits corresponding to set-bits of the number H_xy.
The quantum circuit is as
[scale=0.9] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=3]|x⟩_x [wires=7, nwires=2,5] adder [wires=3]|x⟩_x
⋮ ⋮
[wires=4]|y⟩_y 1 1 [wires=4]|x-y-1⟩_y
⋮ ⋮ ⋮ ⋮
1 1
1 1
[wires=3]|z⟩_z [wires=3, nwires=2]⊕ H_x(x-1) [wires=3, nwires=2]⊕ H_x(x+1) [wires=3]|z ⊕ H_xy⟩_z
⋮ ⋮
⋯ ;
followed by suitable uncomputation, namely
[scale=1] at (-8.66,0) [row sep=20pt,between origins, column sep=10pt]
[wires=3]|x⟩_x [wires=10, nwires=2,5,6,9]O_H [wires=3]|x⟩_x
⋮ ⋮
[wires=4]|y⟩_y [wires=4]|y⟩_y
⋮ ⋮
[wires=3]|z⟩_z [wires=3]|z⊕ H_xy⟩_z
⋮ ⋮
;
at (-6.6,0) =;
[scale=1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=7, nwires=2,5] adder [wires=7, nwires=2,5] adder^†
⋮ ⋮
1 1
⋮ ⋮ ⋮
1 1
1 1
[wires=3, nwires=2]⊕ H_x(x-1) [wires=3, nwires=2]⊕ H_x(x+1)
⋮ ⋮
⋯ ;
again with n-qubit registers x,y and a register z with suitable size to store values.[
For more generic band matrices, one can suitably add corresponding multi-controlled bitwise-XOR gates for H_x(x± 2), H_x(x± 3), and so on
(where there is also a chance that, by sorting multi-controlled operations, adjacent ones can be contracted).
]
This time the circuit consumes 2 × 2n + 2× 2(n-1) = 8n-2 Toffoli gates by a naive counting,
as the multi-controlled gates are decomposed as in Appendix <ref> with 2(n-1) Toffoli gates.[
If the output of the first adder is guaranteed to be either +1 or -1,
one does not need to multi-control the ⊕ H_xy gate and a single-control operation is sufficient.
This reduces the number of Toffoli gates used in the circuit and thus makes the computation faster,
although we stick to the naive implementation for ease of understanding.
]
As already mentioned in Sec. <ref>,
for actual usage, there is no need to first load the value of H_xy and then do the corresponding rotation;
one can directly implement the rotations in-place.
The quantum circuit of the modified oracle is given as
[scale=0.8] at (-9,0.1) [row sep=20pt,between origins, column sep=10pt]
[wires=3]|x⟩_x [wires=8, nwires=2,5,6]O_H^mod. [wires=3]|x⟩_x
⋮ ⋮
[wires=4]|y⟩_y [wires=4]|y⟩_y
⋮ ⋮
|0⟩_z ;
at (-7,0) =;
[scale=0.8] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=7, nwires=2,5] adder [wires=7, nwires=2,5] adder^†
⋮ ⋮
1 1
⋮ ⋮ ⋮ ⋮
1 1
1 1
R_y(2arccos√(H_x(x-1))) R_y(2arccos√(H_x(x+1))) ;
Another point worth mentioning here is about exceptional operations including those concerning “boundary conditions.”
In Eq. (<ref>), the condition for H_xy to be (possibly) non-zero involved taking modulo 2^n (due to the periodic boundary condition of the system),
resulting in a circulant matrix.
However, by inserting a bitwise-XOR (or corresponding rotation) gate controlled on the x register in addition to the y register
between two adders in the above circuit, one can selectively undo the operation and kill unnecessary elements.
For example, by acting controlled-R_y(-2arccos√(H_x(x-1))) gate only when x=0,
one can in effect eliminate the H_0(n-1) element.
Doing the same thing for H_(n-1)0, one can realize an oracle O_H for
a tridiagonal (or more generally a band) version of the original matrix (corresponding to a fixed boundary condition), without much cost.
§.§ Example 1: equal spring constants
Now let us consider a simple system of N=2^n point masses of equal mass m connected by springs with equal spring constant k.
The matrix (<ref>) (up to a constant factor k/m) becomes
H_xy^original =
{[ 1 (|x-y| = 1); -2 (|x-y| = 0); 0 (otherwise); ].
and one can forget about the diagonal elements by considering H = H^original+2I.[
The problem itself is just a special case of that considered in <cit.>,
but the emphasis is put on somewhat complementary aspects, so to speak.
]
Since all the non-zero matrix elements take the same value, things are extremely simplified;
the controlled rotation parts of the algorithm become trivial,
and therefore one can safely skip the steps involving ancillary qubits a_1, a_2, and v
(and thus completely discard them),
making the oracle O_H unnecessary as a result.
This indeed makes sense because all we need is mere information about the position of non-zero matrix elements and not the values themselves.
Anyway, following the description in Sec. <ref>,
an explicit circuit for the unitary U_H is given by concatenating two circuit components corresponding to U_1 and U_2^†
[scale=0.9] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=4]|x'⟩_s [wires=11, nwires=2,6,10] O_F [4]x [wires=4]|x'⟩_s
⋮ ⋮
[wires=4]|0⟩_a_s [4]i [wires=4]|y'⟩_a_s
⋮ ⋮
H
[wires=3]|0⟩ [wires=3]|0⟩
⋮ ⋮
;
[scale=0.9] at (7,0) [row sep=20pt,between origins, column sep=10pt]
[wires=4]|y⟩_s [wires=8, nwires=2,6]swap [wires=11, nwires=2,6,10] O_F^† [4]x [wires=4]|x⟩_s
⋮ ⋮
[wires=4]|x⟩_a_s [4]i [wires=4]|0⟩_a_s
⋮ ⋮
H
[wires=3]|0⟩ [wires=3]|0⟩
⋮ ⋮
;
from which one can immediately obtain a circuit for the unitary U_H^∗ as described in Sec. <ref>.
An important point to note is that
addition of control qubits to U_H is equivalent to that to the “swap” gate alone,
since the net output of the circuit is the same even if the O_F-O_F^† pair and the H-H pair are not controlled on the additional qubits.[
In fact, this is a special case of the trick mentioned in Appendix <ref>.
]
Taking this trick into account, the unitary U_H^∗ is reduced to a (4n+2)-qubit circuit
(including n ancillary qubits for multi-controlled gates (one of which is also used for adders) omitted in the diagram)
[scale=0.9] at (0,0) [row sep=20pt,between origins, column sep=10pt]
|1⟩_sgn Z Z
H -1 H
[wires=4]a_s [wires=10, nwires=2,6,9] O_F [4]i [wires=7, nwires=2,6]swap [wires=10, nwires=2,6,9] O_F^† [4]i -1
⋮ ⋮ ⋮ ⋮
-1
H H -1
[wires=3]s [3]x [3]x
⋮ ⋮ ⋮
[wires=3]|0⟩
⋮ ⋮
;
where the latter half of (controlled-)U_H and the former half of (controlled-)U_H^† which do not involve control qubits are canceled out,
and the two controlled-“swap”s are combined into a single (non-controlled) “swap.”
As a result, a naive counting of Toffoli gates consumed by a controlled-U_H^∗ (to be fed into the QPE) as a whole
(again keeping the trick in mind)
is a sum of
* two O_F's: 2 × 4n
* a controlled-“swap” (cf. Appendix <ref>): n
* a multi-controlled gate at the end (cf. Appendix <ref>): 2[(n+2)-1]
which is equal to 11n+2.
§.§ Example 2: alternating spring constants
Let us next consider a slight generalization where one has alternating two spring constants k_1, k_2.
Without loss of generality, one can assume k_1 < k_2 and
the equation of motion for the i-th point mass to be
m ẍ_i
=
{[ k_1(x_i+1-x_i)-k_2(x_i-x_i-1) (i even),; k_2(x_i+1-x_i)-k_1(x_i-x_i-1) (i odd).; ].
The matrix (<ref>) (up to a constant factor 1/m) becomes[
This is actually equivalent to
a Hamiltonian of the Su-Schrieffer-Heeger model <cit.>,
which describes a polyacetylene as a dimer chain consisting of alternating single (σ) and double (π) bonds,
and suggests potential utility of non-LCU-based qubitization even in quantum chemistry/physics problems.
]
H_xy^original =
{[ k_1 (“x even and x-y = +1” or “x odd and x-y = -1”); -(k_1+k_2) (x-y = 0); k_2 (“x odd and x-y = +1” or “x even and x-y = -1”); 0 (otherwise) ].
and one can consider H = 1/k_2[H^original + (k_1+k_2)I] as a target matrix.
For this problem, we further modify the O_H^mod. oracle;
we add the topmost additional control qubit to the controlled rotation gates to identify the parity of x.
Then, non-trivial rotations are carried out for H_xy = k_1/k_2 (while for H_xy = k_2/k_2=1 rotations are trivial as arccos 1 = 0 and thus unnecessary), which can be implemented as
[scale=0.8] at (-8.4,0.1) [row sep=20pt,between origins, column sep=10pt]
[wires=4]|x⟩_x [wires=9, nwires=2,3,6,7]O_H^mod. [wires=4]|x⟩_x
⋮ ⋮
[wires=4]|y⟩_y [wires=4]|y⟩_y
⋮ ⋮
|0⟩_z ;
at (-6.2,0) =;
[scale=0.8] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[wires=8, nwires=2,6] adder [wires=8, nwires=2,6] adder^†
⋮ ⋮
1 1
[4]x-y-1 1 1
⋮ ⋮ ⋮ ⋮
1 1
1 1
R_y(2arccos√(k_1k_2)) R_y(2arccos√(k_1k_2)) ;
The circuit of the unitary U_H should be given by concatenating U_1 and U_2^† as
[scale=0.9] at (0,0) [row sep=20pt,between origins, column sep=10pt, transparent]
[wires=3]|x⟩_s [wires=12, nwires=2,5,11] O_F [3]x [wires=8, nwires=2,5] O_H^mod. [wires=3]|x⟩_s
⋮ ⋮
[wires=4]|0⟩_a_s [4]i [4]y [wires=4]|y⟩_a_s
⋮ ⋮
H
[]|0⟩_a_1 []z []|∗⟩_a_1
[]|0⟩_a_2 []|0⟩_a_2
[wires=3]|0⟩ [wires=3]|0⟩
⋮ ⋮
;
[scale=0.9] at (9,0) [row sep=20pt,between origins, column sep=10pt, transparent]
[wires=3]|y⟩_s [wires=7, nwires=2,5]swap [wires=9, nwires=2,5] O_H^mod.† [wires=12, nwires=2,6,11] O_F^† [3]x [wires=3]|x⟩_s
⋮ ⋮
[wires=4]|x⟩_a_s [4]i [wires=4]|0⟩_a_s
⋮ ⋮
H
[]|0⟩_a_1 []|0⟩_a_1
[]|∗⟩_a_2 []z []|0⟩_a_2
[wires=3]|0⟩ [wires=3]|0⟩
⋮ ⋮
;
As before, when adding control qubits to U_H, one does not need to add them to the O_F-O_F^† pair, the H-H pair,
and the adder-adder^† pairs inside O_H^mod.'s.
As a result, U_H^∗ is reduced to a (4n+6)-qubit circuit (including (n+2) ancillary qubits (suitably recycled) for the multi-controlled gates) as
[scale=0.66] at (0,0) [row sep=20pt,between origins, column sep=10pt, transparent]
|1⟩_sgn Z Z
1 1 4 4 4 4 1 1 H -1 H
[wires=4]s [wires=13, nwires=2,6,12]O_F [wires=9, nwires=2,6]O_H^mod. [wires=8, nwires=2,6]swap [wires=8, nwires=2,6]adder [wires=8, nwires=2,6]adder^† [wires=8, nwires=2,6]swap [wires=9, nwires=2,6]O_H^mod.† [wires=13, nwires=2,6,12]O_F^†
⋮ ⋮
1 1 1 1
[wires=4]a_s 1 1 1 1 -5
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
1 1 1 1 -1
H 2 2 2 2 H -1
[]a_1 -1
[]a_2 R_y R_y R_y^† R_y^† -1
[wires=3]|0⟩
⋮ ⋮
;
Therefore, a naive counting of Toffoli gates consumed by a controlled-U_H^∗ is a sum of
* two O_F's: 2 × 4n
* two multi-controlled-O_H^mod.'s: 2 ×[2× 2n + 2 × 2{(n+3) - 1}]
* two multi-controlled-“swap”s: 2 ×[n × 2(3-1) ]
* two adders: 2 × 2n
* multi-controlled R_y's (suitably permuted and paired up): 2 × 2[(n+3)-1]
* a multi-controlled gate at the end: 2[(n+4)-1]
which is equal to 42n+30.
The results are summarized as follows:
§.§ Generalization: multiple layers
In passing, we also mention a more abstract level of oracle utilization.
Consider a generic system of K point masses (not necessarily sparsely) connected to each other.
Then, one can fabricate oracles O_F^full, O_H^full for a system of N=K× L point masses
comprised of L copies of the original system layered on top of each other,
if oracles O_F^sub, O_H^sub slightly modified as follows for the original system are available somehow:
For a x-th point mass (x∈{1,...,K}) of a l-th layer (l∈{1,...,L}),
O_F^sub|x,i⟩|0⟩=|x,y_i⟩|Δ l_i⟩
where i labels the spring connecting the x-th point mass to a y_i-th point mass of a (l+Δ l_i)-th layer, and
O_H^sub|x,y⟩|Δ l⟩|z⟩=|x,y⟩|Δ l⟩|z⊕ H_Δ l, xy⟩
where H_Δ l≠ 0,xy describes connections between different layers.
For example for the simplest case, Δ l_i is either -1,0, or +1,
and the underlying matrices satisfy H_-1,xy = -H_+1,yx.
The target oracle O_F^full for the full system
O_F^full|x,i⟩|l,0⟩=|x,y_i⟩|l,l+Δ l_i⟩
can be realized by first applying O_F^sub and then applying an adder.
Similarly, the oracle O_H^full for the full system
O_H^full|x,y⟩|l,l+Δ l⟩|z⟩
=
|x,y⟩|l,l+Δ l⟩|z⊕ H_Δ l, xy⟩
can be realized just by applying O_H^sub.
This allows us to analyze large systems with repetitive structure,
which we expect to be in high demand for many practical applications.
In particular, assuming that each layer is connected to a constant (i.e. L-independent) number of other layers
(so that H_Δ≠ 0 only for a limited number of Δ l's, which is often the case),
this construction is exponentially efficient in terms of L, the number of repetitions,
as dependence on it arises only from the adder of O_F^full.
Also, note that the previous examples can be regarded as special instances of this generic construction
where (K,L) = (1,2^n) or (2, 2^n-1) (instead of (2^n, 1); original |x,y⟩ corresponds to |l,l+Δ l⟩ from this point of view).
§ INITIAL STATES AND THEIR PREPARATION
The remaining necessary ingredient to carry out a quantum phase estimation algorithm is
an initial state upon which a sequence of powers-of-unitary is applied.
If one can prepare an exact eigenstate |ψ_exact⟩ of the unitary,
then the phase estimation outputs the desired eigenvalue
after a single iteration of the whole circuit.
On the other hand, if one can only prepare an approximate eigenstate |ψ_approx.⟩ of the unitary instead,
then the success probability of QPE is given by the overlap |⟨ψ_approx.|ψ_exact⟩|^2,
meaning that the expectation value of the number of iterations necessary to extract a correct eigenvalue is its reciprocal.
While it is almost impossible to prepare the ideal state |ψ_exact⟩ in general,
it is often assumed that some nice approximate state |ψ_approx.⟩ can be efficiently prepared somehow,
without ruining the exponential speedup over classical computation.
In this section,
we propose a naive way of state preparation for the problems of interest,
and examine its effectiveness.
§.§ Proposal
Here we adopt an approximate eigenvector of the 2^n× 2^n matrix H
as the input state to a QPE algorithm, “approximating” the eigenstate (<ref>) of the unitary U_H^∗.[
So the success probability of the QPE algorithm is about 0.5 (which is large enough for our purpose).
]
One way to prepare it is the following:
take small enough 2^m so that one can classically compute (within reasonable time) the exact eigenvector |ψ_m⟩ of an “approximate H” whose size is 2^m× 2^m.
Encoding this state to m qubits by hand
and then adding (n-m) qubits set to |+⟩,
one can create an n-qubit state
|ψ_n^(m)⟩|ψ_m⟩⊗|00…0⟩
+…
+|11… 1⟩√(2)^n-m
which serves as an approximation to the eigenvector |ψ_n⟩ of the target 2^n× 2^n matrix H.
For example, for m=2 and n=4, the procedure can be expressed as
[ |ψ_m⟩ = a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩ ↦ |ψ_n^(m)⟩ = a/2(|0000⟩ + |0001⟩ + |0010⟩ + |0011⟩); + b/2(|0100⟩ + |0101⟩ + |0110⟩ + |0111⟩); + c/2(|1000⟩ + |1001⟩ + |1010⟩ + |1011⟩); + d/2(|1100⟩ + |1101⟩ + |1110⟩ + |1111⟩).; ]
The point of this method is that
the desired approximate state can be prepared in an exponentially efficient manner in the sense that
it only involves insertions of exponentially small number of qubits compared to the size (i.e. 2^n) of the state.
§.§ Numerical experiments
To check how well this method approximates the exact state,
we compute and compare approximate eigenvectors |ψ_n^(m)⟩ for relatively small 2^n.
As a naive choice, here we just take the 2^m× 2^m-matrix version of Eq. (<ref>) as an “approximate H”;
for Example 1, the results are as follows:
For Example 2, the results are basically the same,
but one might want to consider the matrices (<ref>) with k=k_1+k_2/2 instead
in order to regard the approximation as coarse graining.
In either case, eigenvectors for very small 2^m already seem to approximate those for large 2^n quite well.
In fact, the infidelity 1 - |⟨ψ_n | ψ_n^(m)⟩|^2 is about 0.1 for m=3 for any n (≤ 10),
which is reasonably small and the corresponding leaven state |ψ_m⟩ is easy to prepare.
One naive way to do this is by (2^m-1) sequential (controlled) rotations as
[scale=1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
|0⟩ R_y(θ_0) 1 1 1 1 1 1 ⋯
|0⟩ R_y(θ_1) R_y(θ_2) 1 1 1 1 ⋯
|0⟩ R_y(θ_3) R_y(θ_4) R_y(θ_5) R_y(θ_6) ⋯
⋮ ⋱
;
where for |ψ_m⟩ = ∑_i=0^2^m-1a_i |i⟩ (a_i ∈ℝ)
the angles {θ_i} are precomputed as
[ θ_0 = 2arccos√(a_0^2+a_1^2+⋯ + a_2^m-1-1^2a_0^2+a_1^2+⋯ + a_2^m-1-1^2),; θ_1 = 2arccos√(a_0^2+a_1^2+⋯ + a_2^m-2-1^2a_0^2+a_1^2+⋯ + a_2^m-1-1^2),; θ_2 = 2arccos√(a_2^m-1^2+a_2^m-1+1^2+⋯ + a_2^m-1+2^m-2-1^2a_2^m-1^2+a_2^m-1+1^2+⋯ + a_2^m-1+2^m-2-1^2), ]
and so on.
§ RESOURCE ESTIMATION
In order to implement a quantum phase estimation algorithm on actual quantum devices,
one has to employ quantum error-correcting codes and make the circuit fault-tolerant.
One of the popular schemes used in previous resource estimation literature is
the surface code <cit.>
based on the lattice surgery <cit.>,
as the surface code achieves relatively high threshold error rates
using only local interactions between nearest-neighbor qubits on an array,
allowing it to possess high modularity.
The overhead due to this process is mainly twofold:
encoding of logical qubits of the quantum circuit into physical qubits,
and distillation of the magic states to be consumed to realize non-Clifford gates <cit.>.
In this section, we examine them in order and give a very rough estimation[
Precise estimation involves innumerable subtleties (e.g. execution of Clifford gates, routing of qubits),
and here we turn a blind eye to them and content ourselves with a superficial analysis.
The results are hopefully within an order of magnitude of the true values.
] of the number of necessary physical qubits and actual runtime,
following <cit.>.
Notations and details of parameters are summarized below.
parameter 1cdescription 1cassumed value
ϵ_prec. required precision of eigenvalues 10^-4
ϵ_fail target failure rate of the whole circuit 10^-2
p_phys. physical error rate of elementary operations 10^-3
t_cycle code cycle time 1 μs
§.§ Encoding
Given an n_logical-qubit quantum circuit,
one first needs to store the qubits in a larger block designed to enable them to consume magic states via lattice surgery.
There is a trade-off between the block size b(n_logical) and the necessary code cycles c_consume for a magic state consumption,
and the two extreme cases taken up in <cit.> are the following:
block type b(n_logical) c_consume
compact ⌈n_logical + 2/2⌉× 3 9d_code
fast min_1≤ k ≤ n_logical (2k+1)· (⌈n_logicalk⌉+1) d_code
where d_code is the code distance.
Then, each of the b(n_logical) logical qubits is converted to d_code^2 physical qubits,
equipped with another d_code^2 physical qubits used for syndrome measurements.
Since the logical error rate per logical qubit per code cycle (round) is known <cit.> to be approximated as
p_logical = 0.1 · (100 p_phys.)^⌈d_code/2⌉,
the code distance d_code to be adopted is determined by requiring
p_logical· b(n_logical) ·max(
c_consume,
c_produce) · n_T < ϵ_fail
where c_produce is the (net) necessary code cycles for production of a (distilled) magic state at factories built out of n_distill physical qubits in total,
and n_T is the number of T gates in the quantum circuit (after suitably decomposing all the non-Clifford gates).
Making a choice of d_code satisfying Eq. (<ref>),
one can calculate the total number of necessary physical qubits and actual runtime as
[ n_phys. = b(n_logical) · 2d_code^2 + n_distill,; t_total = max(
c_consume,
c_produce) · n_T· t_cycle.; ]
For the analysis below however,
we will focus on Toffoli gates instead of T gates and correspondingly the factor max(
c_consume,
c_produce) · n_T is suitably replaced.
§.§ Realization of Toffoli gates
One way to execute a Toffoli gate is by consuming a Toffoli or CCZ state
which can be synthesized from magic states <cit.>.
According to <cit.>,
one can construct a factory consisting of 0.5× 10^5 physical qubits,
which can ship a CCZ state with small enough infidelity (< 10^-10) every 60 code cycles.
As one typically has d_code≥ 7 (and thus 9d_code > 60),
we adopt the fast block above for encoding and operate n_factory factories so that
[ n_distill = 0.5× 10^5 · n_factory,; max(
c_consume,
c_produce) = max(
d_code, 60/n_factory). ]
§.§ Case study
The quantum circuit to start with is that of quantum phase estimation with unitary U_H^∗ discussed in Sec. <ref> plugged in.
For Example 2, one has
[ n_logical = (4n+6) + ⌈log_21ϵ_prec.⌉,; n_Toffoli = (42n+30) × (2^⌈log_21/ϵ_prec.⌉ - 1). ]
If one takes ϵ_prec. = 10^-4, the latter is n_Toffoli≲ 10^8 for n ≲ 128,
and thus the CCZ state above with infidelity < 10^-10 is reasonable to use.[
Here we also ignore the costs required for (not only Clifford gates but also)
Pauli rotation gates in both the unitary U_H^∗ and the inverse Quantum Fourier Transform component,
whose numbers are independent of the matrix order 2^n.
They can be fault-tolerantly implemented by decomposition into Clifford+T gates, combining the methods of <cit.> and <cit.>,
which roughly uses 3log_2 1/ϵ_prec.· [4 ·1/ϵ_prec. + 1/2(log_2 1/ϵ_prec.)^2] ∼ 10^6 T gates
and are thus indeed negligible compared to the Toffoli gates' cost.
]
Following the preceding argument, some rough estimates are given as follows:
We again stress that they are given only as a demonstration
with many finer points swept under the rug,
and the precise values themselves have no actual meaning at all.
The interested reader is referred to e.g. <cit.> and references therein
for more refined analyses using CCZ state factories.
§ ACKNOWLEDGMENTS
The authors would like to thank Yuya O. Nakagawa and Shoichiro Tsutsui for careful reading of and comments on the draft.
§ QUANTUM ARITHMETIC AND RELATED OPERATIONS
We summarize the necessary modules
to decompose the full circuit into elementary gates.
§.§ Adder
It is known that the sum (modulo 2^n) of two n-bit integers encoded in quantum states can be computed
using only a single ancillary qubit <cit.>.
The basic strategy is to first compute all the carries, and then do the summation.
Correspondingly, the adder consists of two elementary components.
The first component is a majority gate, implemented as
[scale=1.1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[]|c⟩ [wires=3]majority []|∗⟩
[]|a⟩ []|∗⟩
[]|b⟩ []|(a,b,c)⟩;
[] at (4.5,0) =;
[scale=1.1] at (8.8,0) [row sep=20pt,between origins, column sep=10pt]
1 []|c ⊕ b⟩
1 []|a ⊕ b⟩
-1 -2 []|(a,b,c)⟩;
where a,b ∈{0,1} correspond to summand bits and c ∈{0,1} corresponds to a carry from lower bits.
The purpose of the gate is to output a carry of the addition c+a+b, which is equal to (a,b,c).
If b=0, one does not have to do anything to the corresponding qubit |b⟩,
except when (a,b,c)=1, i.e. when both c and a are 1,
which can be realized by a flipping of b conditioned on c=a=1.
This explains the rightmost Toffoli gate.
On the other hand, if b=1, one has to flip this bit when (a,b,c)=0 i.e. when both c and a are 0,
and one can utilize the same Toffoli gate with two preceding CNOT gates to achieve this.
The second component is an un-majority + add gate, implemented as
[scale=1.1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
[]|c⊕ b⟩ [wires=3]majority^† [wires=3]add []|c⟩
[]|a⊕ b⟩ []|a⊕ b⊕ c⟩
[]|(a,b,c)⟩ []|b⟩;
[] at (5.8,0) =;
[black!20, opacity=0.5] (8.0,-1.1) – (9.6,-1.1) – (9.6,0.4) – (8.0,0.4) – (8.0,-1.1);
[scale=1.1] at (8.4,0) [row sep=20pt,between origins, column sep=10pt]
1 1
1
-2 -1 -1 ;
which (undoes the majority operation and then) outputs the bitwise summation result a⊕ b⊕ c.
Here, the two CNOT gates in the shaded box cancel out.
With these two components,
the desired adder of two n-bit integers x,y is realized using only a single ancillary qubit A as follows:
[scale=1.1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
|0⟩_A [wires=3][]90majority ⋯ ⋯ [wires=3][]90majority^† [wires=3][]90add |0⟩_A
|x_0⟩ ⋯ ⋯ |(x+y)_0⟩
|y_0⟩ [wires=3][]90majority ⋯ ⋯ [wires=3][]90majority^† [wires=3][]90add |y_0⟩
|x_1⟩ ⋯ ⋯ |(x+y)_1⟩
|y_1⟩ ⋯ ⋯ |y_1⟩
⋮ []-10⋱ []80⋱
|y_n-2⟩ ⋯ [wires=3][]90majority^† [wires=3][]90majority^† [wires=3][]90add ⋯ |y_n-2⟩
|x_n-1⟩ ⋯ ⋯ |(x+y)_n-1⟩
|y_n-1⟩ ⋯ ⋯ |y_n-1⟩;
where the top qubit of each majority gate always encodes a carry and the other two qubits encode summands.
Here, a naive counting tells us that the number of necessary Toffoli gates for each full adder of n-bit integers is 2n.[
The number of Toffoli gates can be halved at the cost of additional n ancillary qubits (and their measurements) <cit.>,
but for simplicity we will stick to the naive adder explained above.
]
Note it is also known that, rewriting the un-majority + add circuit as
[scale=1.1] at (0,0) [row sep=20pt,between origins, column sep=10pt]
1 1
1
-2 -1 ;
one can reduce the depth of the resultant adder circuit.[
The number of Toffoli gates can also be reduced by one, but (again) for the sake of simplicity we stick to the naive counting.
]
Here, checking the equivalence between this rewritten form and the original circuit
for the top and bottom qubits is straightforward.
For the middle qubit, one can explicitly do the check by hand.
§.§ Controlled unitaries
When adding control qubits to a large quantum gate which consists of multiple (smaller) gates,
there is a useful trick to keep in mind;
if there exists a subsequence of gates which as a whole constitutes a trivial gate,
one does not have to add control qubits to those gates
as the overall action does not change.
An especially important example is the following:
As is well-known (see e.g. a standard textbook <cit.>), using three unitaries
[ A = R_z(β) R_y(γ/2),; B = R_y(-γ/2) R_z(-β+δ/2),; C = R_z(-β-δ/2),; ]
one can construct an arbitrary single-qubit unitary (up to phase) as
U=AXBXC=
(
[ e^-iβ+δ/2cosγ/2 -e^-iβ-δ/2sinγ/2; e^+iβ-δ/2sinγ/2 e^+iβ+δ/2cosγ/2 ]).
Given this form of decomposition, a controlled-U gate can be constructed as
[scale=1.1] at (0,0) [row sep=30pt,between origins, column sep=10pt]
1
U ;
[] at (1.4,0) =;
[scale=1.1] at (4.5,0) [row sep=30pt,between origins, column sep=10pt]
1 1
C B A ;
by adding control qubits only to X gates, since ABC = I.
Of particular interests are the following controlled-rotations:
|
http://arxiv.org/abs/2307.06157v1 | 20230712133050 | Low complexity convergence rate bounds for the synchronous gossip subclass of push-sum algorithms | [
"Balázs Gerencsér",
"Miklós Kornyik"
] | math.PR | [
"math.PR",
"cs.MA",
"37M25 (Primary) 93D50, 93D05 (Secondary)"
] |
Low complexity convergence rate bounds for the synchronous gossip subclass of push-sum algorithms
Balázs Gerencsér
B. Gerencsér is with the Alfréd Rényi Institute of Mathematics, Budapest, Hungary and the Eötvös Loránd University, Department of Probability and Statistics, Budapest, Hungary, [email protected]
[3]
Miklós Kornyik
M. Kornyik is with the Alfréd Rényi Institute of Mathematics, Budapest, Hungary, [email protected]
The research was supported by NRDI (National Research, Development and Innovation Office) grant KKP 137490.
August 12, 2023
===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
We develop easily accessible quantities for bounding the almost sure exponential convergence rate of push-sum algorithms. We analyze the scenario of i.i.d. synchronous gossip, every agent communicating towards its single target at every step. Multiple bounding expressions are developed depending on the generality of the setup, all functions of the spectrum of the network. While the most general bound awaits further improvement, with more symmetries, close bounds can be established, as demonstrated by numerical simulations.
§ INTRODUCTION
Average consensus algorithms have been around for a while <cit.>, <cit.>, with the fundamental goal of computing the average of input values on a network in a distributed manner with only local communication and simple operations. Often some symmetry is imposed on the communication, in terms of the matrix describing the linear update of the vector of values to be either doubly stochastic, or even symmetric. This condition is quite well understood <cit.>, see the survey <cit.> also for applications, further discussion and references.
However, the interest for distributed averaging algorithms capable of handling asynchronous directed communications emerged, naturally driving away the representing update matrix from being doubly stochastic, still with the intent to compute the exact average. As a result, the successful scheme of push-sum was proposed <cit.>, later also
investigated under the name ratio consensus <cit.> and joined by variants such as weighted gossip <cit.>.
The goal of these algorithms is the same,
but now using only local, directed communication and without requiring message passing to happen synchronously or consistently across the network. Given the simple objective of the algorithm, it also serves as a building block for more complex tasks, e.g., the spectral analysis of the network <cit.> or distributed optimization algorithms <cit.>.
With other real-life communication challenges taken into account, the concept has been extended in multiple ways to handle such aspects, including packet loss <cit.> <cit.>, delay <cit.> or even the presence of malicious agents <cit.>.
In the meantime, there is work to better understand the effect of such communication deficiencies for the reference algorithms. The error of the consensus value compared to the true average for the push-sum algorithm has been analyzed in case of packet loss <cit.>, similarly as it has been done for classic (linear) gossip <cit.>, <cit.>.
An essential question in the analysis for usability and efficiency is understanding the asymptotics of the processes, their convergence and the rate at which it happens.
In the cases above, the convergence of the push-sum algorithm (or variants) has been confirmed. Additionally, for the original push-sum scheme, an exponential convergence has been proven <cit.>. However, at that time the focus was not yet on approximating the true rate.
An important step ahead was made in <cit.> providing a convincing upper bound along an unspecified, infinite subset of the timeline for the almost sure (a.s.) rate of convergence.
More recently, the exact rate of a.s. convergence has been identified <cit.> for stationary ergodic updates as the spectral gap in terms of the Lyapunov exponents of random matrix updates with generous applicability. While being a clean representation with the concern that this Lyapunov spectral gap is known to be uncomputable in general <cit.>.
As a follow-up, it was possible to combine the inspiration of <cit.> and the tool-set of <cit.> to obtain an actual upper bound on the a.s. rate for the i.i.d. case <cit.>, now formulated by manipulating the Kronecker square of a single (random) update matrix, thus leading to a computable quantity. The bounds are solid, however for a graph on N vertices, matrices of N^2× N^2 have to be analyzed, quickly increasing in dimension.
Our goal is to provide even simpler convergence rate estimates. For this purpose, we focus our attention to the natural setup, where a weighted network determines the communication scheme driving the consensus process. In particular, we assume synchronized gossip message passing, i.e. every node sending a single packet to a single (random) recipient at each time slot. Convergence of this scheme has been known since the formation of the push-sum concept <cit.>, ensuring that distributed average computation takes place.
The bounds provided can be computed directly once the standard spectral description of the network is available. We are to formulate multiple variants, both to provide general, but more conservative estimates, and also sharper ones for a more restricted setting with stronger symmetries.
The rest of the paper is structured as follows. In the next section we formally define the averaging process and state our results. Section <ref> builds a framework for the proof of the theorems, while Section <ref> completes the proofs. Detailed numerical performance analysis and concluding remarks are provided in Section <ref> and Section <ref>.
§ MAIN RESULTS
Let us introduce the push-sum algorithm along with other concepts that will be used.
Given is a finite graph G = (V,E) with the vertex set V = [N] := {1,2,…,N}, having degree sequence d_1,…,d_N. There is an initial vector of values x(0)∈^N at the vertices to be averaged. The process is also using an auxiliary vector initialized at w(0) = ∈^N.
At each time step, a linear row-stochastic update - representing local communication - is performed to both vectors as
x(t)^⊤ = x(t-1)^⊤ K(t),
w(t)^⊤ = w(t-1)^⊤ K(t).
The average x̅:=1/N∑_i x_i(0) is then locally estimated by x_i(t)/w_i(t).
There is a wide generality of how (K(t))_t≥ 0 can be chosen. In the current paper, we focus on the scenario of i.i.d. K(t), when at each step, every vertex sends a single message to a randomly chosen neighbor with a constant proportion and all these choices independent from one another. Formally, K(t) d= K = (1-q)I + q∑_i e_ie_β_i^T with some fixed q∈ [0,1] and independent β_i. By setting p_ij:=(β_i=j), we obtain an overall transition probability matrix P which by construction has to be compatible with the adjacency matrix of G.
For convenience, we introduce the notation P_q = (1-q)I + qP. It is easy to check that 𝔼 K= P_q. In case P has only real eigenvalues let λ_i denote its i^th largest eigenvalue and let λ_q,i=(1-q) + qλ_i denote that of P_q. Following our setup let us state our main results. Theorem <ref> targets scenarios in more general settings, while Theorem <ref> is designed for more symmetric cases.
Let us consider a push-sum algorithm with message probability matrix P. Then
lim sup1/tmax_ilog|x_i(t)/w_i(t)-x̅| ≤1/2logρ( (I-J) (P_q^⊤ P_q + q^2(Γ -P^⊤ P))(I-J)) a.s.
where Γ is a diagonal matrix with γ_ii = ∑_j p_ji and ρ(·) denotes the spectral radius.
Furthermore, if P is symmetric, then
lim sup1/tmax_ilog|x_i(t)/w_i(t)-x̅| ≤1/2log ((1-q)^2 + 2q(1-q)λ_2 + q^2) a.s.
In case each vertex chooses a recipient uniformly among its neighbors, the bounding quantity in Theorem <ref> will depend only on the graph structure, furthermore the diagonal matrix Γ takes the form Γ_ii=∑_j:(j,i)∈ Ed_j^-1 with d_j denoting the degree of vertex j.
It is easy to check that in this case 𝔼 K = P_q with P=D^-1 A, where D denotes the diagonal matrix consisting of the degrees of the underlying graph's vertices, while A denotes the graph adjacency matrix.
A better bound can be obtained for cases with stronger symmetries. A message probability matrix is said to be transitive if for any pair (i,j) there exists a permutation matrix Π with Π_ij = 1 such that Π P Π^-1 = P.
Suppose that the message probability matrix P is symmetric and transitive. Then
lim sup1/tmax_ilog|x_i(t)/w_i(t)-x̅| ≤1/2logξ_1,
with ξ_1 being the largest root of the polynomial
f(ξ) = ∏_i>1 (ξ-λ_q,i^2) - q^2/N∑_i>1 (1-λ_i^2)∏_i≠ j>1 (ξ-λ_q,i^2).
If G is a transitive graph and each vertex chooses a recipient uniformly among its neighbors, then the corresponding message probability matrix satisfies the assumption of Theorem <ref>.
A special case has been analyzed in
<cit.>, where the underlying topology was given by the complete graph with q=1/2. For this topology and general q Theorem <ref> immediately gives
f(ξ) = (ξ - (1-q)^2)^N-2(ξ-(1-q)^2- q^2(1-N^-1)).
from which the convergence rate bound (1-q)^2 + q^2(1-N^-1) can be easily obtained.
§ TOOLS
Let us first introduce a framework and corresponding tools in a general setting. The alignment to the assumptions of the theorems will be carried out later.
First we remark that the elements w_i(t) are all positive because the diagonal elements of the nonnegative K(t) are strictly positive.
Using the notations H(t)=K(1)K(2)⋯ K(t), J=11^⊤ /N
easy calculation shows
x(t)^⊤ - x̅ w(t)^⊤ = x_0^⊤ H(t)- x̅ w(t)^⊤
= x_0^⊤(JH(t) + (I-J)H(t)) - x̅ w(t)^⊤
=x_0^⊤(I-J)H(t)
meaning that
max_i |x_i(t)/w_i(t)-x̅| ≤ C {min_i(w_i(t))}^-1 ||x_0||_2 ||(I-J)H(t)||_2
≤ C {min_i(w_i(t))}^-1 ||x_0||_2 ||(I-J)H(t)||_F.
from some constant C>0. We are interested in the almost sure convergence rate of the quantity on the left of (<ref>). As we will see the dominant term will be ||(I-J)H(t)||_F. To get a handle on this factor let us analyze
the expectation of || (I-J)H(t)||_F^2.
𝔼||(I-J)H(t)||_F^2 = 𝔼 Tr{(I-J)H(t)H(t)^⊤ (I-J)}
= Tr{ (I-J) 𝔼[H(t)H(t)^⊤] (I-J) }.
According to the definition of H(t) we can write
𝔼[H(t)H(t)^⊤] = 𝔼[𝔼[K(1) H̃(t) H̃(t)^⊤ K(1)^⊤| H̃(t) ] ] = 𝔼[𝔼. [K(1)XK(1)^⊤]|_X=H̃(t)H̃(t)^⊤],
where H̃(t) = K(2)K(3)⋯ K(t), thus by the i.i.d. nature of the updates H̃(t) d= H(t-1).
This motivates the following definition of the linear operator Φ: ℝ^N× N→ℝ^N× N acting on matrices:
Φ(X):=𝔼[KXK^⊤],
which we will need to understand for further developing (<ref>).
For satisfactory notation, before we progress let us introduce the linear operator Ψ: ℝ^N× N→ℝ^N
(Ψ(X))_i = x_ii
and its pseudo-inverse Ψ^-:ℝ^N→ℝ^N× N
(Ψ^-(v))_ij =
v_i i=j,
0
Following the pattern of (<ref>) we can prove the following
For any matrix X, we have
Φ(X) = P_qXP_q^⊤ +q^2{Ψ^- [P Ψ(X)] - Ψ^- Ψ(PXP^⊤) }
reminding that P_q = (1-q)I + qP.
Let L := ∑_i e_i e_β_i^⊤
then
𝔼[KXK^⊤] = (1-q)^2X + q(1-q)𝔼[XL^⊤ ] + q(1-q)𝔼[LX] + q^2 𝔼[LXL^⊤]
= (1-q)^2 X + q(1-q)XP^⊤ + q(1-q)PX + q^2 𝔼[LXL^⊤]
= P_q X P_q^⊤ - q^2PXP^⊤ + q^2 𝔼[LXL^⊤].
Next we will compute the term 𝔼[LXL^⊤] as
𝔼[LXL^⊤] = ∑_i,i'𝔼[e_ie_β_i^⊤ Xe_β_i'e_i'^⊤ ] = ∑_i≠ i'𝔼x_β_i,β_i'e_ie_i'^⊤ + ∑_i 𝔼x_β_i,β_i e_ie_i^⊤
= ∑_i≠ i'
j,j' p_ijp_i'j' x_jj' e_ie_i'^⊤ + ∑_i p_ij x_jj e_i e_i^⊤
= PXP^⊤ - ∑_i,j,j' p_ijp_ij' x_jj' e_ie_i^⊤ + ∑_i,jp_ijx_jje_ie_i^⊤
= PXP^⊤ - Ψ^-Ψ(PXP^⊤) + Ψ^- (P Ψ(X)).
Thus putting together the two parts gives
Φ(X) = P_qXP_q^⊤ + q^2{Ψ^-(PΨ(X)) -Ψ^- Ψ (PXP^⊤)}
and this concludes the proof.
In order to obtain a bound on { (I-J)𝔼[H(t)H(t)^⊤ ](I-J) } it is enough to understand Φ, since
{ (I-J)𝔼[H(t)H(t)^⊤](I-J) } = {(I-J)Φ^t(I)(I-J)} ,
where Φ^t(I) denotes the application of Φ on I t times, i.e. Φ∘Φ∘⋯∘Φ_t times (I).
The map Φ has the following fundamental properties:
(P1) Φ is linear,
(P2) Φ(X^⊤) = Φ(X)^⊤,
(P3) for any skew-symmetric matrix X, Φ(X) = P_qXP_q^⊤,
(P4) if X≥ 0 then Φ(X)≥ 0, i.e. Φ keeps the positive semi-definite property,
(P5) if x_kl≥ 0 ∀ (k,l), then Φ(X)_kl≥ 0 ∀ (k,l),
(P6) J is an eigenmatrix of Φ with eigenvalue 1, i.e. Φ(J) = J,
(P7) for X≥ 0, and P=P^⊤
we have Φ(X) ≤ X.
For the adjoint map Φ^* the following observations can be added:
(P*1) Φ^*(Y) = P_q^⊤ Y P_q +q^2 {Ψ^-[P^⊤Ψ(Y)] - P^⊤ (Ψ^-Ψ Y) P},
(P*2) if X≥ 0 then Φ^*(X)≥ 0, i.e. Φ^* also keeps the positive semi-definite property,
(P*3) if x_kl≥ 0 ∀ (k,l), then Φ^*(X)≥ 0 ∀ (k,l),
(P*4) if X1 = 0 then Φ^*(X)1=0.
The first three properties follow directly from the definition of Φ, hence their proofs are left to the respected reader.
Property (P4) can be proved as follows. Let X≥ 0 and let w be an arbitrary vector, then
w^⊤Φ(X)w = w^⊤𝔼[KXK^⊤] w= 𝔼[w^⊤ K X K^⊤ w] ≥ 0
(P5) is analogous to (P4), namely
Φ(X)_kl = (𝔼[KXK^⊤])_kl≥ 0.
Property (P6) is the result of a short series of calculations.
Φ(J) = PJP^⊤ + q^2(Ψ^- PΨ J - Ψ^-Ψ(PJP^⊤))
= J + q^2(1/N· I - 1/N· I) = J,
due to the facts PJ = JP^⊤ = J and Ψ J = 1/N.
Before proving (P7) let us note that due to X≥0 and the linearity of Φ it is enough to prove this property for X=xx^⊤. Using the definition of K = (1-q)I + q^2 ∑_i e_i e_β_i^⊤ we have
Φ(xx^⊤) = 𝔼[Kxx^⊤ K^⊤ ˘] =
=𝔼{(1-q)^2 xx^⊤ + q(1-q)(Lxx^⊤ + xx^⊤ L^⊤) + q^2Lxx^⊤ L^⊤}
= (1-q)^2||x||_2^2 + 2q(1-q) x^⊤ P x +q^2𝔼 ||Lx||_2^2
= (1-q)^2||x||_2^2 + 2q(1-q) x^⊤ P x + q^2∑_i,j p_ij x_j^2
≤ ||x||^2_2
where in the last step we used the facts P=P^⊤, P1 = 1 and x^⊤ P x ≤λ_1(P) ||x||_2^2 = ||x||_2^2.
Now we proceed with proving the properties of the adjoint.
The proof of (P*1) is based on the following series of calculations:
due to the equivalences
(Ψ^-(P Ψ X )) Y^⊤ = ∑_i,k p_ikx_kk y_ii = ∑_k x_kk∑_i p_iky_ii = {X Ψ^-(P^⊤ Y^⊤)} ,
Ψ^-Ψ(PXP^⊤) Y^⊤ = ∑_i,k,l p_ikx_klp_ily_ii = ∑_k,l x_kl∑_i p_iky_iip_il = {X P^⊤Ψ(Y^⊤) P}
we have
⟨Φ(X), Y ⟩ = Φ(X)Y^⊤ = P_qXP_q^⊤ Y^⊤ + q^2{ [Ψ^-(P Ψ(X)) ] Y^⊤ - [Ψ^-Ψ(PXP^⊤)] Y^⊤}
= (X P_q^⊤ Y^⊤ P_q) + q^2{X Ψ^-(P^⊤Ψ Y^⊤) -X P^⊤Ψ (Y^⊤) P}
= ⟨ X,Φ^*(Y) ⟩.
Properties (P*2) and (P*3) can be confirmed analogously to (P4), (P5).
(P*4) is a result of the short derivation
Φ^*(X)1 = P_q^⊤ XP_q1 + q^2(Ψ^- P Ψ X1 - P^⊤ (Ψ^-Ψ X) P 1)
= 0 + q^2( Ψ^- (P^⊤Ψ(X)) 1 - P^⊤ (Ψ^-Ψ X) 1)
since P_q1 = P1 = 1 and we assumed X1 = 0. For the second term we have
(Ψ^- (P^⊤Ψ(X)) 1)_i = ∑_j p_jix_jj
(P^⊤ (Ψ^-Ψ X) 1)_i = ∑_j p_ji x_jj,
so Φ^*(X)1 = 0.
This concludes the proof.
Properties (P1), (P2), (P4), (P7)
mean that Φ describes a quantum operation.
According to the previously listed properties we have
The cone of positive semi-definite matrices is invariant under the action of Φ.
The following statement is going to help us in proving Theorem <ref> as our focus is on the speed of convergence and not the limit x̅1, which is of constant order.
Since (I-J)H(t)H(t)^⊤ (I-J) = H(t)^⊤ (I-J)H(t) and the adjoint of the linear map f:X ↦ AXA^⊤ is the map f^* : X ↦ A^⊤ X A, it is easy to show that Φ^*(X) = 𝔼[K^⊤ X K] for any symmetric matrix X.
According to the definition of K, we have KJ=J, and so
(I-J)K(I-J) = (I-J)(K-J) = K-JK = (I-J)K
whence
𝔼[(I-J) KXK^T(I-J)] = 𝔼[(I-J)K(I-J)X(I-J)K^T (I-J)].
Let us define the operator Φ as
Φ: X↦ (I-J)Φ(X)(I-J) ∈End({Y∈ℝ^N× N : Y=Y^⊤, YJ=0}).
According to properties of Φ combined with Remark <ref>, we have
Φ^t (X) = (I-J) Φ^t(X) (I-J), X∈{Y∈ℝ^N× N : Y=Y^⊤, YJ=0}
furthermore
Φ((I-J)X(I-J)) = Φ(X) + JXP_q^T + P_qXJ -JXJ.
§ PROOFS
The next proposition is the final step before proving Theorem <ref>.
Let P be a row stochastic matrix. Then
(Φ^*)^t(I-J) ≤
N ρ((I-J)B_q(I-J) )^t
where B_q = P_q^⊤ P_q + q^2(Γ- P^⊤ P) is a positive definite matrix, and D is a diagonal matrix with d_ii = ∑_jp_ji on its diagonal.
Due to its properties Φ^* : 𝒳_0 →𝒳_0 = { X ∈ℝ^N× N : X=X^⊤, XJ = 0 }, and
X≥ 0 Φ^*(X) ≥ 0, we have
(Φ^*)^t(I-J) ≤ N ρ((Φ^*)^t(I-J)) = N max{ v^⊤ (Φ^*)^t(I-J) v : v = 1 , v⊥1}
= N (Φ^*)^t(I-J)_2 ≤ N (Φ^*)^t_𝒳_0 →𝒳_0(I-J)_2
≤ N Φ^*^t_𝒳_0 →𝒳_0
where
Φ^*_𝒳_0→𝒳_0 = max{Φ^*(X)_2 : X_2≤ 1, X ∈𝒳_0}.
It is not hard to show that maxΦ^*(X)_2 ≥ 0, since let X = maxΦ^*(X)_2 and let us consider its decomposition X=X^* - X^- with X^+, X^-≥ 0. Then
v^⊤Φ^*(X) v = v^⊤Φ^*(X^+)v - v^⊤Φ^*(X^-)v ≤ v^⊤Φ^*(X^+)v + v^⊤Φ^*(X^-) v ,
and this would lead to a contradiction if X^- was not 0. This implies
Φ^*_𝒳_0 →𝒳_0 = max{v^⊤Φ^*(X) v : X∈𝒳_0,X≥ 0, X_2≤ 1, v≤ 1, v ⊥1}.
meaning that it is enough to bound v^⊤Φ^*(X) v from above. Let v ⊥1 and X∈𝒳_0 with X_2 ≤ 1, then
v^⊤Φ^*(X) v = (P_qv)^⊤ X P_qv + q^2(∑_i,j p_ijx_ii v_j^2 - ∑_i (Pv)_i^2 x_ii).
Due to the conditions imposed on X, we have x_ii∈ [0,1], furthermore ∑_jp_ij - (Pv)_i^2 ≥ 0 for any i, thus
v^⊤Φ^*(X)v ≤P_qv^2 + q^2(∑_i,j p_ijv_j^2 - ∑_i (Pv)_i^2)
= v^⊤(P_q^⊤ P_q + q^2(Γ - P^⊤ P))v ≤ρ((I-J)(P^⊤_qP_q + q^2(Γ-P^⊤ P)(I-J))
where Γ is a diagonal matrix with diagonal elements γ_ii=∑_jp_ji. Note that for symmetric P we have Γ = I and so the upper bound above becomes (1-q)^2 + 2q(1-q)λ_2 + q^2.
With all the tools at our hands we can prove Theorem <ref>.
According to Lemma 10 in <cit.> whose assumptions are clearly satisfied we have
lim sup_t 1/tlog1/min_i w_i(t)≤ 0
thus considering the quantity in (<ref>) we can infer that
lim sup_t 1/tlog({min_i w_i(t)}^-1||x_0||· ||(I-J) H(t)||_F )≤lim sup_t 1/tlog ||(I-J)H(t)||_F .
The matrix (I-J)H(t) can we written as a product of the i.i.d. random matrices (I-J)K(t), therefore due to the Fürstenberg-Kesten theorem it follows that
lim sup_t 1/tlog ||(I-J)H(t)||_F = lim_t 1/tlog ||(I-J)H(t)||_F = lim_t1/t𝔼log||(I-J)H(t)||_F
= lim_t 1/2t𝔼log (I-J)H(t) _F^2 .
Using Jensen's inequality yields
𝔼log ||(I-J)H(t)||^2_F ≤log𝔼||(I-J)H(t)||_F^2
thus
lim sup_t 1/tlog(I-J)H(t)_F ≤lim_t 1/2tlog𝔼(I-J)H(t)_F^2.
By taking the expectation we obtain
𝔼 ||(I-J)H(t)||_F^2 = (Φ^*)^t(I-J).
Combining the series of calculations above with Proposition <ref> we arrive at
lim sup_t 1/2tlog (Φ^*)^t(I-J)≤1/2ρ((I-J)(P_q^⊤ P_q + q^2(Γ - P^⊤ P))(I-J))
(1-q)^2 + 2q(1-q)λ_2 + q^2 P=P^⊤.
Now we will turn to the case when the underlying graph is transitive. In this scenario it is possible to give stronger bounds, but in order to do this we need to reformulate the problem. Rearranging our main quantity of interest as
𝔼||(I-J)H(t)||_F^2 = 𝔼 [H(t)^⊤(I-J)H(t)],
therefore
𝔼||(I-J)H(t)||_F^2 = {(Φ^*)^t(I-J)}.
Let us define X_t =(Φ^*)^t(I-J). The following two lemmas will help us in our progress.
Assume that P is a kernel of the transitive Markov chain, implying that it is symmetric and any diagonal element p^(k)_ii of P^k depends solely on k and not on i. Then we have
* X_t ∈𝒫_t: =Span{P^k, J ; 0≤ k≤ 2t}, hence the diagonal of X_t is also constant,
* X_t+1 = P_q X_t P_q + q^2 r_t (I-P^2),
where r_t denotes the common diagonal element of X_t.
We prove by induction. For t=0 X_0 = I-J, which trivially is a polynomial of P and J.
For the induction step t→ t+1 assume that X_t ∈𝒫_t, then
X_t+1 = Φ^*(X_t) = P_qX_tP_q + q^2{(Ψ^- P Ψ X_t) - P(Ψ^-Ψ X_t )P}
since X_t is a polynomial of P and J it is also transitive. Noting Ψ X_t = r_t 1 and Ψ^- Ψ X_t = r_t I, where r_t denotes the common diagonal element of X_t, we can derive the recursion
X_t+1 = P_qX_tP_q + q^2 r_t(I-P^2)
showing that X_t+1∈𝒫_t+1.
If P is symmetric and transitive then X_t and P possess the same eigenvectors. If v is an eigenvector to P and X_t corresponding to the eigenvalue λ and μ_t respectively then the following recursion holds for μ_t:
μ_t+1 = λ_q^2 μ_t + q^2 r_t (1-λ^2).
Recall that r_t is the common diagonal element of X_t.
Furthermore the largest eigenvalue of X_t is asymptotically bounded by the second largest root ξ_2 of the polynomial
p(x) = ∏_i (x-λ_q,i^2)(1+q^2/N∑_j>11-λ_j^2/x-λ_q,j^2)
in the following sense:
lim sup_t 1/tlogmax_i μ_t,i≤logξ_2 .
Using r_t = 1/N X_t = 1/N∑_i μ_t,i we can write the recursion described in (<ref>) as
y_t+1 = (D+q^2/N𝐛1^⊤)y_t
with the vectors (y_t)_i = μ_t,i, 𝐛_i = 1-λ_i^2, and D= (λ_q,1^2,…,λ_q,N^2), where λ_q,i denotes the i^th largest eigenvalue of P_q and μ_t,i denotes the eigenvalue of X_t corresponding to λ_q,i. By correspondence we mean in the sense of defined by the recursion (<ref>), in which case μ_t,1 = 0, since λ_q,1 = 1 and X_t 1 = 0 for any t.
Let (λ,v) denote an eigen-pair of P. According to the previous lemma X_t is a polynomial of P and J, hence v is an eigenvector of X_t, furthermore, due to the symmetry of P, we have JP=PJ=J. Recursion (<ref>) then yields
X_t+1 v = P_qX_t P_q v + q^2 r_t(I-P^2)v
= λ_q^2 μ_t v + q^2 r_t(1-λ^2)v
and we have
μ_t+1 = λ_q^2μ_t + q^2 r_t(1-λ^2),
proving the first part.
Before continuing with the proof of the second part, let us note that e_1 is a left-eigenvector of the matrix D + q^2 𝐛1^⊤/N corresponding to the eigenvalue 1, meaning that the right-eigenvectors corresponding to a different eigenvalue are orthogonal to e_1. Now let us choose the following vectors as basis: f_1 = e_1, f_i = e_i-e_1, i>1, then for i>1 we have f_i ⊥1.
Writing equation (<ref>) in basis {f_i} yields
Df_1 = De_1 = λ_q,1^2 e_1,
Df_i = De_i - De_1 = λ_q,i^2e_i - λ_q,1^2 e_1 =
= λ_q,1^2 f_i +(λ_q,i^2-λ_q,1^2)f_1, i>1,
𝐛1^⊤ f_1 = 𝐛 = b_1 f_1 + ∑_i>1 b_i(f_i + f_1),
𝐛1^⊤ f_i = 0 i>1,
thus rewriting the matrix D+q^2 𝐛1^⊤ /N in this new basis gives us
[ (q^2/N)∑_i b_i + λ_q,1^2 -(λ_q,1^2-λ_q,2^2) -(λ_q,1^2 -λ_q,3^2) … -(λ_q,1^2 - λ_q,N^2); (q^2/N) b_2 λ_q,2^2 0 … 0; (q^2/N) b_3 0 λ_q,3^2 … 0; ⋮ ⋱ ; (q^2/N)b_N 0 … 0 λ_q,N^2 ]
The characteristic polynomial of the matrix in (<ref>) can be computed via expanding along the first column, leading to
p(x) = (x-q^2/N∑_i b_i - λ_q,1^2)∏_i>1(x-λ_q,i^2) + ∑_i>1q^2/N b_i(λ_q,1^2- λ_q,i^2) ∏_1<j≠ i(x-λ_q,j^2)
= ∏_i>1 (x-λ_q,i^2) {x - λ_q,1^2-q^2/N∑_i (1-λ_i^2) + q^2/N∑_j>1(1-λ_j^2)(1 - λ_q,j^2)/x-λ_q,j^2} .
Exploiting the fact λ_1 = λ_q,1= 1 we obtain
p(x) = ∏_i>1(x-λ_q,i^2){ x-1 + q^2/N∑_j>1[ (1-λ_j^2)(1-λ_q,j^2/x-λ_q,j^2 -1) ] }
= ∏_i>1 (x-λ_q,i^2) {x-1 + q^2/N∑_j>1 (1-λ_j^2)1-x/x-λ_q,j^2}
= ∏_i (x-λ_q,i^2)(1-q^2/N∑_j>11-λ_j^2/x-λ_q,j^2).
We note here that due the initialization μ_0 = (0,1,…, 1) and the fact that e_1 is a left-eigenvector of the recursion, we have for each subsequent vector μ_t ⊥ e_1. This means that we are only interested in the second largest root η_2 of the characteristic polynomial to obtain the asymptotic growth rate of μ_t, i.e.
||μ_t||_∞≤ Cη_2^t ||μ_0||_∞
for some C>0.
The first part of the proof, up until the point where we have to bound the quantity 𝔼 ||(I-J)H(t)||_F^2 from above, is analogous to the proof of Theorem <ref>, hence it will be omitted here. Before proceeding with the actual proof we note that
||(I-J)H(t)||_F^2 = (I-J)H(t)H(t)^⊤(I-J) = H(t)^⊤(I-J)H(t),
whence
𝔼 ||(I-J)H(t)||_F^2 = {(Φ^*)^t(I-J)} = ∑_jμ_t,j≤ N max_j μ_t,j,
where μ_t,j denotes the eigenvalue of X_t = (Φ^*)^t(I-J) corresponding to the j^th largest eigenvalue λ_j of P. Due to X_0 = I-J and λ_1= 1 with v_1 = c 1, we have μ_0,1 = 0 and due to the recursion (<ref>), μ_t,1 = 0 for any t>0. In Proposition <ref> it was shown that
||μ_t||_∞≤ C η_2^t ||μ_0|| = C η_2^t,
where η_2 denoted the second largest root of the polynomial
p(x) = ∏_i (x-λ_q,i^2) - q^2/N∑_i>1 (1-λ_i^2)∏_i≠ j>1 (x-λ_q,i^2).
Exploiting the fact that 1 is a root of p we have
p_1(x) := p(x)/(x-1)=∏_i>1 (x-λ_q,i^2) - q^2/N∑_i>1 (1-λ_i^2)∏_i≠ j>1 (x-λ_q,i^2)
thus by denoting ξ_1 the largest root of p_1(x) we can write ||μ_t||_∞≤ C ξ_1^t. Altogether
lim sup_t 1/2tlog𝔼||(I-J)H(t)||_F^2 ≤1/2logξ_1 ,
and this concludes the proof.
§ NUMERICAL EXPERIMENTS
In order to evaluate the performance of the bounds obtained on the convergence rate, we perform a detailed numerical comparison of the available quantities. For setups with N≤ 120 the synchronous gossip process is realized for t=500 steps and the approximate rate is expressed as
1/tlog1/√(N) (I-J)H(t) Ψ^-(w(t))^-1_F.
For N>120 we set t=1000 and, for complexity and memory usage reduction, we use the modified approximation
1/tlog1/√(M) X^M(0)H(t) Ψ^-(w(t))^-1_F
where X^M(0) is an M× N matrix of uniform random independent rows in 1^⊥ with unit norm, representing various initializations, reaching the principal rate with high probability. We choose M=⌊√(N)⌋.
This is compared both with η := 1/2logρ((I-J)^⊗ 2 (K(1)^⊗ 2)) which is the bound of <cit.> and the bounds developed in the current paper.
For the context of Theorem <ref>, more precisely its specialized version Corollary <ref> we build a Barabási-Albert random graph with 2 edges added with each new vertex, and assign uniform probabilities for choosing gossip recipients. See Figure <ref> for the resulting approximate rates and bounds as q varies in (0,1). We can see that η provides a very close fit, the current bound is only moderately usable for small q. However, note that for N=100 handling the N^2× N^2 matrices needed for the tensor product based bound is getting computationally heavy, thus we only plot the simulations versus the current bound. We also conducted simulation experiments for N=5000 in which case our bound was mostly nonnegative and hence did not carry any useful information.
In the case of symmetric P, the second part of Theorem <ref> provides a more usable bound as shown in Figure <ref>. We use random regular graphs and uniform recipient probabilities again. The resulting bound captures the linear trend when q is near 0, it is a factor ≈ 2 off from the numerical value when q≤ 0.5 and then deteriorates.
For the more refined bound of Theorem <ref>, we generate a random Cayley-graph of S_k. This time we see a substantially better fit as shown in Figure <ref>. In fact, it recovers η which is expected from the exact analysis carried out in its proof, but circumvents the necessity of working with the large matrices of the Kronecker-product.
§ SUMMARY AND FUTURE PLANS
In this work we have presented bounds of various accuracy depending on the level of symmetry of the underlying topology.
Along the way proving our main results we have developed a framework as described in Section <ref> relying on matrix operators Φ, Φ^* that we hope can be useful when analyzing similar dynamics.
The computational cost of these bounds is orders of magnitude less than that of the simulations or computation of η from <cit.> confirming their usefulness in assessing the efficiency of various push-sum algorithms.
Concerning our future plans, we observed in some of our numerical experiments that the expression in (<ref>) was also a valid bound for regular graphs. This lead us to state the following
The bound given in <ref> remains valid in the simple symmetric case, e.g. for regular graphs without transitivity.
siam
|
http://arxiv.org/abs/2307.04343v1 | 20230710045405 | Hierarchical Semantic Tree Concept Whitening for Interpretable Image Classification | [
"Haixing Dai",
"Lu Zhang",
"Lin Zhao",
"Zihao Wu",
"Zhengliang Liu",
"David Liu",
"Xiaowei Yu",
"Yanjun Lyu",
"Changying Li",
"Ninghao Liu",
"Tianming Liu",
"Dajiang Zhu"
] | cs.CV | [
"cs.CV"
] |
Hierarchical Semantic Tree Concept Whitening for Interpretable Image Classification
Haixing Dai*, Lu Zhang*, Lin Zhao, Zihao Wu, Zhengliang Liu, David Liu,
Xiaowei Yu, Yanjun Lyu, Changying Li, Ninghao Liu, Tianming Liu, Dajiang Zhu.
* Co-first authors.
Haixing Dai, Lin Zhao, Zihao Wu, Zhengliang Liu, Ninghao Liu, Tianming Liu and Changying Li are with the
Department of Computer Science, University of Georgia, Athens, GA, USA.
(e-mail: hd54134, lin.zhao, zw63397,zl18864, ninghao.liu, [email protected], [email protected]).
Lu Zhang, Xiaowei Yu, Yanjun Lyu and Dajiang Zhu are with the Department of Computer Science
and Engineering, The University of Texas at Arlington, Arlington, TX, USA.
(e-mail: lu.zhang2, xxy1302, [email protected], [email protected])
David Weizhong Liu is with Athens Academy, Athens, GA, USA.(e-mail:
[email protected])
August 12, 2023
=====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
With the popularity of deep neural networks (DNNs), model interpretability is becoming a critical concern. Many approaches have been developed to tackle the problem through post-hoc analysis, such as explaining how predictions are made or understanding the meaning of neurons in middle layers. Nevertheless, these methods can only discover the patterns or rules that naturally exist in models. In this work, rather than relying on post-hoc schemes, we proactively instill knowledge to alter the representation of human-understandable concepts in hidden layers. Specifically, we use a hierarchical tree of semantic concepts to store the knowledge, which is leveraged to regularize the representations of image data instances while training deep models. The axes of the latent space are aligned with the semantic concepts, where the hierarchical relations between concepts are also preserved. Experiments on real-world image datasets show that our method improves model interpretability, showing better disentanglement of semantic concepts, without negatively affecting model classification performance.
Explainable AI (XAI), hierarchical tree of semantic concepts, image embedding, image interpretation.
§ INTRODUCTION
Machine learning interpretability has recently received considerable attention in various domains <cit.>. An important challenge that arises with deep neural networks (DNNs) is the opacity of semantic meanings of data representations in hidden layers. Several types of methods have been proposed to tackle the problem. First, recent works have shown that some neurons could be aligned with certain high-level semantic patterns in data <cit.>. Second, it is possible to extract concept vectors <cit.> or clusters <cit.> to identify semantic meanings from latent representations. However, these methods are built upon the assumption that semantic patterns are already learned by DNNs, and the models would admit the post-hoc method of a specific form. There is no guarantee that the assumption holds true for any model, especially when meaningful patterns or rules may not be manifested in the model, thus leading to over-interpretation <cit.>. Meanwhile, although many post-hoc explanation methods are proposed with the expectation of improving or debugging models, it is challenging to achieve this goal in practice. Although we could collect human annotations to guide prediction explanations and improve model credibility <cit.>, manually labeling or checking semantic concepts is rather difficult. Unlike explaining individual predictions, which is a local and instance-level task, extracting concepts provides a global understanding of models, where manual inspection of such interpretation is time-consuming and much harder, if not impossible.
Instead of relying on post-hoc approaches, we aim to instill interpretability as a constraint into model establishment. For example, explanation regularization is proposed in <cit.>, but it constrains gradient magnitude instead of focusing on semantic concepts. Meanwhile, β-VAE and its variants <cit.> add independence constraints to learn disentangled factors in latent representations, but it is difficult to explicitly specify and align latent dimensions with semantic meanings. Ideally, we want to construct DNNs whose latent space could tell us how it is encoding concepts. The recent decorrelated batch normalization (DBN) method <cit.> normalizes representations, providing an end-to-end technique for manipulating representations, but it is not directly related to interpretability.
In this work, we propose a novel Hierarchical Semantic Tree Concept Whitening (HaST-CW) model to decorrelate the latent representations in image classification for disentangling concepts with hierarchical relations. The idea of our work is illustrated in Fig. <ref>. Specifically, we define each concept as one class of objects, where the concepts are of different granularities and form a hierarchical tree structure. We decorrelate the activations of neural network layers, so that each concept is aligned with one or several latent dimensions. Unlike the traditional DBN method (Fig. <ref>a), which treats different concepts as independent, our method is able to leverage the hierarchically related organization of label concepts inherent in domain knowledge (Fig. <ref>b).
The consideration of relations between different concepts is crucial in many real-world applications <cit.>. For example, in the healthcare domain, the relationship of different disease stages (concepts) may reflect the progression of the disease, which is significant for reversing pathology <cit.>. Also, in the precision agriculture domain <cit.>, real-time monitoring of interactions of multiple agricultural objects (concepts) with each other and with the environment are crucial in maintaining agro-ecological balance <cit.>.
In our model, a novel semantic constraint (SC) loss function is designed to regularize representations. As a result, the data representations of two concepts with higher semantic similarity will be closer with each other in the latent space. Moreover, a new hierarchical concept whitening (HCW) method is proposed to decorrelate different label concepts hierarchically. We evaluated the proposed HaST-CW model using a novel agriculture image dataset called Agri-ImageNet. The results suggest that our model could preserve the semantic relationship between the label concepts, and provide a clear understanding of how the network gradually learns the concept in different layers, without hurting classification performance.
§ RELATED WORK
Post-Hoc Interpretation. Post-Hoc interpretation can be divided into approaches that explain predictions or models <cit.>. Prediction-oriented interpretation aims to develop faithful and robust measures to quantify feature importance towards individual predictions for identifying those features (e.g., pixels, super-pixels, words) that made most contributions <cit.>. Model-oriented interpretation analyzes behaviors of neural networks either by characterizing the function of model components <cit.> or analyzing semantic concepts from latent representations <cit.>. The proposed method also targets concept-level interpretation in deep neural networks. Different from post-hoc techniques that focus on discovering existing patterns in models, the newly proposed HaST-CW proactively injects concept-related knowledge into training and disentangles different concepts to promote model interpretability.
Inherently Interpretable Models. Another school of thought favors building inherently explainable machine learning models <cit.>. Some approaches design models that highlight prototypical features of samples as interpretation. For example, Chen et al. <cit.> classifies images by dissecting images into parts and comparing these components to similar prototypes towards prediction. Li et al. <cit.> designs an encoder-decoder framework to allow comparisons between inputs and the learned prototypes in latent space. Some other works such as β-VAE and its variants <cit.> regularize representation learning for autoencoders to produce disentangled factors in representation dimensions, but the semantic meaning of each dimension remains unknown without further manual inspection. In contrast, our method attempts to explicitly align latent dimensions with specific semantic concepts contained in external knowledge. A recent technique called Concept Whitening (CW) <cit.> constrains the latent space, after revising Batch Whitening <cit.>, such that it aligns with predefined classes. Our method attempts to infuse more complex knowledge of concept relations into representation learning.
Applying Whitening to Computer Vision. Whitening is a standard image preprocessing technique, which refers to transforming the covariance matrix of input vectors into the identity matrix. In fact, the well-known Batch Normalization <cit.> can be regarded as a variant of whitening where only the normalization process is retained. There are many works in deep learning that describe the effectiveness of whitening <cit.> and the process of finding the whitening matrix <cit.>. Our work further takes semantics into consideration during the whitening process towards more interpretable representation learning.
§ METHODOLOGY
§.§ Overview
The proposed HaST-CW model aims to preserve the underlying hierarchical relationship of label concepts, as well as to disentangle these concepts by decorrelating their latent representations. To achieve this goal, we leverage the hierarchical tree structure of the label concepts extracted from specific domain knowledge (<ref>). Then, the obtained structure of label concepts is used as prior knowledge to be instilled into the model for guiding the representation learning process. There are two key components in the knowledge instillation process – the hierarchical concept whitening (HCW) module and the semantic constraint (SC) loss, which will be elaborated in <ref> and <ref>, respectively.
§.§ The Hierarchical Semantic Tree of Concepts
In this work, we used a newly collected and curated Agri-ImageNet dataset to develop and evaluate the HaST-CW model. There are 9173 high quality images in Agri-ImageNet, covering 21 different types of agricultural objects. Taking each type of agricultural object as one class, we have 21 label concepts in total. Some pairs of agriculture objects have the supertype-subtype relationship between them, so we obtain the parent-child relationship between the corresponding labels. As a result, a tree structure is built to represent the underlying hierarchically related organization of label concepts, which is shown in <ref>. Two concepts connected in the tree structure means they have parent-child relationship, where the parent is located at the lower hierarchy level. Besides the parent-child relation, we further introduce two notions – brother and cousin. If two concepts have the same parent, then they are brothers. If the parents of two concepts are brothers, then the two concepts are cousins. According to the laws of inheritance: (1) objects with the parent-child relation should be more similar than those with the uncle-child relation (vertical parent-child relationship); and (2) the traits of brothers should be more similar than cousins (horizontal brother-cousin relationship). An effective model should be able to capture both of the vertical relationship and horizontal relationship, so that the representation of any concept in the latent space should be closer to its parent than uncles, and closer to brothers than cousins. For our HaST-CW model shown in <ref>, a new HCW module (<ref>) is proposed to preserve the vertical relationship, and a novel SC loss (<ref>) is proposed to preserve the horizontal relationship.
§.§ Hierarchical Concept Whitening
The hierarchical concept whitening (HCW) module is one of the key components in the HaST-CW model, which aims to disentangle different label concepts while preserving their underlying hierarchical relationship. Specifically, in this work, the set of label concepts were denoted by C={C_i}_i=1^N_c, where C_i represents the i^th concept and N_c = 21 is the number of concepts. For C_i, its parent, children, brothers and cousins were denoted as C_i.𝒫, {C_i.children}, {C_i.ℬ} and {C_i.𝒞}, respectively. A dataset is denoted as 𝒟{x_i,y_i} ^n_i=1. We use X^C_i={x_j^C_i}_j=1^n_i to denote the set of i^th-class samples labeled by C_i.
In traditional whitening transformation <cit.>, during the training process, data samples are first fed into the model in mini-batches to obtain the latent representation matrix Z_d× n, where n is the mini-batch size and d is the dimension of latent representation. We use ResNet as the model backbone in this work. Then a transformation ψ is applied to decorrelate and standardize Z_d× n:
ψ(Z)=W(Z-μ1_n× 1^T),
where W_d× d is the orthogonal whitening matrix, and μ=1/n∑^n_i=1z_i is the sample mean. A property of representation whitening is that Q^TW is still a valid whitening matrix if Q is an orthogonal matrix. We leverage this property for interpretable representation learning. In our model, besides decorrelation and standardization, we expect that the transformed representation of samples from concept C_i, namely Q^Tψ(Z^C_i), can align well with the i^th axis of latent space. Meanwhile, the underlying hierarchical relationship of concepts should also be preserved in their latent representations. That is, we need to find an orthogonal matrix Q= [q_1, q_2, …, q_N_c] with two requirements: (1) Z^C_i should be most activated by q_i, i.e., the i^th column of Q; (2) Z^C_i should also be activated by {q_c}, where c∈{C_i.children} is the child of concept C_i. The first constraint makes the representation align together with the corresponding concept dimension, and the second one maintains the vertical parent-child relationship between concepts. To this end, the optimization problem can be formulated as:
max_q_1,…q_N_c ∑^N_c_i=1[ 1/n_iq^T_iψ(Z^C_i)1_n_i ×1
+
∑_c∈{C_i.children}1/n_i× N_cd(q_c)^Tψ(Z^C_i)1_n_i ×1],
s.t. Q^TQ= I_d ,
where N_cd = |{C_i.children}| is the number of child concepts of C_i. To solve this optimization problem with the orthogonality constraint, a gradient descent method with the curvilinear search algorithm <cit.> is adopted. With the whitening matrix W and rotation orthogonality matrix Q, HaST-CW can replace any batch normalization layer in deep neural networks. The details of representation whitening for HaST-CW is summarized in Algorithm <ref>.
The overall training pipeline of our HaST-CW model is shown in <ref>. We adopt an alternative training scheme. In the first stage, the deep neural network is trained with the traditional classification loss. In the second stage, we solve for Q to align representation dimension with semantic concepts. The two stages work alternatively during the training process. The classification loss of the first stage is defined as:
min_θ,ω,W,μ,1/m∑^m_i=1ℓ(g(Q^Tψ( Φ(x_i;θ);W,μ);ω);y_i),
where Φ(·) and g(·) are layers before and after the HaST-CW module parameterized by θ and ω, respectively. ψ(·) is the whitening transformation parameterized by the sample mean μ and whitening matrix W. The rotation orthogonal matrix Q will be updated according to <ref> in the second stage. The operation of Q^Tψ(·) forms the HCW module. During the first training stage, Q will be fixed and other parameters (θ,ω,W,μ) will be optimized according to <ref> to minimize the classification error. The first stage will take T_thre mini batches (we set T_thre=30 in experiments). After that, Q will be updated by the Cayley transform <cit.>:
Q^' = (I+η/2A)^-1(I-η/2A)Q,
A = GQ^T-QG^T,
where A is a skew-symmetric matrix. G is the gradient of the concept alignment loss, which is defined in <ref>. η is the learning rate. At the end of the second stage, an updated Q^' will participate in the first training stage of the next iteration.
[tb]
The Overall Framework of HaST-CW
§.§ Semantic Constraint Loss
Besides preserving the vertical parent-child relationship of concepts, we further model the horizontal relation between concepts that are at the same hierarchy level (i.e., brothers or cousins). Different from the HCW in <ref> that focuses on concept alignment, here we directly control the distance between representations of different concepts with the horizontal relation <cit.>. To this end, we propose a Semantic Constraint (SC) loss to model the horizontal brother-cousin relationship as below:
ℒ_SC = αℒ_ℬ + βℒ_𝒞,
ℒ_ℬ=∑_j ∑_ℬ_i∈{C_i.ℬ}∑_k max{0,m_ℬ-d(z^C_i_j,z^ℬ_i_k)},
ℒ_𝒞 =∑_j ∑_ℬ_i∈{C_i.ℬ}∑_𝒞_i∈{C_i.𝒞}∑_k∑_l max{0,d(z^C_i_j,z^ℬ_i_k)
-d (z^C_i_j,z^𝒞_i_l)+m_𝒞}.
There are two components in the SC loss and their contributions are controlled by two hyperparameters – α and β. The first term ℒ_ℬ is a contrastive loss, which takes a pair of image representations labeled by two brother concepts as input and enlarges the distance between them. It uses a hyperparameter m_ℬ to control the distance. The distance between two concepts increases when m_ℬ is set larger. ℬ_i∈{C_i.ℬ} denotes one of the brothers of concept C_i. The second term ℒ_𝒞 is a triplet loss. It takes three inputs: the anchor image representation z^C_i_j, the image representation z^ℬ_i_k labeled by brother concept of the anchor, and the image representation z^𝒞_i_l labeled by cousin concept of the anchor. 𝒞_i∈{C_i.𝒞} denotes the cousins of concept C_i. The triplet loss encourages the anchor-brother distance to be smaller compared with the anchor-cousin distance in representation space. In this way, the distance of image representations from brother classes tends to be smaller than the distance of image representations from cousin classes. The gap between the two types of distance is controlled by the margin value m_𝒞. Consequently, the hierarchical concept whitening module, together with the SC loss, enables the latent representations of concepts with similar semantics to be close with each other in the latent space.
§.§ Latent Feature Maps Activation
The proposed HaST-CW model can generate latent representations (ẑ_i) for input images (x_i) at each neural network layer by ẑ_i=Q^Tψ( Φ(x_i;θ);W,μ). The latent representation can be used to assess the interpretability of the learning process by measuring the degree of activation of ẑ_i at different concept dimensions (i.e. {q_i}). In the implementation, Φ(·) is a CNN based deep network, whose convolution output z_i= Φ(x_i;θ) is a tensor with the dimension z_i∈ R^d× h× w. Since ẑ_i is calculated by ẑ_i = Q^Tψ(z_i) where Q^T∈ R^d× d, we obtain ẑ_i∈ R^d× h× w, where d is the channel dimension and h× w is the feature map dimension. The hierarchical concept whitening operation Q^Tψ(·) is conducted upon the d feature maps. Therefore, different feature maps contain the information of whether and where the concept patterns exist in the image. However, as a tensor the feature map cannot directly measure the degree of concept activation. To solve this problem and at the same time to reserve both of the high-level and low-level information, we first apply the max pooling on the feature map and then use the mean value of the downsteam feature map to represent the original one. By this way, we reshape the original feature map z_i∈ R^d× h× w to z_i^'∈ R^d× 1. Finally, z_i^' is used to measure the activation of image x_i at each concept dimension.
§ EXPERIMENTS
In the experiments section, we first visually demonstrate how our method can effectively learn and hierarchically organize concepts in the latent space (<ref>). We also show that (<ref>), compared to existing concept whitening methods, HaST-CW not only separates concepts, but also can separate groups of semantically related concepts in the latent space. After that, we discuss the advantages offered by our method with quantitative results and intuitive examples (<ref>) compared with baselines, including the CW module and ablated versions of our method.
§.§ Experimental Setting
§.§.§ Data Preparation
In this work, we use a newly collected and curated Agri-ImageNet dataset to evaluate the proposed HaST-CW model. In total, 9173 images from 21 classes are used in our experiments. Each image is labeled with the class at the highest possible hierarchy level. For example, an image of Melrose apple will be labeled as "Melrose" rather than the superclass "Apple". Then we divide images per class into three parts by 60%/20%/20% for a standardized training/validation/test splitting. Because the resolution of the original images can range from 300 to 5000, we adopt the following steps to normalize the image data: 1) we first lock aspect ratio and resize the images to make the short edge to be 256; 2) During each training epoch, the images in the training and validation datasets are randomly cropped into 224×224; 3) During testing process, images in the test dataset are center cropped to be of size 224×224; 4) After cropping, the pixel values of images are normalized to [0,1]. Then, the whole training dataset is divided into two parts (𝒟_T and 𝒟_C in <ref>). 𝒟_C is the concept dataset used to update the matrix Q in the second stage (<ref>). It is created by randomly selecting 64 images from each class in the training dataset. The remaining images in the training dataset 𝒟_T are used in the first stage to train the model parameters (<ref>).
§.§.§ Model Setting
In this work, we use several ResNet structures <cit.> to extract features from images, including ResNet18 and ResNet50. During the training process, the two-stage training scheme adopts a 30-to-1 ratio to alternatively train the whole framework. In this case, after 30 mini batches of continuous training, the model will pause and the rotation orthogonal matrix Q will be optimized at the next mini batch. Two hyper-parameters α and β in the SC loss are set to be 1.0. Adam optimizer is used to train the whole model with a learning rate of 0.1, a batch size of 64, a weight decay of 0.01, and a momentum rate of 0.9.
§.§ Visualization of Semantic Map
To illustrate the learned semantic hierarchical structure, we show the representations extracted from the latent hidden layer of all the samples in <ref>. For better visualization, we use Uniform Manifold Approximation and Projection (UMAP) <cit.> to project the representations to a two-dimensional space. All the images are color coded using the 17 sub-concepts which are defined on the left of <ref>. The top panel shows the result using CW method. In general, all the concepts are assembled as small groups, but neither semantic relations nor hierarchical structures have been learned. We highlight the super-concept of “Weed" (black) and three sub-concepts ( “Apple Golden" - green, “Apple Fuji" - red and “Apple Melrose" - blue) in the right column. We can see that the three types of apple (sub-concepts) are evenly distributed along with other fruits samples. The bottom panel shows our HaST-CW results. All the different concepts successfully keep their distinct cluster patterns as CW result. After our two-stage training process to instill the semantic and hierarchical knowledge, the three types of apple images have been pulled together and form a new concept (“Apple" with orange circle) at a higher level. Moreover, the newly learned concept of “Apple" simultaneously possesses sufficient distance to “Weed" (different super-concept) and maintains relatively close relations to “Strawberry", “Orange", “Mango" as well as other types of “Fruit". This result demonstrates the effectiveness of our hierarchical semantic concept learning framework, without negatively affecting the overall classification performance.
§.§ Efficiency and Accuracy of Concept Alignment
In this section, we compare the learning efficiency and accuracy of the proposed HaST-CW with that of the conventional CW method. We track the alignment between image representations and their corresponding concepts at each layer. Specifically, we randomly select six concepts, and for each concept we sort and select the top five images whose representations show the strongest activation at the corresponding concept axis. We show the results at both shallow and deep layers (layer 4 vs. layer 8) in <ref>. From the results of layer 4 (the left column) we can see that most of the top five images obtained by conventional CW (the rows marked by green box) are mismatched with the corresponding concepts. For example, the five images under the concept of “Apple-Melrose" obtained by CW are from the “Weed" class. The five images under the concept of “Snake Weed" are actually from other subclass of “Weed". Moreover, this situation continues in the following layers and has not been changed until layer 8. On the contrary, with the help of our designed semantic constraint loss, our HaST-CW (the rows marked by orange boxes) can learn the intrinsic concept faster and achieves the best performance at an earlier training stage (e.g., at a shallow layer). This result demonstrates that by paralleling multiple HCW layers the proposed HaST-CW model can capture the high-level features more efficiently.
To further demonstrate the alignment between images and the corresponding concepts, we project each image in the test dataset into a latent space where each concept can be represented by an axis. To visualize the alignments at different concept hierarchies (<ref>), we show three pairs of concepts which belong to different hierarchical levels as examples: “Apple-Melrose"-“Apple-Fuji" is from hierarchy 3 (H-3), “Snake Weed"-“Parkinsonia" is from hierarchy 2 (H-2), and “Weed"-“Apple" crosses hierarchies 1 and 2 (“Weed": H-1, “Apple": H-2). Within each concept pair, a two-dimensional space has been built by taking the two concepts as axes. Thus, each image can be mapped into the space by calculating the similarity between image representation and the two concept representations. The results are shown in <ref>. Different rows correspond to different methods and the concept axes (space) are defined at the bottom.
The first column of <ref> shows the data distribution in the two-dimensional space of “Apple-Melrose"-“Apple-Fuji" concept pair. The images belonging to Apple-Melrose class should have the highest similarity with the concept of “Apple-Melrose", and thereby they should be located at the right-bottom corner. Similarly, the images of Apple-Fuji class should be located at the left-top corner. The other images should distribute in the space according to the similarity with the two concepts. For example, compared to images of fruit-related classes, images of weed-related classes will have lower semantic similarity with the two concepts, so they should locate near the origin point (left-bottom corner). As shown in the first column, the two models which adopt the HaST-CW method (the second and third rows) can better follow the above-mentioned patterns. While in the CW model (the first row), nearly all the images are gathered at the right-bottom corner. This may be due to the high similarity between the two concepts considered, since they share the same super-class of “Apple". As a result, CW model may be limited in distinguishing different classes with high semantic similarity. A similar situation happens in the second column with the concept pair of “Snake Weed"-“Parkinsonia". These results suggest that compared to CW method, HaST-CW can better capture the subtle differences of semantic-related classes.
The third column shows the results of the concept pair of two super-classes: “Weed" and “Apple". As each of the super-class concept contains multiple sub-classes, the intra-class variability is greater. Our proposed HaST-CW, together with the SC loss (the third row), can effectively capture the common visual features and project the “Weed" and “Apple" images to the left-top and right-bottom, respectively. At the same time, the images belonging to different sub-classes under “Weed" and “Apple" are assembled as blocks instead of scattered along the diagonal line. In the other two methods, especially in the CW method (the first row), the images of “Weed" class spread out over a wide range along the vertical axis. This result suggests that the proposed HaST-CW with SC loss can effectively model both the inter- and intra- class similarity.
§.§ Interpretable Image Classification
In this section, we compare the classification performance of the proposed HaST-CW method and the SC loss function with the conventional CW method using different backbones: ResNet18 and ResNet50. The results are summarized in <Ref>. Different rows correspond to different model settings. Within each model setting, we repeat the experiments for five times to reduce the effect of random noise. The mean and variance of accuracy (ACC.) are reported in the fourth column. From the results, we can see that the classification performance is slightly better than the other three model settings. This result indicates that the proposed HaST-CW model can improve the interpretability without hurting predictive performance.
To track and visualize the classification process, we randomly select two images from Apple-Melrose class and Snake Weed class. The activation values between each image with the six relevant concepts are calculated and normalized to [0, 1]. The images, concepts and activation values are organized into a hierarchical activation tree. The results are shown in <ref>. We could observe that the activation values of each image correctly represent the semantic relationship between the images and the concepts. For example, in <ref> (a), the image located at the root is from Snake Weed class which is a subclass of Weed. The activation values of the image are consistent with this relationship and possess the highest activation values on the two concepts – “Weed" and “Snake Weed".
§ CONCLUSION AND FUTURE WORK
In this study, we propose a new HaST-CW and demonstrate its superiority over Concept Whitening <cit.>. HaST-CW decorrelates representations in the latent space and aligns concepts with corresponding dimensions. In addition, it correctly groups concepts at different granularity levels in the latent space and preserves hierarchical structures of concepts of interest. By doing so, we can interpret concepts better and observe the semantic relationships among concepts.
We believe there are many possibilities for future work. One promising direction is automatically learning concepts from data. In this scenario, we can jointly learn possible concepts from common abstract features among images and how to represent these learned concepts in the latent space. For example, it might be possible to develop unsupervised or weakly-supervised methods to automatically learn the concept tree from data. By jointly learning concepts, their representations, and relations, the model may discover more data-driven semantic structures.
HaST-CW can also be extended with post-hoc interpretability strategies (such as saliency-based methods that highlight focused areas used for classification). Such explanations at the concept level can provide a more global view of model behaviors.
In addition, while this work focuses on the the natural image domain, the idea of leveraging hierarchical knowledge to guide representation learning is generalizable to other domains such as natural language processing <cit.> and medical image analysis <cit.>. Exploring knowledge-infused learning in different domains <cit.> and tasks <cit.>, including innovative applications <cit.>, is an interesting future direction.
In conclusion, as deep learning models become increasingly complex, model interpretability is crucial for understanding behaviors, gaining trust, and enabling human-AI collaboration. Our work complements previous work and lays a solid foundation for further exploration.
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